Pengguna:Dedhert.Jr/Uji halaman 01/11

Artikel teori himpunan pertama Cantor memuat teorema pertama Georg Cantor mengenai teori himpunan transfinit, yang mempelajari himpunan takhingga beserta sifat-sifatnya. Salah satu teorema yang merupakan "penemuan revolusioner"-nya mengatakan bahwa himpunan dari semua bilangan real adalah taktercacahkan, bukan tercacahkan.^[1] Teorema ini dibuktikan melalui bukti ketaktercacahkan pertama Cantor, yang berbeda dengan bukti-bukti yang terkenal lainnya, yaitu argumen diagram Cantor. Artikel yang berjudulkan dalam Jerman: "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"code: de is deprecated , merujuk ke teorema pertamanya yang mengatakan bahwa himpunan bilangan aljabar real adalah tercacahkan. Artikel Cantor diterbitkan pada tahun 1874, dan pada tahun 1879 ia mengubah bukti ketaktercacahkan dengan menggunakan gagasan topologi mengenai himpunan rapat dalam sebuah interval.

Artikel Cantor juga memuat bukti keberadaan bilangan transendental. Bukti konstruktif dan nonkonstruktif telah disajikan sebagai "bukti Cantor". Kepopuleran dalam menyajikan bukti nonkonstrukif mengakibatkan kesalahpahaman bahwa argumen Cantor berupa nonkonstruktif. Sebab bukti yang Cantor terbitkan bersifat membangun bilangan transendental atau tidak, maka analisis dari artikelnya dapat menentukan apakah buktinya bersifat konstruktif atau tidak.^[2] Dalam surat Cantor kepada Richard Dedekind, ia memperlihatkan pengembangan tentang gagasannya, dan memperlihatkan bahwa ia memilih di antara dua bukti tersebut: bukti nonkonstruktif yang menggunakan ketaktercacahkan dari bilangan real, dan bukti konstruktif yang tidak menggunakan ketaktercacahkan.

Para sejarawan matematika telah memeriksa artikel Cantor beserta isinya yang ia tulis. Sebagai contoh, para sejarawan matematika menemukan bahwa Cantor disarankan untuk menghapus teorema ketaktercacahkan miliknya dalam artikel yang ia serahkan— dia menambahkannya saat mengoreksi. They have traced this and other facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Para sejarawan juga mempelajari kontribusi Dedekind mengenai artikel ini, seperti kontribusinya dalam teorema tentang ketercacahkan dari bilangan real aljabar. Selain itu, para sejarawan telah mengakui peran yang dimainkan oleh teorema ketaktercacahan dan konsep ketercacahkan dalam pengembangan teori himpunan, teori ukuran, dan integral Lebesgue.

Isi artikel sunting

Artikel Cantor memuat empat setengah halaman,^[A] yang dimulai dari pembahasan bilangan aljabar real dan pernyataan teorema pertamanya sebagai berikut: Himpunan bilangan aljabar real dapat korespondensi satu-ke-satu dengan himpunan bilangan bulat positif.^[3] Cantor menulis pernyataan ulang mengenai teoremanya dalam bentuk yang lebih dikenal para matematikawan pada masanya: Himpunan bilangan aljabar real dapat ditulis sebagai barisan takhingga yang setiap bilangan hanya muncul sekali.^[4]

Teorema kedua Cantor bekerja pada himpunan bilangan real yang lebih besar sama dengan $a$ dan lebih kecil sama dengan $b$ , yakni pada interval tertutup $[a, b]$ . Teorema ini dinyatakan sebagai berikut: Diberikan suatu barisan bilangan real $x 1, x 2, x 3, ...$ dan suatu interval $[a, b]$ , maka terdapat sebuah bilangan di $[a, b]$ yang tidak memuat di dalam barisan bilangan real tersebut. Karena itu, ada tak berhingga banyaknya bilangan tersebut .^[5]

Cantor mengamati bahwa dengan menggabungkan dua teorema tersebut akan menghasilkan bukti baru tentang teorema Liouville. Teorema tersebut mengatakan bahwa setiap interval $[a, b]$ memuat tak berhingga banyaknya bilangan transendental.^[5]

Cantor kemudian mengatakan tentang teorema keduanya:

[ada] alasan mengapa kumpulan dari bilangan real membentuk sesuatu yang disebut kontinum (seperti, semua bilangan real yang lebih besar dari 0 dan lebih kecil dari 1) tidak dapat korespondensi satu-ke-satu dengan kumpulan [dari semua bilangan bulat positif]; karena itu saya telah menemukan perbedaan yang jelas antara sesuatu yang disebut kontinum dan kumpulan yang bagaikan totalitas dari bilangan aljabar real.^[6]

Apa yang dikatakannya mengandung teorema ketaktercacahkan Cantor, yang hanya mengatakan bahwa sebarang interval $[a, b]$ tak dapat diletakkan menjadi himpunan yang korespondensi satu-dengan-satu dengan himpunan dari bilangan bulat positif, tetapi tidak mengatakan bahwa interval tersebut adalah himpunan takhingga dari kardinalitas yang lebih besar daripada himpunan dari bilangan bulat positif. Kardinalitas didefinisikan dalam artikel Cantor selanjutnya, yang diterbitikan pada tahun 1878.^[7]

Bukti teorema ketaktercacahkan Cantor
Cantor tidak membuktikan teoerma ketaktercacahannya secara eksplisit melalui teorema keduanya dengan mudah. Teorema tersebut dapat dibuktikan melalui kontradiksi. Asumsi bahwa interval $[a, b]$ dapat diletakkan ke himpunan yang korespondensi satu-ke-satu dengan himpunan dari bilangan bulat positif, atau dalam pernyataan yang serupa: Bilangan real di $[a, b]$ dapat ditulis sebagai sebuah barisan, dengan masing-masing bilangan real yang hanya muncul sekali. Dengan menerapkan teorema kedua Cantor ke barisan ini dan $[a, b]$ menghasilkan bilangan real di $[a, b]$ yang bukan merupakan milik barisan tersebut. Hal ini kontradiksi dengan asumsi sebelumnya dan teorema ketaktercacahan telah terbukti.^[8]

Cantor hanya menyatakan teorema ketaktercacahannya. Ia tidak menggunakannya dalam bukti apapun.^[3]

Bukti sunting

Teorema pertama sunting

Pemberian warna pada bilangan aljabar di bidang kompleks berdasarkan tingkat derajat polinomial. Warna merah menunjukkan derajat 1, hijau menunjukkan 2, biru menunjukkan 3, dan kuning menunjukkan 4. Titik-titik tersebut menjadi semakin kecil ketika koefisien polinomial bilangan bulat semakin besar.

Untuk membuktikan bahwa himpunan bilangan aljabar real adalah tercacahkan, tinggi polinomial berderajat $n$ dengan koefisien bilangan bulat didefinisikan sebagai $n - 1 + | a 0 | + | a 1 | + ... + | a n |$ , dengan $a 0, a 1, ... a n$ adalah koefisien dari polinomial. Urutkan polinomial berdasarkan ketinggian, dan urutkan akar real polinomial dengan tinggi derajat yang sama berdasarkan urutan numerik. Karena ada jumlah akar terhingga dari polinomial dengan tinggi yang diberikan, urutannya meletakkan bilangan aljabar real menjadi sebuah barisan. Cantor menuju langkah selanjutnya dan menghasilkan sebuah barisan yang setiap bilangan aljabar real hanya muncul sekali. Ia membuktikannya hanya menggunakan polinomial yang tidak tereduksi atas bilangan bulat. Tabel berikut menunjukkan enumerasi Cantor.^[9]

Enumerasi Cantor dari bilangan aljabar real
Bilangan aljabar real	Poinomial	Tinggi polinomial
x₁ = 0	x	1
x₂ = −1	x + 1	2
x₃ = 1	x − 1	2
x₄ = −2	x + 2	3
x₅ = −12	2x + 1	3
x₆ = 12	2x − 1	3
x₇ = 2	x − 2	3
x₈ = −3	x + 3	4
x₉ = −1 − √52	x² + x − 1	4
x₁₀ = −√2	x² − 2	4
x₁₁ = −1√2	2x² − 1	4
x₁₂ = 1 − √52	x² − x − 1	4
x₁₃ = −13	3x + 1	4
x₁₄ = 13	3x − 1	4
x₁₅ = −1 + √52	x² + x − 1	4
x₁₆ = 1√2	2x² − 1	4
x₁₇ = √2	x² − 2	4
x₁₈ = 1 + √52	x² − x − 1	4
x₁₉ = 3	x − 3	4

Teorema kedua sunting

Hanya bagian pertama dari teorema kedua Cantor yang harus dibuktikan. Teorema tersebut mengatakan: Diberikan suatu barisan bilangan real $x 1, x 2, x 3, ...$ dan setiap interval $[a, b]$ , maka ada sebuah bilangan di $[a, b]$ yang tidak memuat di dalam barisan tersebut.^[B]

Untuk mencari bilangan di $[a, b]$ yang tidak memuat di barisan tersebut, konstruksilah dua barisan bilangan real dikonstruksi seperti berikut: Carilah dua bilangan dari barisan tertentu yang ada di interval terbuka $(a, b)$ . Kedua bilangan tersebut dilambangkan dengan $a 1$ untuk yang lebih kecil dan $b 1$ untuk yang lebih besar. Mirip dengan cara sebelumnya, carilah dua bilangan dari barisan tertentu yang ada di $(a 1, b 1)$ . Lambangkan bilangan yang lebih kecil dengan $a 2$ dan bilangan yang lebih besar dengan $b 2$ . Dengan melanjutkan prosedur ini menghasilkan barisan interval $(a 1, b 1), (a 2, b 2), (a 3, b 3), ...$ sehingga setiap interval dalam barisan memuat semua interval sebelumnya — dalam artian, barisan tersebut menghasilkan barisan interval bersarang. Prosedur ini menyiratkan bahwa barisan $a 1, a 2, a 3, ...$ menaik dan barisan $b 1, b 2, b 3, ...$ menurun.^[10]

Jumlah interval yang dihasilkan adalah terhingga atau takterhingga. Jika terhingga, misalkan $(a L, b L)$ adalah interval terakhir. Jika tak terhingga, ambil limit $a \infty = lim n \to \infty$ dan $lim n \to \infty b n$ . Karena $a n < b n$ untuk semua $n$ , baik $a \infty = b \infty$ ataupun $a \infty < b \infty$ , maka ada tiga kasus yang ditinjau:

Kasus pertama: Interval terakhir

(a L, b L)

Kasus pertama: Terdapat interval terakhir

(a L, b L)

. Sebab pada interval dapat memuat sebanyak satu

x n

, maka setiap

y

di interval tersebut kecuali

x n

(jika ada) tidak termuat di barisan.

Kasus kedua:

a \infty = b \infty

Kasus kedua:

a \infty = b \infty

. Maka

a \infty

tidak termuat di barisan yang diberikan sebab untuk semua

n

:

a \infty

merupakan milik interval

(a n, b n)

tetapi

x n

bukan milik interval

(a n, b n)

. Secara simbolis,

a \infty \in (a n, b n)

tetapi

x n \notin (a n, b n)

.

Bukti bahwa untuk semua $n$ : $x n \notin (a n, b n)$
Lema ini dipakai pada kasus kedua dan ketiga, yang disiratkan dengan lema yang lebih kuat: Untuk semua $n$ , interval $(a n, b n)$ tidak memuat $x 1, ..., x 2 n$ . Lema ini dibuktikan melalui induksi. Langkah dasar: karena titik akhir dari $(a 1, b 1)$ adalah $x 1$ dan $x 2$ , serta interval terbuka tidak memuat titik akhir, maka $(a 1, b 1)$ tidak memuat $x 1, x 2$ . Langkah induktif: Asumsi bahwa $(a n, b n)$ tidak memuat $x 1, ..., x 2 n$ . Sebab $(a n +1, b n +1)$ adalah subhimpunan dari (a_n, b_n)}} serta titik akhirnya $x 2 n +1$ dan $x 2 n +2$ , maka $(a n +1, b n +1)$ tidak memuat $x 1, ..., x 2 n$ dan $x 2 n +1, x 2 n +2$ . Jadi, untuk semua $n$ , $(a n, b n)$ tidak memuat $x 1, ..., x 2 n$ . Oleh karena itu, untuk semua n, x_n ∉ (a_n, b_n)}}.^[C]

Kasus ketiga:

a \infty < b \infty

Kasus ketiga:

a \infty < b \infty

. Maka setiap

y

di interval

[a \infty, b \infty]

tidak termuat di barisan tersebut sebab untuk semua

n

:

y

merupakan milik

(a n, b n)

tetapi

x n

bukan miliknya.^[11]

Bukti di atas telah lengkap karena pada semua kasus, sudah ditemukan setidaknya ada satu bilangan real di $[a, b]$ yang tidak termuat di barisan.^[D]

Bukti Cantor bersifat konstruktif dan telah dipakai untuk menulis program komputer yang menghasilkan digit dari bilangan transendental. Program tersebut menerapkan konstruksi Cantor ke sebuah barisan yang mengandung semua bilangan aljabar real di antara 0 dan 1. Artikel yang membahas program tersebut memberikan nilai keluaran (bahasa Inggris: output), yang memperlihatkan cara kerja konstruksi tersebut menghasilkan bilangan transendental.^[12]

Contoh untuk konstruksi Cantor sunting

Ada contoh yang mengilustrasikan cara kerja konstruksi Cantor. Tinjau barisan: $12, 13, 23, 14, 34, 15, 25, 35, 45, ...$ . Barisan ini diperoleh dengan mengurutkan bilangan rasionals di $(0, 1)$ berdasarkan nilai penyebut yang besar, mengurutkannya dengan penyebut yang sama berdasarkan nilai pembilang yang menaik, dan menyederhanakan pecahan. Tabel berikut memperlihatkan lima langkah pertama konstruksi. Pada tabel, kolom yang pertama memuat interval $(a n, b n)$ , sedangkan kolom yang kedua menulis suku-suku pada barisan selama mencari dua suku pertama di $(a n, b n)$ , yang ditandai dengan warna merah.^[13]

**Menghasilkan bilangan melalui konstruksi Cantor**
Interval	Mencari interval berikutnya	Interval (desimal)
$\left(\;{\frac {1}{3}},\;\;\!{\frac {1}{2}}\;\right)$	$\;{\frac {2}{3}},\;\!{\frac {1}{4}},\;\!{\frac {3}{4}},\;\!{\frac {1}{5}},\;\!{\color {red}{\frac {2}{5}},}\;\!\;\!{\frac {3}{5}},\;\!{\frac {4}{5}},\;\!{\frac {1}{6}},\;\!{\frac {5}{6}},\;\!{\frac {1}{7}},\;\!{\frac {2}{7}},\;\!{\color {red}{\frac {3}{7}}}$	$\left(0,3333\dots ,\;0,5000\dots \right)$
$\left(\;{\frac {2}{5}},\;\;\!{\frac {3}{7}}\;\right)$	$\;{\frac {4}{7}},\;\dots ,\;{\frac {1}{12}},{\color {red}{\frac {5}{12}},}\;\!{\frac {7}{12}},\;\dots ,\;{\frac {6}{17}},{\color {red}{\frac {7}{17}}}$	$\left(0,4000\dots ,\;0,4285\dots \right)$
$\left({\frac {7}{17}},{\frac {5}{12}}\right)$	${\frac {8}{17}},\;\!\dots ,\;\!{\frac {11}{29}},{\color {red}{\frac {12}{29}},}\;\!{\frac {13}{29}},\;\dots ,\;{\frac {16}{41}},{\color {red}{\frac {17}{41}}}$	$\left(0,4117\dots ,\;0,4166\dots \right)$
$\left({\frac {12}{29}},{\frac {17}{41}}\right)$	${\frac {18}{41}},\;\!\dots ,\;\!{\frac {27}{70}},{\color {red}{\frac {29}{70}},}\;\!{\frac {31}{70}},\;\dots ,\;{\frac {40}{99}},{\color {red}{\frac {41}{99}}}$	$\left(0,4137\dots ,\;0,4146\dots \right)$
$\left({\frac {41}{99}},{\frac {29}{70}}\right)$	${\frac {43}{99}},\dots ,{\frac {69}{169}},{\color {red}{\frac {70}{169}},}\;\!{\frac {71}{169}},\,\dots ,{\frac {98}{239}},{\color {red}{\frac {99}{239}}}$	$\left(0,4141\dots ,\;0,4142\dots \right)$

Karena barisan memuat semua bilangan rasional di $(0, 1)$ , maka konstruksi tersebut menghasilkan sebuah bilangan irasional, yaitu $\sqrt 2 - 1$ .^[14]

Bukti bahwa bilangan yang dihasilkan bernilai $\sqrt 2 - 1$
The proof uses Farey sequences and continued fractions. The Farey sequence $F_{n}$ is the increasing sequence of completely reduced fractions whose denominators are $\leq n.$ If ${\frac {a}{b}}$ and ${\frac {c}{d}}$ are adjacent in a Farey sequence, the lowest denominator fraction between them is their mediant ${\frac {a+c}{b+d}}.$ This mediant is adjacent to both ${\frac {a}{b}}$ and ${\frac {c}{d}}$ in the Farey sequence $F_{b+d}.$ ^[15] Cantor's construction produces mediants because the rational numbers were sequenced by increasing denominator. The first interval in the table is $({\frac {1}{3}},{\frac {1}{2}}).$ Since ${\frac {1}{3}}$ and ${\frac {1}{2}}$ are adjacent in $F_{3},$ their mediant ${\frac {2}{5}}$ is the first fraction in the sequence between ${\frac {1}{3}}$ and ${\frac {1}{2}}.$ Hence, ${\frac {1}{3}}<{\frac {2}{5}}<{\frac {1}{2}}.$ In this inequality, ${\frac {1}{2}}$ has the smallest denominator, so the second fraction is the mediant of ${\frac {2}{5}}$ and ${\frac {1}{2}},$ which equals ${\frac {3}{7}}.$ This implies: ${\frac {1}{3}}<{\frac {2}{5}}<{\frac {3}{7}}<{\frac {1}{2}}.$ Therefore, the next interval is $({\frac {2}{5}},{\frac {3}{7}}).$ We will prove that the endpoints of the intervals converge to the continued fraction $[0;2,2,\dots ].$ This continued fraction is the limit of its convergents: ${\frac {p_{n}}{q_{n}}}=[0;2,\dots ,2]\;\;(n\;2{\text{'s}}).$ The $p_{n}$ and $q_{n}$ sequences satisfy the equations:^[16] $p_{0}=0\;\;\;\;\;\;\;p_{1}=1\;\;\;\;\;\;\;p_{n+1}=2p_{n}+p_{n-1}{\text{ for }}n\geq 1$ $q_{0}=1\;\;\;\;\;\;\;q_{1}=2\;\;\;\;\;\;\;q_{n+1}=2q_{n}+q_{n-1}{\text{ for }}n\geq 1$ First, we prove by induction that for odd n, the n-th interval in the table is: $\left({\frac {p_{n}+p_{n-1}}{q_{n}+q_{n-1}}},{\frac {p_{n}}{q_{n}}}\right)\!,$ and for even n, the interval's endpoints are reversed: $\left({\frac {p_{n}}{q_{n}}},{\frac {p_{n}+p_{n-1}}{q_{n}+q_{n-1}}}\right)\!.$ This is true for the first interval since: $\left({\frac {p_{1}+p_{0}}{q_{1}+q_{0}}},{\frac {p_{1}}{q_{1}}}\right)=\left({\frac {1}{3}},{\frac {1}{2}}\right)\!.$ Assume that the inductive hypothesis is true for the k-th interval. If k is odd, this interval is: $\left({\frac {p_{k}+p_{k-1}}{q_{k}+q_{k-1}}},{\frac {p_{k}}{q_{k}}}\right)\!.$ The mediant of its endpoints ${\frac {2p_{k}+p_{k-1}}{2q_{k}+q_{k-1}}}={\frac {p_{k+1}}{q_{k+1}}}$ is the first fraction in the sequence between these endpoints. Hence, ${\frac {p_{k}+p_{k-1}}{q_{k}+q_{k-1}}}<{\frac {p_{k+1}}{q_{k+1}}}<{\frac {p_{k}}{q_{k}}}.$ In this inequality, ${\frac {p_{k}}{q_{k}}}$ has the smallest denominator, so the second fraction is the mediant of ${\frac {p_{k+1}}{q_{k+1}}}$ and ${\frac {p_{k}}{q_{k}}},$ which equals ${\frac {p_{k+1}+p_{k}}{p_{k+1}+q_{k}}}.$ This implies: ${\frac {p_{k}+p_{k-1}}{q_{k}+q_{k-1}}}<{\frac {p_{k+1}}{q_{k+1}}}<{\frac {p_{k+1}+p_{k}}{p_{k+1}+q_{k}}}<{\frac {p_{k}}{q_{k}}}.$ Therefore, the (k + 1)-st interval is $\left({\frac {p_{k+1}}{q_{k+1}}},{\frac {p_{k+1}+p_{k}}{p_{k+1}+q_{k}}}\right)\!.$ This is the desired interval; ${\frac {p_{k+1}}{q_{k+1}}}$ is the left endpoint because k + 1 is even. Thus, the inductive hypothesis is true for the (k + 1)-st interval. For even k, the proof is similar. This completes the inductive proof. Since the right endpoints of the intervals are decreasing and every other endpoint is ${\frac {p_{2n-1}}{q_{2n-1}}},$ their limit equals $\lim _{n\to \infty }{\frac {p_{n}}{q_{n}}}.$ The left endpoints have the same limit because they are increasing and every other endpoint is ${\frac {p_{2n}}{q_{2n}}}.$ As mentioned above, this limit is the continued fraction $[0;2,2,\dots ],$ which equals ${\sqrt {2}}-1.$ ^[17]

Bukti ketaktercacahkan Cantor tahun 1879 sunting

Rapat dimana-mana sunting

Pada tahun 1879, Cantor memperbaharui bukti 1874 miliknya dengan menerbitkan bukti ketercacahan yang baru. Ia, pertama-tama, mendefinisikan gagasan topologis dari himpunan titik dari $P$ yang "rapat dimana-mana di interval":^[E]

Jika $P$ berada di sebagian atau seluruh interval $[α, β]$ , maka kasus yang luar biasa dapat terjadi bahwa setiap interval $[γ, δ]$ termuat di interval $[α, β]$ yang memuat titik dari $P$ tidak peduli seberapa kecil. Pada kasus ini, kita katakan bahwa $P$ rapat dimana-mana di interval $[α, β]$ .^[F]

Dalam pembahasan bukti Cantor, $a, b, c, d$ dipakai ketimbang $α, β, γ, δ$ . Selain itu, Cantor hanya memakai notasi interval jika titik akhir pertama lebih kecil dari titik akhir yang kedua. Hal ini mengartikan bahwa $(a, b)$ menyiratkan $a < b$ .

Semenjak pembahasan bukti Cantor pada tahun 1874 disederhanakan menggunakan interval terbuka alih-alih interval tertutup, penyederhanaan yang sama berlaku dipakai pula. Hal ini memerlukan definisi yang ekuivalen mengenai rapat dimana-mana: Himpunan $P$ rapat dimana-mana di interval $[a, b]$ jika dan hanya jika setiap subinterval terbuka $(c, d)$ dari $[a, b]$ memuat setidaknya satu titik dari $P$ .^[18]

Cantor tidak menjelaskan ada berapa banyak $P$ yang termuat di subinterval terbuka $(c, d)$ , karena dengan mengasumsi bahwa setiap subinterval terbuka memuat setidaknya satu titik $P$ menyiratkan bahwa setiap subinterval terbuka memuat ada tak berhingga banyaknya titik dari $P$ .^[G]

Bukti Cantor tahun 1879 sunting

Cantor memperbaharui bukti miliknya yang dibuat pada tahun 1874 dengan menambahkan bukti baru yang memuat teorema kedua, yang mengatakan: Diberikan setiap barisan $P$ dari bilangan real $x 1, x 2, x 3, ...$ dan setiap interval $[a, b]$ , terdapat sebuah bilangan di $[a, b]$ yang tidak termuat di $P$ . Bukti barunya hanya mempunyai dua kasus. Yang pertama adalah menangani kasus untuk $P$ tidak rapat di interval, dan selanjutnya untuk kasus yang sulit, yaitu kasus untuk $P$ yang rapat di interval. Tujuan dalam membaginya menjadi beberapa kasus bukan hanya barisan manakah yang sulit untuk ditangani, tetapi juga memperlihatkan pentingnya kerapatan dalam pembuktian.^{[proof 1]}

Pada kasus pertama, $P$ tidak rapat di $[a, b]$ . Menurut definisi, $P$ rapat di $[a, b]$ jika dan hanya jika untuk semua subinterval $(c, d)$ dari $[a, b]$ , terdapat $x \in P$ sehingga $x \in (c, d)$ . Dengan mengambil negasi dari masing-masing sisi dari "jika dan hanya jika", maka dihasilkanlah definisi berikut: P tidak rapat di $[a, b]$ jika dan hanya jika terdapat subinterval $(c, d)$ dari $[a, b]$ sehingga untuk semua $x \in P : x \notin (c, d)$ . Oleh karena itu, setiap bilangan di $(c, d)$ tidak termuat di barisan P.^{[proof 1]} Kasus pertama menangani kasus 1 dan kasus 3 dari bukti Cantor tahun 1874.

In the second case, which handles case 2 of Cantor's 1874 proof, P is dense in [a, b]. The denseness of sequence P is used to recursively define a sequence of nested intervals that excludes all the numbers in P and whose intersection contains a single real number in [a, b]. The sequence of intervals starts with (a, b). Given an interval in the sequence, the next interval is obtained by finding the two numbers with the least indices that belong to P and to the current interval. These two numbers are the endpoints of the next open interval. Since an open interval excludes its endpoints, every nested interval eliminates two numbers from the front of sequence P, which implies that the intersection of the nested intervals excludes all the numbers in P.^{[proof 1]} Details of this proof and a proof that this intersection contains a single real number in [a, b] are given below.

Definition and proofs for the nested intervals
The denseness of sequence P is used to recursively define a nested sequence of intervals that excludes all of the numbers in P. The base case starts with the interval (a, b). Since P is dense in [a, b], there are infinitely many numbers of P in (a, b). Let x_k₁ be the number with the least index and x_k₂ be the number with the next larger index, and let a₁ be the smaller and b₁ be the larger of these two numbers. Then, k₁ < k₂, a < a₁ < b₁ < b, and (a₁, b₁) is a proper subinterval of (a, b). Also, x_m ∉ (a₁, b₁) for m ≤ k₂ since these x_m are the endpoints of (a₁, b₁). Repeating the above proof with the interval (a₁, b₁) produces k₃, k₄, a₂, b₂ such that k₁ < k₂ < k₃ < k₄ and a < a₁ < a₂ < b₂ < b₁ < b and x_m ∉ (a₂, b₂) for m ≤ k₄.^{[proof 1]} The recursive step starts with the interval (a_n–1, b_n–1), the inequalities k₁ < k₂ < . . . < k_2n–2 < k_2n–1 and a < a₁ < . . . < a_n–1 < b_n–1 . . . < b₁ < b, and the fact that the interval (a_n–1, b_n–1) excludes the first 2n –2 members of the sequence P — that is, x_m ∉ (a_n–1, b_n–1) for m ≤ k_2n–2. Since P is dense in [a, b], there are infinitely many numbers of P in (a_n–1, b_n–1). Let x_{k_2n –1} be the number with the least index and x_{k_2n} be the number with the next larger index, and let a_n be the smaller and b_n be the larger of these two numbers. Then, k_2n –1 < k_2n, a_n–1 < a_n < b_n < b_n–1, and (a_n, b_n) is a proper subinterval of (a_n–1, b_n–1). Combining these inequalities with the ones for step n –1 of the recursion produces k₁ < k₂ < . . . < k_2n–1 < k_2n and a < a₁ < . . . < a_n < b_n . . . < b₁ < b. Also, x_m ∉ (a_n, b_n) for m = k_2n–1 and m = k_2n since these x_m are the endpoints of (a_n, b_n). This together with (a_n–1, b_n–1) excluding the first 2n –2 members of sequence P implies that the interval (a_n, b_n) excludes the first 2n members of P — that is, x_m ∉ (a_n, b_n) for m ≤ k_2n. Therefore, for all n, x_n ∉ (a_n, b_n) since n ≤ k_2n.^{[proof 1]} The sequence a_n is increasing and bounded above by b, so the limit A = lim_n → ∞ a_n exists. Similarly, the limit B = lim_n → ∞ b_n exists since the sequence b_n is decreasing and bounded below by a. Also, a_n < b_n implies A ≤ B. If A < B, then for every n: x_n ∉ (A, B) because x_n is not in the larger interval (a_n, b_n). This contradicts P being dense in [a, b]. Hence, A = B. For all n, A ∈ (a_n, b_n) but x_n ∉ (a_n, b_n). Therefore, A is a number in [a, b] that is not contained in P.^{[proof 1]}

Pengembangan ide Cantor sunting

The development leading to Cantor's 1874 article appears in the correspondence between Cantor and Richard Dedekind. On November 29, 1873, Cantor asked Dedekind whether the collection of positive integers and the collection of positive real numbers "can be corresponded so that each individual of one collection corresponds to one and only one individual of the other?" Cantor added that collections having such a correspondence include the collection of positive rational numbers, and collections of the form (a_{n₁, n₂, . . . , n_ν}) where n₁, n₂, . . . , n_ν, and ν are positive integers.^[19]

Dedekind replied that he was unable to answer Cantor's question, and said that it "did not deserve too much effort because it has no particular practical interest." Dedekind also sent Cantor a proof that the set of algebraic numbers is countable.^[20]

On December 2, Cantor responded that his question does have interest: "It would be nice if it could be answered; for example, provided that it could be answered no, one would have a new proof of Liouville's theorem that there are transcendental numbers."^[21]

On December 7, Cantor sent Dedekind a proof by contradiction that the set of real numbers is uncountable. Cantor starts by assuming that the real numbers in $[0,1]$ can be written as a sequence. Then, he applies a construction to this sequence to produce a number in $[0,1]$ that is not in the sequence, thus contradicting his assumption.^[22] Together, the letters of December 2 and 7 provide a non-constructive proof of the existence of transcendental numbers.^[23] Also, the proof in Cantor's December 7th letter shows some of the reasoning that led to his discovery that the real numbers form an uncountable set.^[24]

Cantor's December 7, 1873 proof
The proof is by contradiction and starts by assuming that the real numbers in $[0,1]$ can be written as a sequence: $(\mathrm {I} )\;\;\omega _{1},\,\omega _{2},\,\omega _{3},\,\ldots ,\,\omega _{n},\,\ldots$ An increasing sequence is extracted from this sequence by letting $\omega _{1}^{1}=$ the first term $,\ \omega _{1}^{2}=$ the next largest term following $\omega _{1}^{1},\ \omega _{1}^{3}=$ the next largest term following $\omega _{1}^{2},$ and so forth. The same procedure is applied to the remaining members of the original sequence to extract another increasing sequence. By continuing this process of extracting sequences, one sees that the sequence $(\mathrm {I} )$ can be decomposed into the infinitely many sequences:^[22] $(1)\;\;\omega _{1}^{1},\,\omega _{1}^{2},\,\omega _{1}^{3},\,\ldots ,\,\omega _{1}^{n},\,\ldots$ $(2)\;\;\omega _{2}^{1},\,\omega _{2}^{2},\,\omega _{2}^{3},\,\ldots ,\,\omega _{2}^{n},\,\ldots$ $(3)\;\;\omega _{3}^{1},\,\omega _{3}^{2},\,\omega _{3}^{3},\,\ldots ,\,\omega _{3}^{n},\,\ldots$ $\ \ \vdots$ Let $[p,q]$ be an interval such that no term of sequence (1) lies in it. For example, let $p$ and $q$ satisfy $\omega _{1}^{1}<p<q<\omega _{1}^{2}.$ Then $\omega _{1}^{1}<p<q<\omega _{1}^{n}$ for $n\geq 2,$ so no term of sequence (1) lies in $[p,q].$ ^[22] Now consider whether the terms of the other sequences lie outside $[p,q].$ All terms of some of these sequences may lie outside of $[p,q];$ however, there must be some sequence such that not all its terms lie outside $[p,q].$ Otherwise, the numbers in $[p,q]$ would not be contained in sequence $(\mathrm {I} ),$ contrary to the initial hypothesis. Let sequence $(k)$ be the first sequence that contains a term in $[p,q]$ and let $\omega _{k}^{n}$ be the first term. Since $p<\omega _{k}^{n}<q,$ let $p_{1}$ and $q_{1}$ satisfy $p<p_{1}<q_{1}<\omega _{k}^{n}<q.$ Then $[p,q]$ is a proper superset of $[p_{1},q_{1}]$ (in symbols, $[p,q]\supsetneq [p_{1},q_{1}]$ ). Also, the terms of sequences $(1),(2),\ldots ,(k-1)$ lie outside of $[p_{1},q_{1}].$ ^[22] Repeat the above argument starting with $[p_{1},q_{1}]\!:\,$ Let sequence $(k_{1})$ be the first sequence containing a term in $[p_{1},q_{1}]$ and let $\omega _{k_{1}}^{n}$ be the first term. Since $\ p_{1}<\omega _{k_{1}}^{n}<q_{1},$ let $p_{2}$ and $q_{2}$ satisfy $p_{1}<p_{2}<q_{2}<\omega _{k_{1}}^{n}<q_{1}.$ Then $[p_{1},q_{1}]\supsetneq [p_{2},q_{2}]$ and the terms of sequences $(k_{1}),\ldots ,(k_{2}-1)$ lie outside of $[p_{2},q_{2}].$ ^[22] One sees that it is possible to form an infinite sequence of nested intervals $[p,q]\supsetneq [p_{1},q_{1}]\supsetneq [p_{2},q_{2}]\supsetneq \ldots$ such that: the members of the $1^{st},2^{nd},\ldots ,(k-1)^{st}$ sequence lie outside $[p,q];$ the members of the $k^{th},\ldots ,(k_{1}-1)^{st}$ sequence lie outside $[p_{1},q_{1}];$ the members of the $(k_{1})^{th},\ldots ,(k_{2}-1)^{st}$ sequence lie outside $[p_{2},q_{2}];$ $\ldots ;$ ^[22] Since $p_{n}$ and $q_{n}$ are bounded monotonic sequences, the limits $\lim _{n\to \infty }p_{n}$ and $\lim _{n\to \infty }q_{n}$ exist. Also, $p_{n}<q_{n}$ for all $n$ implies $\lim _{n\to \infty }p_{n}\leq \lim _{n\to \infty }q_{n}.$ Hence, there is at least one number $\eta$ in $(0,1)$ that lies in all the intervals $[p,q]$ and $[p_{n},q_{n}].$ Namely, $\eta$ can be any number in $[\lim _{n\to \infty }p_{n},\lim _{n\to \infty }q_{n}].$ This implies that $\eta$ lies outside all the sequences $(1),(2),(3),\ldots ,$ contradicting the initial hypothesis that sequence $(\mathrm {I} )$ contains all the real numbers in $[0,1].$ Therefore, the set of all real numbers is uncountable.^[22]

Dedekind received Cantor's proof on December 8. On that same day, Dedekind simplified the proof and mailed his proof to Cantor. Cantor used Dedekind's proof in his article.^[25] The letter containing Cantor's December 7th proof was not published until 1937.^[26]

On December 9, Cantor announced the theorem that allowed him to construct transcendental numbers as well as prove the uncountability of the set of real numbers:

I show directly that if I start with a sequence

(1) ω₁, ω₂, ... , ω_n, ...

I can determine, in every given interval [α, β], a number η that is not included in (1).^[27]

This is the second theorem in Cantor's article. It comes from realizing that his construction can be applied to any sequence, not just to sequences that supposedly enumerate the real numbers. So Cantor had a choice between two proofs that demonstrate the existence of transcendental numbers: one proof is constructive, but the other is not. These two proofs can be compared by starting with a sequence consisting of all the real algebraic numbers.

The constructive proof applies Cantor's construction to this sequence and the interval [a, b] to produce a transcendental number in this interval.^[5]

The non-constructive proof uses two proofs by contradiction:

The proof by contradiction used to prove the uncountability theorem (see Proof of Cantor's uncountability theorem).
The proof by contradiction used to prove the existence of transcendental numbers from the countability of the real algebraic numbers and the uncountability of real numbers. Cantor's December 2nd letter mentions this existence proof but does not contain it. Here is a proof: Assume that there are no transcendental numbers in [a, b]. Then all the numbers in [a, b] are algebraic. This implies that they form a subsequence of the sequence of all real algebraic numbers, which contradicts Cantor's uncountability theorem. Thus, the assumption that there are no transcendental numbers in [a, b] is false. Therefore, there is a transcendental number in [a, b].^[H]

Cantor chose to publish the constructive proof, which not only produces a transcendental number but is also shorter and avoids two proofs by contradiction. The non-constructive proof from Cantor's correspondence is simpler than the one above because it works with all the real numbers rather than the interval [a, b]. This eliminates the subsequence step and all occurrences of [a, b] in the second proof by contradiction.^[5]

Kesalahpahaman tentang karya Cantor sunting

Akihiro Kanamori, who specializes in set theory, stated that "Accounts of Cantor's work have mostly reversed the order for deducing the existence of transcendental numbers, establishing first the uncountability of the reals and only then drawing the existence conclusion from the countability of the algebraic numbers. In textbooks the inversion may be inevitable, but this has promoted the misconception that Cantor's arguments are non-constructive."^[29]

Cantor's published proof and the reverse-order proof both use the theorem: Given a sequence of reals, a real can found that is not in the sequence. By applying this theorem to the sequence of real algebraic numbers, Cantor produced a transcendental number. He then proved that the reals are uncountable: Assume that there is a sequence containing all the reals. Applying the theorem to this sequence produces a real not in the sequence, contradicting the assumption that the sequence contains all the reals. Hence, the reals are uncountable.^[5] The reverse-order proof starts by first proving the reals are uncountable. It then proves that transcendental numbers exist: If there were no transcendental numbers, all the reals would be algebraic and hence countable, which contradicts what was just proved. This contradiction proves that transcendental numbers exist without constructing any.^[29]

Oskar Perron, c. 1948

The correspondence containing Cantor's non-constructive reasoning was published in 1937. By then, other mathematicians had rediscovered his non-constructive, reverse-order proof. As early as 1921, this proof was called "Cantor's proof" and criticized for not producing any transcendental numbers.^[30] In that year, Oskar Perron gave the reverse-order proof and then stated: "… Cantor's proof for the existence of transcendental numbers has, along with its simplicity and elegance, the great disadvantage that it is only an existence proof; it does not enable us to actually specify even a single transcendental number."^[31]^[I]

Abraham Fraenkel, between 1939 and 1949

As early as 1930, some mathematicians have attempted to correct this misconception of Cantor's work. In that year, the set theorist Abraham Fraenkel stated that Cantor's method is "… a method that incidentally, contrary to a widespread interpretation, is fundamentally constructive and not merely existential."^[32] In 1972, Irving Kaplansky wrote: "It is often said that Cantor's proof is not 'constructive,' and so does not yield a tangible transcendental number. This remark is not justified. If we set up a definite listing of all algebraic numbers … and then apply the diagonal procedure …, we get a perfectly definite transcendental number (it could be computed to any number of decimal places)."^[33]^[J] Cantor's proof is not only constructive, it is also simpler than Perron's proof, which requires the detour of first proving that the set of all reals is uncountable.^[34]

Cantor's diagonal argument has often replaced his 1874 construction in expositions of his proof. The diagonal argument is constructive and produces a more efficient computer program than his 1874 construction. Using it, a computer program has been written that computes the digits of a transcendental number in polynomial time. The program that uses Cantor's 1874 construction requires at least sub-exponential time.^[35]^[K]

The presentation of the non-constructive proof without mentioning Cantor's constructive proof appears in some books that were quite successful as measured by the length of time new editions or reprints appeared—for example: Oskar Perron's Irrationalzahlen (1921; 1960, 4th edition), Eric Temple Bell's Men of Mathematics (1937; still being reprinted), Godfrey Hardy and E. M. Wright's An Introduction to the Theory of Numbers (1938; 2008 6th edition), Garrett Birkhoff and Saunders Mac Lane's A Survey of Modern Algebra (1941; 1997 5th edition), and Michael Spivak's Calculus (1967; 2008 4th edition).^[36]^[L] Since 2014, at least two books have appeared stating that Cantor's proof is constructive,^[37] and at least four have appeared stating that his proof does not construct any (or a single) transcendental.^[38]

Asserting that Cantor gave a non-constructive argument without mentioning the constructive proof he published can lead to erroneous statements about the history of mathematics. In A Survey of Modern Algebra, Birkhoff and Mac Lane state: "Cantor's argument for this result [Not every real number is algebraic] was at first rejected by many mathematicians, since it did not exhibit any specific transcendental number."^[39] The proof that Cantor published produces transcendental numbers, and there appears to be no evidence that his argument was rejected. Even Leopold Kronecker, who had strict views on what is acceptable in mathematics and who could have delayed publication of Cantor's article, did not delay it.^[4] In fact, applying Cantor's construction to the sequence of real algebraic numbers produces a limiting process that Kronecker accepted—namely, it determines a number to any required degree of accuracy.^[M]

Pengaruh Weierstrass dan Kronecker tentang artikel Cantor sunting

Karl Weierstrass

Leopold Kronecker, 1865

Historians of mathematics have discovered the following facts about Cantor's article "On a Property of the Collection of All Real Algebraic Numbers":

Cantor's uncountability theorem was left out of the article he submitted. He added it during proofreading.^[43]
The article's title refers to the set of real algebraic numbers. The main topic in Cantor's correspondence was the set of real numbers.^[44]
The proof of Cantor's second theorem came from Dedekind. However, it omits Dedekind's explanation of why the limits a_∞ and b_∞ exist.^[45]
Cantor restricted his first theorem to the set of real algebraic numbers. The proof he was using demonstrates the countability of the set of all algebraic numbers.^[20]

To explain these facts, historians have pointed to the influence of Cantor's former professors, Karl Weierstrass and Leopold Kronecker. Cantor discussed his results with Weierstrass on December 23, 1873.^[46] Weierstrass was first amazed by the concept of countability, but then found the countability of the set of real algebraic numbers useful.^[47] Cantor did not want to publish yet, but Weierstrass felt that he must publish at least his results concerning the algebraic numbers.^[46]

From his correspondence, it appears that Cantor only discussed his article with Weierstrass. However, Cantor told Dedekind: "The restriction which I have imposed on the published version of my investigations is caused in part by local circumstances …"^[46] Cantor biographer Joseph Dauben believes that "local circumstances" refers to Kronecker who, as a member of the editorial board of Crelle's Journal, had delayed publication of an 1870 article by Eduard Heine, one of Cantor's colleagues. Cantor would submit his article to Crelle's Journal.^[48]

Weierstrass advised Cantor to leave his uncountability theorem out of the article he submitted, but Weierstrass also told Cantor that he could add it as a marginal note during proofreading, which he did.^[43] It appears in a remark at the end of the article's introduction. The opinions of Kronecker and Weierstrass both played a role here. Kronecker did not accept infinite sets, and it seems that Weierstrass did not accept that two infinite sets could be so different, with one being countable and the other not.^[49] Weierstrass changed his opinion later.^[50] Without the uncountability theorem, the article needed a title that did not refer to this theorem. Cantor chose "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"), which refers to the countability of the set of real algebraic numbers, the result that Weierstrass found useful.^[51]

Kronecker's influence appears in the proof of Cantor's second theorem. Cantor used Dedekind's version of the proof except he left out why the limits a_∞ = lim_n → ∞ a_n and b_∞ = lim_n → ∞ b_n exist. Dedekind had used his "principle of continuity" to prove they exist. This principle (which is equivalent to the least upper bound property of the real numbers) comes from Dedekind's construction of the real numbers, a construction Kronecker did not accept.^[52]

Cantor restricted his first theorem to the set of real algebraic numbers even though Dedekind had sent him a proof that handled all algebraic numbers.^[20] Cantor did this for expository reasons and because of "local circumstances."^[53] This restriction simplifies the article because the second theorem works with real sequences. Hence, the construction in the second theorem can be applied directly to the enumeration of the real algebraic numbers to produce "an effective procedure for the calculation of transcendental numbers." This procedure would be acceptable to Weierstrass.^[54]

Dedekind's contributions to Cantor's article sunting

Richard Dedekind, c. 1870

Since 1856, Dedekind had developed theories involving infinitely many infinite sets—for example: ideals, which he used in algebraic number theory, and Dedekind cuts, which he used to construct the real numbers. This work enabled him to understand and contribute to Cantor's work.^[55]

Dedekind's first contribution concerns the theorem that the set of real algebraic numbers is countable. Cantor is usually given credit for this theorem, but the mathematical historian José Ferreirós calls it "Dedekind's theorem." Their correspondence reveals what each mathematician contributed to the theorem.^[56]

In his letter introducing the concept of countability, Cantor stated without proof that the set of positive rational numbers is countable, as are sets of the form (a_{n₁, n₂, ..., n_ν}) where n₁, n₂, ..., n_ν, and ν are positive integers.^[57] Cantor's second result uses an indexed family of numbers: a set of the form (a_{n₁, n₂, ..., n_ν}) is the range of a function from the ν indices to the set of real numbers. His second result implies his first: let ν = 2 and a_n₁, n₂ = n₁n₂. The function can be quite general—for example, a_{n₁, n₂, n₃, n₄, n₅} = (n₁n₂)^1n₃ + tan(n₄n₅).

Dedekind replied with a proof of the theorem that the set of all algebraic numbers is countable.^[20] In his reply to Dedekind, Cantor did not claim to have proved Dedekind's result. He did indicate how he proved his theorem about indexed families of numbers: "Your proof that (n) [the set of positive integers] can be correlated one-to-one with the field of all algebraic numbers is approximately the same as the way I prove my contention in the last letter. I take n₁² + n₂² + ··· + n_ν² = ${\mathfrak {N}}$ and order the elements accordingly."^[58] However, Cantor's ordering is weaker than Dedekind's and cannot be extended to $n$ -tuples of integers that include zeros.^[59]

Dedekind's second contribution is his proof of Cantor's second theorem. Dedekind sent this proof in reply to Cantor's letter that contained the uncountability theorem, which Cantor proved using infinitely many sequences. Cantor next wrote that he had found a simpler proof that did not use infinitely many sequences.^[60] So Cantor had a choice of proofs and chose to publish Dedekind's.^[61]

Cantor thanked Dedekind privately for his help: "… your comments (which I value highly) and your manner of putting some of the points were of great assistance to me."^[46] However, he did not mention Dedekind's help in his article. In previous articles, he had acknowledged help received from Kronecker, Weierstrass, Heine, and Hermann Schwarz. Cantor's failure to mention Dedekind's contributions damaged his relationship with Dedekind. Dedekind stopped replying to his letters and did not resume the correspondence until October 1876.^[62]^[N]

The legacy of Cantor's article sunting

Cantor's article introduced the uncountability theorem and the concept of countability. Both would lead to significant developments in mathematics. The uncountability theorem demonstrated that one-to-one correspondences can be used to analyze infinite sets. In 1878, Cantor used them to define and compare cardinalities. He also constructed one-to-one correspondences to prove that the n-dimensional spaces Rⁿ (where R is the set of real numbers) and the set of irrational numbers have the same cardinality as R.^[63]^[O]

In 1883, Cantor extended the positive integers with his infinite ordinals. This extension was necessary for his work on the Cantor–Bendixson theorem. Cantor discovered other uses for the ordinals—for example, he used sets of ordinals to produce an infinity of sets having different infinite cardinalities.^[65] His work on infinite sets together with Dedekind's set-theoretical work created set theory.^[66]

The concept of countability led to countable operations and objects that are used in various areas of mathematics. For example, in 1878, Cantor introduced countable unions of sets.^[67] In the 1890s, Émile Borel used countable unions in his theory of measure, and René Baire used countable ordinals to define his classes of functions.^[68] Building on the work of Borel and Baire, Henri Lebesgue created his theories of measure and integration, which were published from 1899 to 1901.^[69]

Countable models are used in set theory. In 1922, Thoralf Skolem proved that if conventional axioms of set theory are consistent, then they have a countable model. Since this model is countable, its set of real numbers is countable. This consequence is called Skolem's paradox, and Skolem explained why it does not contradict Cantor's uncountability theorem: although there is a one-to-one correspondence between this set and the set of positive integers, no such one-to-one correspondence is a member of the model. Thus the model considers its set of real numbers to be uncountable, or more precisely, the first-order sentence that says the set of real numbers is uncountable is true within the model.^[70] In 1963, Paul Cohen used countable models to prove his independence theorems.^[71]

Lihat pula sunting

Teorema Cantor

Catatan sunting

^ Dalam suratnya kepada Dedekind pada 25 Desember 1873, Cantor mengatakan bahwa ia menulis dan menyerahkan "makalah kecil" yang berjudulkan On a Property of the Set of All Real Algebraic Numbers. (Noether & Cavaillès 1937, hlm. 17)
^ Hal ini menyiratkan sisa teorema — yang mengatakan bahwa ada tak berhingga banyaknya bilangan di $[a, b]$ yang tidak termuat di dalam barisan yang dinyatakan. Sebagai contoh, misalkan $[0, 1]$ adalah interval dan misalkan ada subinterval, katakanlah $[0, 12], [34, 78], [1516, 3132], ...$ Sebab subinterval tersebut saling lepas berpasangan, maka dengan menerapkan bagian pertama dari teorema ke masing-masing subinterval menghasilkan bilangan yang tak berhingga banyaknya di interval $[0, 1]$ yang tidak termuat di barisan. Secara umum, untuk interval $[a, b]$ , dengan menerapkan bagian pertama dari teorema ke subinterval $[a, a + 12 (b - a)], [a + 34 (b - a), a + 78 (b - a)], [a + 1516 (b - a), a + 3132 (b - a)], ...$
^ Cantor tidak membuktikan lema ini. Dalam catatan kaki untuk kasus kedua, ia mengatakan bahwa $x n$ tidak terletak di dalam interval $[a n, b n]$ ,^[11] yang berasal dari buktinya yang dibuat pada tahun 1879. Hal ini memuat bukti induktif yang lebih rumit, yang membuktikan beberapa sifat-sifat dari interval dihasilkan, di antaranya sifat yang dibuktikan di sini.
^ The main difference between Cantor's proof and the above proof is that he generates the sequence of closed intervals [a_n, b_n]. To find a_n + 1 and b_n + 1, he uses the interior of the interval [a_n, b_n], which is the open interval (a_n, b_n). Generating open intervals combines Cantor's use of closed intervals and their interiors, which allows the case diagrams to depict all the details of the proof.
^ Cantor bukanlah pertama kali yang mendefinisikan "rapat dimana-mana", melainkan terminologinya dipakai dengan atau tanpa "dimana-mana" (rapat dimana-mana: Arkhangel'skii & Fedorchuk 1990, hlm. 15; rapat: Kelley 1991, hlm. 49). Pada tahun 1870, Hermann Hankel telah mendefinisikan konsep tersebut melalui terminologi lain: "banyaknya titik … mengisi garis segmen jika tiada interval, tidak peduli seberapa kecil, dapat dinyatakan dalam bentuk segmen tanpa mencari setidaknya satu titik dari banyaknya titik tersebut." (Ferreirós 2007, hlm. 155). Hankel mengembangkan artikel tahun 1829 milik Dirichlet yang memuat fungsi Dirichlet, fungsi terintegralkan non-(Riemann) yang bernilai 0 untuk bilangan rasional dan bernilai 1 untuk bilangan irasional. (Ferreirós 2007, hlm. 149.)
^ Diterjemahkan dari Cantor 1879, hlm. 2: Liegt $P$ theilweise oder ganz im Intervalle $(α . . . β)$ , so kann der bemerkenswerthe Fall eintreten, dass jedes noch so kleine in $(α . . . β)$ enthaltene Intervall $(γ . . . δ)$ Punkte von $P$ enthält. In einem solchen Falle wollen wir sagen, dass $P$ im Intervalle $(α . . . β)$ überall-dicht sei.code: de is deprecated
^ Ini dibuktikan dengan menghasilkan barisan dari titik yang merupakan anggota dari $P$ dan $(c, d)$ . Karena $P$ rapat di $[a, b]$ , maka subinterval $(c, d)$ memuat setidaknya satu titik $x 1$ dari $P$ . Berdasarkan asumsi, subinterval $(x 1, d)$ memuat setidaknya satu titik $x 2$ dari $P$ dan $x 2 > x 1$ sebab $x 2$ adalah anggota dari subinterval tersebut. Setelah menghasilkan $x n$ , subinterval $(x n, d)$ secara umum dipakai menghasilkan titik $x n + 1$ sehingga memenuhi $x n + 1 > x n$ . Ada tak berhingga banyaknya titik $x n$ yang termuat di $P$ dan $(c, d)$ .
^ The beginning of this proof is derived from the proof below by restricting its numbers to the interval [a, b] and by using a subsequence since Cantor was using sequences in his 1873 work on countability.
German text: Satz 68. Es gibt transzendente Zahlen.
Gäbe es nämlich keine transzendenten Zahlen, so wären alle Zahlen algebraisch, das Kontinuum also identisch mit der Menge aller algebraischen Zahlen. Das ist aber unmöglich, weil die Menge aller algebraischen Zahlen abzählbar ist, das Kontinuum aber nicht.code: de is deprecated ^[28]
Translation: Theorem 68. There are transcendental numbers.
If there were no transcendental numbers, then all numbers would be algebraic. Hence, the continuum would be identical to the set of all algebraic numbers. However, this is impossible because the set of all algebraic numbers is countable, but the continuum is not.
^ By "Cantor's proof," Perron does not mean that it is a proof published by Cantor. Rather, he means that the proof only uses arguments that Cantor published. For example, to obtain a real not in a given sequence, Perron follows Cantor's 1874 proof except for one modification: he uses Cantor's 1891 diagonal argument instead of his 1874 nested intervals argument to obtain a real. Cantor never used his diagonal argument to reprove his theorem. In this case, both Cantor's proof and Perron's proof are constructive, so no misconception can arise here. Then, Perron modifies Cantor's proof of the existence of a transcendental by giving the reverse-order proof. This converts Cantor's 1874 constructive proof into a non-constructive proof which leads to the misconception about Cantor's work.
^ This proof is the same as Cantor's 1874 proof except for one modification: it uses his 1891 diagonal argument instead of his 1874 nested intervals argument to obtain a real.
^ The program using the diagonal method produces $n$ digits in ${\color {Blue}O}(n^{2}\log ^{2}n\log \log n)$ steps, while the program using the 1874 method requires at least $O(2^{\sqrt[{3}]{n}})$ steps to produce $n$ digits. (Gray 1994, hlm. 822–823.)
^ Starting with Hardy and Wright's book, these books are linked to Perron's book via their bibliographies: Perron's book is mentioned in the bibliography of Hardy and Wright's book, which in turn is mentioned in the bibliography of Birkhoff and Mac Lane's book and in the bibliography of Spivak's book. (Hardy & Wright 1938, hlm. 400; Birkhoff & Mac Lane 1941, hlm. 441; Spivak 1967, hlm. 515.)
^ Kronecker's opinion was: "Definitions must contain the means of reaching a decision in a finite number of steps, and existence proofs must be conducted so that the quantity in question can be calculated with any required degree of accuracy."^[40] So Kronecker would accept Cantor's argument as a valid existence proof, but he would not accept its conclusion that transcendental numbers exist. For Kronecker, they do not exist because their definition contains no means for deciding in a finite number of steps whether or not a given number is transcendental.^[41] Cantor's 1874 construction calculates numbers to any required degree of accuracy because: Given a k, an n can be computed such that b_n – a_n ≤ 1k where (a_n, b_n) is the n-th interval of Cantor's construction. An example of how to prove this is given in Gray 1994, hlm. 822. Cantor's diagonal argument provides an accuracy of 10⁻ⁿ after n real algebraic numbers have been calculated because each of these numbers generates one digit of the transcendental number.^[42]
^ Ferreirós has analyzed the relations between Cantor and Dedekind. He explains why "Relations between both mathematicians were difficult after 1874, when they underwent an interruption…" (Ferreirós 1993, hlm. 344, 348–352.)
^ Cantor's method of constructing a one-to-one correspondence between the set of irrational numbers and R can be used to construct one between the set of transcendental numbers and R.^[64] The construction begins with the set of transcendental numbers T and removes a countable subset {t_n} (for example, t_n = en). Let this set be T₀. Then T = T₀ ∪ {t_n} = T₀ ∪ {t_{2n – 1}} ∪ {t_2n}, and R = T ∪ {a_n} = T₀ ∪ {t_n} ∪ {a_n} where a_n is the sequence of real algebraic numbers. So both T and R are the union of three pairwise disjoint sets: T₀ and two countable sets. A one-to-one correspondence between T and R is given by the function: g(t) = t if t ∈ T₀, g(t_{2n – 1}) = t_n, and g(t_2n) = a_n.

Catatan tentang bukti Cantor tahun 1879 sunting

^ ^a ^b ^c ^d ^e ^f Since Cantor's proof has not been published in English, an English translation is given alongside the original German text, which is from Cantor 1879, hlm. 5–7. The translation starts one sentence before the proof because this sentence mentions Cantor's 1874 proof. Cantor states it was printed in Borchardt's Journal. Crelle’s Journal was also called Borchardt’s Journal from 1856-1880 when Carl Wilhelm Borchardt edited the journal (Audin 2011, hlm. 80). Square brackets are used to identify this mention of Cantor's earlier proof, to clarify the translation, and to provide page numbers. Also, "Mannichfaltigkeit" (manifold) has been translated to "set" and Cantor's notation for closed sets (α . . . β) has been translated to [α, β]. Cantor changed his terminology from Mannichfaltigkeit to Menge (set) in his 1883 article, which introduced sets of ordinal numbers (Kanamori 2012, hlm. 5). Currently in mathematics, a manifold is type of topological space.

English translation	German text
[Page 5] . . . But this contradicts a very general theorem, which we have proved with full rigor in Borchardt's Journal, Vol. 77, page 260; namely, the following theorem: "If one has a simply [countably] infinite sequence ω₁, ω₂, . . . , ω_ν, . . . of real, unequal numbers that proceed according to some rule, then in every given interval [α, β] a number η (and thus infinitely many of them) can be specified that does not occur in this sequence (as a member of it)." In view of the great interest in this theorem, not only in the present discussion, but also in many other arithmetical as well as analytical relations, it might not be superfluous if we develop the argument followed there [Cantor's 1874 proof] more clearly here by using simplifying modifications. Starting with the sequence: ω₁, ω₂, . . . , ω_ν, . . . (which we give [denote by] the symbol (ω)) and an arbitrary interval [α, β], where α < β, we will now demonstrate that in this interval a real number η can be found that does not occur in (ω). I. We first notice that if our set (ω) is not everywhere dense in the interval [α, β], then within this interval another interval [γ, δ] must be present, all of whose numbers do not belong to (ω). From the interval [γ, δ], one can then choose any number for η. It lies in the interval [α, β] and definitely does not occur in our sequence (ω). Thus, this case presents no special considerations and we can move on to the more difficult case. II. Let the set (ω) be everywhere dense in the interval [α, β]. In this case, every interval [γ,δ] located in [α,β], however small, contains numbers of our sequence (ω). To show that, nevertheless, numbers η in the interval [α, β] exist that do not occur in (ω), we employ the following observation. Since some numbers in our sequence: ω₁, ω₂, . . . , ω_ν, . . .	[Seite 5] . . . Dem widerspricht aber ein sehr allgemeiner Satz, welchen wir in Borchardt's Journal, Bd. 77, pag. 260, mit aller Strenge bewiesen haben, nämlich der folgende Satz: "Hat man eine einfach unendliche Reihe ω₁, ω₂, . . . , ω_ν, . . . von reellen, ungleichen Zahlen, die nach irgend einem Gesetz fortschreiten, so lässt sich in jedem vorgegebenen, Intervalle (α . . . β) eine Zahl η (und folglich lassen sich deren unendlich viele) angeben, welche nicht in jener Reihe (als Glied derselben) vorkommt." In Anbetracht des grossen Interesses, welches sich an diesen Satz, nicht blos bei der gegenwärtigen Erörterung, sondern auch in vielen anderen sowohl arithmetischen, wie analytischen Beziehungen, knüpft, dürfte es nicht überflüssig sein, wenn wir die dort befolgte Beweisführung [Cantors 1874 Beweis], unter Anwendung vereinfachender Modificationen, hier deutlicher entwickeln. Unter Zugrundelegung der Reihe: ω₁, ω₂, . . . , ω_ν, . . . (welcher wir das Zeichen (ω) beilegen) und eines beliebigen Intervalles (α . . . β), wo α < β ist, soll also nun gezeigt werden, dass in diesem Intervalle eine reelle Zahl η gefunden werden kann, welche in (ω) nicht vorkommt. I. Wir bemerken zunächst, dass wenn unsre Mannichfaltigkeit (ω) in dem Intervall (α . . . β) nicht überall-dicht ist, innerhalb dieses Intervalles ein anderes (γ . . . δ) vorhanden sein muss, dessen Zahlen sämmtlich nicht zu (ω) gehören; man kann alsdann für η irgend eine Zahl des Intervalls (γ . . . δ) wählen, sie liegt im Intervalle (α . . . β) und kommt sicher in unsrer Reihe (ω) nicht vor. Dieser Fall bietet daher keinerlei besondere Umstände; und wir können zu dem schwierigeren übergehen. II. Die Mannichfaltigkeit (ω) sei im Intervalle (α . . . β) überall-dicht. In diesem Falle enthält jedes, noch so kleine in (α . . . β) gelegene Intervall (γ . . . δ) Zahlen unserer Reihe (ω). Um zu zeigen, dass nichtsdestoweniger Zahlen η im Intervalle (α . . . β) existiren, welche in (ω) nicht vorkommen, stellen wir die folgende Betrachtung an. Da in unserer Reihe: ω₁, ω₂, . . . , ω_ν, . . . code: de is deprecated
[Page 6] definitely occur within the interval [α, β], one of these numbers must have the least index, let it be ω_κ₁, and another: ω_κ₂ with the next larger index. Let the smaller of the two numbers ω_κ₁, ω_κ₂ be denoted by α', the larger by β'. (Their equality is impossible because we assumed that our sequence consists of nothing but unequal numbers.) Then according to the definition: α < α' < β' < β , furthermore: κ₁ < κ₂ ; and all numbers ω_μ of our sequence, for which μ ≤ κ₂, do not lie in the interior of the interval [α', β'], as is immediately clear from the definition of the numbers κ₁, κ₂. Similarly, let ω_κ₃ and ω_κ₄ be the two numbers of our sequence with smallest indices that fall in the interior of the interval [α', β'] and let the smaller of the numbers ω_κ₃, ω_κ₄ be denoted by α'', the larger by β''. Then one has: α' < α'' < β'' < β' , κ₂ < κ₃ < κ₄ ; and one sees that all numbers ω_μ of our sequence, for which μ ≤ κ₄, do not fall into the interior of the interval [α'', β'']. After one has followed this rule to reach an interval [α^{(ν - 1)}, β^{(ν - 1)}], the next interval is produced by selecting the first two (i. e. with lowest indices) numbers of our sequence (ω) (let them be ω_{κ_{2ν - 1}} and ω_{κ_2ν}) that fall into the interior of [α^{(ν - 1)}, β^{(ν - 1)}]. Let the smaller of these two numbers be denoted by α^(ν), the larger by β^(ν). The interval [α^(ν), β^(ν)] then lies in the interior of all preceding intervals and has the specific relation with our sequence (ω) that all numbers ω_μ, for which μ ≤ κ_2ν, definitely do not lie in its interior. Since obviously: κ₁ < κ₂ < κ₃ < . . . , ω_{κ_{2ν – 2}} < ω_{κ_{2ν – 1}} < ω_{κ_2ν} , . . . and these numbers, as indices, are whole numbers, so: κ_2ν ≥ 2ν , and hence: ν < κ_2ν ; thus, we can certainly say (and this is sufficient for the following): That if ν is an arbitrary whole number, the [real] quantity ω_ν lies outside the interval [α^(ν) . . . β^(ν)].	[Seite 6] sicher Zahlen innerhalb des Intervalls (α . . . β) vorkommen, so muss eine von diesen Zahlen den kleinsten Index haben, sie sei ω_κ₁, und eine andere: ω_κ₂ mit dem nächst grösseren Index behaftet sein. Die kleinere der beiden Zahlen ω_κ₁, ω_κ₂ werde mit α', die grössere mit β' bezeichnet. (Ihre Gleichheit ist ausgeschlossen, weil wir voraussetzten, dass unsere Reihe aus lauter ungleichen Zahlen besteht.) Es ist alsdann der Definition nach: α < α' < β' < β , ferner: κ₁ < κ₂ ; und ausserdem ist zu bemerken, dass alle Zahlen ω_μ unserer Reihe, für welche μ ≤ κ₂, nicht im Innern des Intervalls (α' . . . β') liegen, wie aus der Bestimmung der Zahlen κ₁, κ₂ sofort erhellt. Ganz ebenso mögen ω_κ₃, ω_κ₄ die beiden mit den kleinsten Indices versehenen Zahlen unserer Reihen [see note 1 below] sein, welche in das Innere des Intervalls (α' . . . β') fallen und die kleinere der Zahlen ω_κ₃, ω_κ₄ werde mit α'', die grössere mit β'' bezeichnet. Man hat alsdann: α' < α'' < β'' < β' , κ₂ < κ₃ < κ₄ ; und man erkennt, dass alle Zahlen ω_μ unserer Reihe, für welche μ ≤ κ₄ nicht in das Innere des Intervalls (α'' . . . β'') fallen. Nachdem man unter Befolgung des gleichen Gesetzes zu einem Intervall (α^{(ν - 1)}, . . . β^{(ν - 1)}) gelangt ist, ergiebt sich das folgende Intervall dadurch aus demselben, dass man die beiden ersten (d. h. mit niedrigsten Indices versehenen) Zahlen unserer Reihe (ω) aufstellt (sie seien ω_{κ_{2ν – 1}} und ω_{κ_2ν}), welche in das Innere von (α^{(ν – 1)} . . . β^{(ν – 1)}) fallen; die kleinere dieser beiden Zahlen werde mit α^(ν), die grössere mit β^(ν) bezeichnet. Das Intervall (α^(ν) . . . β^(ν)) liegt alsdann im Innern aller vorangegangenen Intervalle und hat zu unserer Reihe (ω) die eigenthümliche Beziehung, dass alle Zahlen ω_μ, für welche μ ≤ κ_2ν sicher nicht in seinem Innern liegen. Da offenbar: κ₁ < κ₂ < κ₃ < . . . , ω_{κ_{2ν – 2}} < ω_{κ_{2ν – 1}} < ω_{κ_2ν} , . . . und diese Zahlen, als Indices, ganze Zahlen sind, so ist: κ_2ν ≥ 2ν , und daher: ν < κ_2ν ; wir können daher, und dies ist für das Folgende ausreichend, gewiss sagen: Dass, wenn ν eine beliebige ganze Zahl ist, die Grösse ω_ν ausserhalb des Intervalls (α^(ν) . . . β^(ν)) liegt. code: de is deprecated
[Page 7] Since the numbers α', α'', α''', . . ., α^(ν), . . . are continually increasing by value while simultaneously being enclosed in the interval [α, β], they have, by a well-known fundamental theorem of the theory of magnitudes [see note 2 below], a limit that we denote by A, so that: A Templat:Eq Lim α^(ν) for ν Templat:Eq ∞. The same applies to the numbers β', β'', β''', . . ., β^(ν), . . ., which are continually decreasing and likewise lying in the interval [α, β]. We call their limit B, so that: B Templat:Eq Lim β^(ν) for ν Templat:Eq ∞. Obviously, one has: α^(ν) < A ≤ B < β^(ν). But it is easy to see that the case A < B can not occur here since otherwise every number ω_ν of our sequence would lie outside of the interval [A, B] by lying outside the interval [α^(ν), β^(ν)]. So our sequence (ω) would not be everywhere dense in the interval [α, β], contrary to the assumption. Thus, there only remains the case A Templat:Eq B and now it is demonstrated that the number: η Templat:Eq A Templat:Eq B does not occur in our sequence (ω). If it were a member of our sequence, such as the ν^th, then one would have: η Templat:Eq ω_ν. But the latter equation is not possible for any value of ν because η is in the interior of the interval [α^(ν), β^(ν)], but ω_ν lies outside of it.	[Seite 7] Da die Zahlen α', α'', α''', . . ., α^(ν), . . . ihrer Grösse nach fortwährend wachsen, dabei jedoch im Intervalle (α . . . β) eingeschlossen sind, so haben sie, nach einem bekannten Fundamentalsatze der Grössenlehre, eine Grenze, die wir mit A bezeichnen, so dass: A Templat:Eq Lim α^(ν) für ν Templat:Eq ∞. Ein Gleiches gilt für die Zahlen β', β'', β''', . . ., β^(ν), . . . welche fortwährend abnehmen und dabei ebenfalls im Intervalle (α . . . β) liegen; wir nennen ihre Grenze B, so dass: B Templat:Eq Lim β^(ν) für ν Templat:Eq ∞. Man hat offenbar: α^(ν) < A ≤ B < β^(ν). Es ist aber leicht zu sehen, dass der Fall A < B hier nicht vorkommen kann; da sonst jede Zahl ω_ν, unserer Reihe ausserhalb des Intervalles (A . . . B) liegen würde, indem ω_ν, ausserhalb des Intervalls (α^(ν) . . . β^(ν)) gelegen ist; unsere Reihe (ω) wäre im Intervall (α . . . β) nicht überalldicht, gegen die Voraussetzung. Es bleibt daher nur der Fall A Templat:Eq B übrig und es zeigt sich nun, dass die Zahl: η Templat:Eq A Templat:Eq B in unserer Reihe (ω) nicht vorkommt. Denn, würde sie ein Glied unserer Reihe sein, etwa das ν^te, so hätte man: η Templat:Eq ω_ν. Die letztere Gleichung ist aber für keinen Werth von v möglich, weil η im Innern des Intervalls [α^(ν), β^(ν)], ω_ν aber ausserhalb desselben liegt. code: de is deprecated
Note 1: This is the only occurrence of "unserer Reihen" ("our sequences") in the proof. There is only one sequence involved in Cantor's proof and everywhere else "Reihe" ("sequence") is used, so it is most likely a typographical error and should be "unserer Reihe" ("our sequence"), which is how it has been translated. Note 2: Grössenlehre, which has been translated as "the theory of magnitudes", is a term used by 19th century German mathematicians that refers to the theory of discrete and continuous magnitudes. (Ferreirós 2007, hlm. 41–42, 202.)

Referensi sunting

^ Dauben 1993, hlm. 4.
^ Gray 1994, hlm. 819–821.
^ ^a ^b Cantor 1874
^ ^a ^b Gray 1994, hlm. 828.
^ ^a ^b ^c ^d ^e Cantor 1874, hlm. 259.
^ Cantor 1874, hlm. 259
^ Cantor 1878, hlm. 242.
^ Gray 1994, hlm. 820.
^ Cantor 1874, hlm. 259–260. English translation: Ewald 1996, hlm. 841.
^ Cantor 1874, hlm. 260–261. English translation: Ewald 1996, hlm. 841–842.
^ ^a ^b Cantor 1874, hlm. 261.
^ Gray 1994, hlm. 822.
^ Havil 2012, hlm. 208–209.
^ Havil 2012, hlm. 209.
^ LeVeque 1956, hlm. 154–155.
^ LeVeque 1956, hlm. 174.
^ Weisstein 2003, hlm. 541.
^ Arkhangel'skii & Fedorchuk 1990, hlm. 16.
^ Noether & Cavaillès 1937, hlm. 12–13. English translation: Gray 1994, hlm. 827; Ewald 1996, hlm. 844.
^ ^a ^b ^c ^d Noether & Cavaillès 1937, hlm. 18. English translation: Ewald 1996, hlm. 848.
^ Noether & Cavaillès 1937, hlm. 13. English translation: Gray 1994, hlm. 827.
^ ^a ^b ^c ^d ^e ^f ^g Noether & Cavaillès 1937, hlm. 14–15. English translation: Ewald 1996, hlm. 845–846.
^ Gray 1994, hlm. 827
^ Dauben 1979, hlm. 51.
^ Noether & Cavaillès 1937, hlm. 19. English translation: Ewald 1996, hlm. 849.
^ Ewald 1996, hlm. 843.
^ Noether & Cavaillès 1937, hlm. 16. English translation: Gray 1994, hlm. 827.
^ Perron 1921, hlm. 162.
^ ^a ^b Kanamori 2012, hlm. 4.
^ Gray 1994, hlm. 827–828.
^ Perron 1921, hlm. 162
^ Fraenkel 1930, hlm. 237. English translation: Gray 1994, hlm. 823.
^ Kaplansky 1972, hlm. 25.
^ Gray 1994, hlm. 829–830.
^ Gray 1994, hlm. 821–824.
^ Bell 1937, hlm. 568–569; Hardy & Wright 1938, hlm. 159 (6th ed., pp. 205–206); Birkhoff & Mac Lane 1941, hlm. 392, (5th ed., pp. 436–437); Spivak 1967, hlm. 369–370 (4th ed., pp. 448–449).
^ Dasgupta 2014, hlm. 107; Sheppard 2014, hlm. 131–132.
^ Jarvis 2014, hlm. 18; Chowdhary 2015, hlm. 19; Stewart 2015, hlm. 285; Stewart & Tall 2015, hlm. 333.
^ Birkhoff & Mac Lane 1941, hlm. 392, (5th ed., pp. 436–437).
^ Burton 1995, hlm. 595.
^ Dauben 1979, hlm. 69.
^ Gray 1994, hlm. 824.
^ ^a ^b Ferreirós 2007, hlm. 184.
^ Noether & Cavaillès 1937, hlm. 12–16. English translation: Ewald 1996, hlm. 843–846.
^ Dauben 1979, hlm. 67.
^ ^a ^b ^c ^d Noether & Cavaillès 1937, hlm. 16–17. English translation: Ewald 1996, hlm. 847.
^ Grattan-Guinness 1971, hlm. 124.
^ Dauben 1979, hlm. 67, 308–309.
^ Ferreirós 2007, hlm. 184–185, 245.
^ Ferreirós 2007, hlm. 185
^ Ferreirós 2007, hlm. 177.
^ Dauben 1979, hlm. 67–68.
^ Ferreirós 2007, hlm. 183.
^ Ferreirós 2007, hlm. 185.
^ Ferreirós 2007, hlm. 109–111, 172–174.
^ Ferreirós 1993, hlm. 349–350.
^ Noether & Cavaillès 1937, hlm. 12–13. English translation: Ewald 1996, hlm. 844–845.
^ Noether & Cavaillès 1937, hlm. 13. English translation: Ewald 1996, hlm. 845.
^ Ferreirós 2007, hlm. 179.
^ Noether & Cavaillès 1937, hlm. 14–16, 19. English translation: Ewald 1996, hlm. 845–847, 849.
^ Ferreirós 1993, hlm. 358–359.
^ Ferreirós 1993, hlm. 350.
^ Cantor 1878, hlm. 245–254.
^ Cantor 1879, hlm. 4.
^ Ferreirós 2007, hlm. 267–273.
^ Ferreirós 2007, hlm. xvi, 320–321, 324.
^ Cantor 1878, hlm. 243.
^ Hawkins 1970, hlm. 103–106, 127.
^ Hawkins 1970, hlm. 118, 120–124, 127.
^ Ferreirós 2007, hlm. 362–363.
^ Cohen 1963, hlm. 1143–1144.

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Spivak, Michael (1967), Calculus, London: W. A. Benjamin, ISBN 978-0914098911 .
Stewart, Ian (2015), Galois Theory (edisi ke-4th), Boca Raton, Florida: CRC Press, ISBN 978-1-4822-4582-0 .
Stewart, Ian; Tall, David (2015), The Foundations of Mathematics (edisi ke-2nd), New York: Oxford University Press, ISBN 978-0-19-870644-1 .
Weisstein, Eric W., ed. (2003), "Continued Fraction", CRC Concise Encyclopedia of Mathematics, Boca Raton, Florida: Chapman & Hall/CRC, ISBN 978-1-58488-347-0 .

[3] Dalam suratnya kepada Dedekind pada 25 Desember 1873, Cantor mengatakan bahwa ia menulis dan menyerahkan "makalah kecil" yang berjudulkan On a Property of the Set of All Real Algebraic Numbers. (Noether & Cavaillès 1937, hlm. 17)

[11] Hal ini menyiratkan sisa teorema — yang mengatakan bahwa ada tak berhingga banyaknya bilangan di $[a, b]$ yang tidak termuat di dalam barisan yang dinyatakan. Sebagai contoh, misalkan $[0, 1]$ adalah interval dan misalkan ada subinterval, katakanlah $[0, 12], [34, 78], [1516, 3132], ...$ Sebab subinterval tersebut saling lepas berpasangan, maka dengan menerapkan bagian pertama dari teorema ke masing-masing subinterval menghasilkan bilangan yang tak berhingga banyaknya di interval $[0, 1]$ yang tidak termuat di barisan. Secara umum, untuk interval $[a, b]$ , dengan menerapkan bagian pertama dari teorema ke subinterval $[a, a + 12 (b - a)], [a + 34 (b - a), a + 78 (b - a)], [a + 1516 (b - a), a + 3132 (b - a)], ...$

[14] Cantor tidak membuktikan lema ini. Dalam catatan kaki untuk kasus kedua, ia mengatakan bahwa $x n$ tidak terletak di dalam interval $[a n, b n]$ ,^[11] yang berasal dari buktinya yang dibuat pada tahun 1879. Hal ini memuat bukti induktif yang lebih rumit, yang membuktikan beberapa sifat-sifat dari interval dihasilkan, di antaranya sifat yang dibuktikan di sini.

[15] The main difference between Cantor's proof and the above proof is that he generates the sequence of closed intervals [a_n, b_n]. To find a_n + 1 and b_n + 1, he uses the interior of the interval [a_n, b_n], which is the open interval (a_n, b_n). Generating open intervals combines Cantor's use of closed intervals and their interiors, which allows the case diagrams to depict all the details of the proof.

[22] Cantor bukanlah pertama kali yang mendefinisikan "rapat dimana-mana", melainkan terminologinya dipakai dengan atau tanpa "dimana-mana" (rapat dimana-mana: Arkhangel'skii & Fedorchuk 1990, hlm. 15; rapat: Kelley 1991, hlm. 49). Pada tahun 1870, Hermann Hankel telah mendefinisikan konsep tersebut melalui terminologi lain: "banyaknya titik … mengisi garis segmen jika tiada interval, tidak peduli seberapa kecil, dapat dinyatakan dalam bentuk segmen tanpa mencari setidaknya satu titik dari banyaknya titik tersebut." (Ferreirós 2007, hlm. 155). Hankel mengembangkan artikel tahun 1829 milik Dirichlet yang memuat fungsi Dirichlet, fungsi terintegralkan non-(Riemann) yang bernilai 0 untuk bilangan rasional dan bernilai 1 untuk bilangan irasional. (Ferreirós 2007, hlm. 149.)

[23] Diterjemahkan dari Cantor 1879, hlm. 2: Liegt $P$ theilweise oder ganz im Intervalle $(α . . . β)$ , so kann der bemerkenswerthe Fall eintreten, dass jedes noch so kleine in $(α . . . β)$ enthaltene Intervall $(γ . . . δ)$ Punkte von $P$ enthält. In einem solchen Falle wollen wir sagen, dass $P$ im Intervalle $(α . . . β)$ überall-dicht sei.code: de is deprecated

[25] Ini dibuktikan dengan menghasilkan barisan dari titik yang merupakan anggota dari $P$ dan $(c, d)$ . Karena $P$ rapat di $[a, b]$ , maka subinterval $(c, d)$ memuat setidaknya satu titik $x 1$ dari $P$ . Berdasarkan asumsi, subinterval $(x 1, d)$ memuat setidaknya satu titik $x 2$ dari $P$ dan $x 2 > x 1$ sebab $x 2$ adalah anggota dari subinterval tersebut. Setelah menghasilkan $x n$ , subinterval $(x n, d)$ secara umum dipakai menghasilkan titik $x n + 1$ sehingga memenuhi $x n + 1 > x n$ . Ada tak berhingga banyaknya titik $x n$ yang termuat di $P$ dan $(c, d)$ .

[37] The beginning of this proof is derived from the proof below by restricting its numbers to the interval [a, b] and by using a subsequence since Cantor was using sequences in his 1873 work on countability.
German text: Satz 68. Es gibt transzendente Zahlen.
Gäbe es nämlich keine transzendenten Zahlen, so wären alle Zahlen algebraisch, das Kontinuum also identisch mit der Menge aller algebraischen Zahlen. Das ist aber unmöglich, weil die Menge aller algebraischen Zahlen abzählbar ist, das Kontinuum aber nicht.code: de is deprecated ^[28]
Translation: Theorem 68. There are transcendental numbers.
If there were no transcendental numbers, then all numbers would be algebraic. Hence, the continuum would be identical to the set of all algebraic numbers. However, this is impossible because the set of all algebraic numbers is countable, but the continuum is not.

[41] By "Cantor's proof," Perron does not mean that it is a proof published by Cantor. Rather, he means that the proof only uses arguments that Cantor published. For example, to obtain a real not in a given sequence, Perron follows Cantor's 1874 proof except for one modification: he uses Cantor's 1891 diagonal argument instead of his 1874 nested intervals argument to obtain a real. Cantor never used his diagonal argument to reprove his theorem. In this case, both Cantor's proof and Perron's proof are constructive, so no misconception can arise here. Then, Perron modifies Cantor's proof of the existence of a transcendental by giving the reverse-order proof. This converts Cantor's 1874 constructive proof into a non-constructive proof which leads to the misconception about Cantor's work.

[44] This proof is the same as Cantor's 1874 proof except for one modification: it uses his 1891 diagonal argument instead of his 1874 nested intervals argument to obtain a real.

[47] The program using the diagonal method produces $n$ digits in ${\color {Blue}O}(n^{2}\log ^{2}n\log \log n)$ steps, while the program using the 1874 method requires at least $O(2^{\sqrt[{3}]{n}})$ steps to produce $n$ digits. (Gray 1994, hlm. 822–823.)

[49] Starting with Hardy and Wright's book, these books are linked to Perron's book via their bibliographies: Perron's book is mentioned in the bibliography of Hardy and Wright's book, which in turn is mentioned in the bibliography of Birkhoff and Mac Lane's book and in the bibliography of Spivak's book. (Hardy & Wright 1938, hlm. 400; Birkhoff & Mac Lane 1941, hlm. 441; Spivak 1967, hlm. 515.)

[56] Kronecker's opinion was: "Definitions must contain the means of reaching a decision in a finite number of steps, and existence proofs must be conducted so that the quantity in question can be calculated with any required degree of accuracy."^[40] So Kronecker would accept Cantor's argument as a valid existence proof, but he would not accept its conclusion that transcendental numbers exist. For Kronecker, they do not exist because their definition contains no means for deciding in a finite number of steps whether or not a given number is transcendental.^[41] Cantor's 1874 construction calculates numbers to any required degree of accuracy because: Given a k, an n can be computed such that b_n – a_n ≤ 1k where (a_n, b_n) is the n-th interval of Cantor's construction. An example of how to prove this is given in Gray 1994, hlm. 822. Cantor's diagonal argument provides an accuracy of 10⁻ⁿ after n real algebraic numbers have been calculated because each of these numbers generates one digit of the transcendental number.^[42]

[77] Ferreirós has analyzed the relations between Cantor and Dedekind. He explains why "Relations between both mathematicians were difficult after 1874, when they underwent an interruption…" (Ferreirós 1993, hlm. 344, 348–352.)

[80] Cantor's method of constructing a one-to-one correspondence between the set of irrational numbers and R can be used to construct one between the set of transcendental numbers and R.^[64] The construction begins with the set of transcendental numbers T and removes a countable subset {t_n} (for example, t_n = en). Let this set be T₀. Then T = T₀ ∪ {t_n} = T₀ ∪ {t_{2n – 1}} ∪ {t_2n}, and R = T ∪ {a_n} = T₀ ∪ {t_n} ∪ {a_n} where a_n is the sequence of real algebraic numbers. So both T and R are the union of three pairwise disjoint sets: T₀ and two countable sets. A one-to-one correspondence between T and R is given by the function: g(t) = t if t ∈ T₀, g(t_{2n – 1}) = t_n, and g(t_2n) = a_n.

[p-26] ^ ^a ^b ^c ^d ^e ^f Since Cantor's proof has not been published in English, an English translation is given alongside the original German text, which is from Cantor 1879, hlm. 5–7. The translation starts one sentence before the proof because this sentence mentions Cantor's 1874 proof. Cantor states it was printed in Borchardt's Journal. Crelle’s Journal was also called Borchardt’s Journal from 1856-1880 when Carl Wilhelm Borchardt edited the journal (Audin 2011, hlm. 80). Square brackets are used to identify this mention of Cantor's earlier proof, to clarify the translation, and to provide page numbers. Also, "Mannichfaltigkeit" (manifold) has been translated to "set" and Cantor's notation for closed sets (α . . . β) has been translated to [α, β]. Cantor changed his terminology from Mannichfaltigkeit to Menge (set) in his 1883 article, which introduced sets of ordinal numbers (Kanamori 2012, hlm. 5). Currently in mathematics, a manifold is type of topological space.

English translation German text

[Page 5]

. . . But this contradicts a very general theorem, which we have proved with full rigor in Borchardt's Journal, Vol. 77, page 260; namely, the following theorem:
"If one has a simply [countably] infinite sequence
ω₁, ω₂, . . . , ω_ν, . . .
of real, unequal numbers that proceed according to some rule, then in every given interval [α, β] a number η (and thus infinitely many of them) can be specified that does not occur in this sequence (as a member of it)."
In view of the great interest in this theorem, not only in the present discussion, but also in many other arithmetical as well as analytical relations, it might not be superfluous if we develop the argument followed there [Cantor's 1874 proof] more clearly here by using simplifying modifications.
Starting with the sequence:
ω₁, ω₂, . . . , ω_ν, . . .
(which we give [denote by] the symbol (ω)) and an arbitrary interval [α, β], where α < β, we will now demonstrate that in this interval a real number η can be found that does not occur in (ω).
I. We first notice that if our set (ω) is not everywhere dense in the interval [α, β], then within this interval another interval [γ, δ] must be present, all of whose numbers do not belong to (ω). From the interval [γ, δ], one can then choose any number for η. It lies in the interval [α, β] and definitely does not occur in our sequence (ω). Thus, this case presents no special considerations and we can move on to the more difficult case.
II. Let the set (ω) be everywhere dense in the interval [α, β]. In this case, every interval [γ,δ] located in [α,β], however small, contains numbers of our sequence (ω). To show that, nevertheless, numbers η in the interval [α, β] exist that do not occur in (ω), we employ the following observation.
Since some numbers in our sequence:
ω₁, ω₂, . . . , ω_ν, . . .

[Seite 5]
. . . Dem widerspricht aber ein sehr allgemeiner Satz, welchen wir in Borchardt's Journal, Bd. 77, pag. 260, mit aller Strenge bewiesen haben, nämlich der folgende Satz:
"Hat man eine einfach unendliche Reihe
ω₁, ω₂, . . . , ω_ν, . . .
von reellen, ungleichen Zahlen, die nach irgend einem Gesetz fortschreiten, so lässt sich in jedem vorgegebenen, Intervalle (α . . . β) eine Zahl η (und folglich lassen sich deren unendlich viele) angeben, welche nicht in jener Reihe (als Glied derselben) vorkommt."
In Anbetracht des grossen Interesses, welches sich an diesen Satz, nicht blos bei der gegenwärtigen Erörterung, sondern auch in vielen anderen sowohl arithmetischen, wie analytischen Beziehungen, knüpft, dürfte es nicht überflüssig sein, wenn wir die dort befolgte Beweisführung [Cantors 1874 Beweis], unter Anwendung vereinfachender Modificationen, hier deutlicher entwickeln.
Unter Zugrundelegung der Reihe:
ω₁, ω₂, . . . , ω_ν, . . .
(welcher wir das Zeichen (ω) beilegen) und eines beliebigen Intervalles (α . . . β), wo α < β ist, soll also nun gezeigt werden, dass in diesem Intervalle eine reelle Zahl η gefunden werden kann, welche in (ω) nicht vorkommt.
I. Wir bemerken zunächst, dass wenn unsre Mannichfaltigkeit (ω) in dem Intervall (α . . . β) nicht überall-dicht ist, innerhalb dieses Intervalles ein anderes (γ . . . δ) vorhanden sein muss, dessen Zahlen sämmtlich nicht zu (ω) gehören; man kann alsdann für η irgend eine Zahl des Intervalls (γ . . . δ) wählen, sie liegt im Intervalle (α . . . β) und kommt sicher in unsrer Reihe (ω) nicht vor. Dieser Fall bietet daher keinerlei besondere Umstände; und wir können zu dem schwierigeren übergehen.
II. Die Mannichfaltigkeit (ω) sei im Intervalle (α . . . β) überall-dicht. In diesem Falle enthält jedes, noch so kleine in (α . . . β) gelegene Intervall (γ . . . δ) Zahlen unserer Reihe (ω). Um zu zeigen, dass nichtsdestoweniger Zahlen η im Intervalle (α . . . β) existiren, welche in (ω) nicht vorkommen, stellen wir die folgende Betrachtung an.
Da in unserer Reihe:
ω₁, ω₂, . . . , ω_ν, . . .
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[Page 6]

definitely occur within the interval [α, β], one of these numbers must have the least index, let it be ω_κ₁, and another: ω_κ₂ with the next larger index.
Let the smaller of the two numbers ω_κ₁, ω_κ₂ be denoted by α', the larger by β'. (Their equality is impossible because we assumed that our sequence consists of nothing but unequal numbers.)
Then according to the definition:
α < α' < β' < β ,
furthermore:
κ₁ < κ₂ ;
and all numbers ω_μ of our sequence, for which μ ≤ κ₂, do not lie in the interior of the interval [α', β'], as is immediately clear from the definition of the numbers κ₁, κ₂. Similarly, let ω_κ₃ and ω_κ₄ be the two numbers of our sequence with smallest indices that fall in the interior of the interval [α', β'] and let the smaller of the numbers ω_κ₃, ω_κ₄ be denoted by α'', the larger by β''.
Then one has:
α' < α'' < β'' < β' ,
κ₂ < κ₃ < κ₄ ;
and one sees that all numbers ω_μ of our sequence, for which μ ≤ κ₄, do not fall into the interior of the interval [α'', β''].
After one has followed this rule to reach an interval [α^{(ν - 1)}, β^{(ν - 1)}], the next interval is produced by selecting the first two (i. e. with lowest indices) numbers of our sequence (ω) (let them be ω_{κ_{2ν - 1}} and ω_{κ_2ν}) that fall into the interior of [α^{(ν - 1)}, β^{(ν - 1)}]. Let the smaller of these two numbers be denoted by α^(ν), the larger by β^(ν).
The interval [α^(ν), β^(ν)] then lies in the interior of all preceding intervals and has the specific relation with our sequence (ω) that all numbers ω_μ, for which μ ≤ κ_2ν, definitely do not lie in its interior. Since obviously:
κ₁ < κ₂ < κ₃ < . . . , ω_{κ_{2ν – 2}} < ω_{κ_{2ν – 1}} < ω_{κ_2ν} , . . .
and these numbers, as indices, are whole numbers, so:
κ_2ν ≥ 2ν ,
and hence:
ν < κ_2ν ;
thus, we can certainly say (and this is sufficient for the following):
That if ν is an arbitrary whole number, the [real] quantity ω_ν lies outside the interval [α^(ν) . . . β^(ν)].

[Seite 6]
sicher Zahlen innerhalb des Intervalls (α . . . β) vorkommen, so muss eine von diesen Zahlen den kleinsten Index haben, sie sei ω_κ₁, und eine andere: ω_κ₂ mit dem nächst grösseren Index behaftet sein.
Die kleinere der beiden Zahlen ω_κ₁, ω_κ₂ werde mit α', die grössere mit β' bezeichnet. (Ihre Gleichheit ist ausgeschlossen, weil wir voraussetzten, dass unsere Reihe aus lauter ungleichen Zahlen besteht.)
Es ist alsdann der Definition nach:
α < α' < β' < β ,
ferner:
κ₁ < κ₂ ;
und ausserdem ist zu bemerken, dass alle Zahlen ω_μ unserer Reihe, für welche μ ≤ κ₂, nicht im Innern des Intervalls (α' . . . β') liegen, wie aus der Bestimmung der Zahlen κ₁, κ₂ sofort erhellt. Ganz ebenso mögen ω_κ₃, ω_κ₄ die beiden mit den kleinsten Indices versehenen Zahlen unserer Reihen [see note 1 below] sein, welche in das Innere des Intervalls (α' . . . β') fallen und die kleinere der Zahlen ω_κ₃, ω_κ₄ werde mit α'', die grössere mit β'' bezeichnet.
Man hat alsdann:
α' < α'' < β'' < β' ,
κ₂ < κ₃ < κ₄ ;
und man erkennt, dass alle Zahlen ω_μ unserer Reihe, für welche μ ≤ κ₄ nicht in das Innere des Intervalls (α'' . . . β'') fallen.
Nachdem man unter Befolgung des gleichen Gesetzes zu einem Intervall (α^{(ν - 1)}, . . . β^{(ν - 1)}) gelangt ist, ergiebt sich das folgende Intervall dadurch aus demselben, dass man die beiden ersten (d. h. mit niedrigsten Indices versehenen) Zahlen unserer Reihe (ω) aufstellt (sie seien ω_{κ_{2ν – 1}} und ω_{κ_2ν}), welche in das Innere von (α^{(ν – 1)} . . . β^{(ν – 1)}) fallen; die kleinere dieser beiden Zahlen werde mit α^(ν), die grössere mit β^(ν) bezeichnet.
Das Intervall (α^(ν) . . . β^(ν)) liegt alsdann im Innern aller vorangegangenen Intervalle und hat zu unserer Reihe (ω) die eigenthümliche Beziehung, dass alle Zahlen ω_μ, für welche μ ≤ κ_2ν sicher nicht in seinem Innern liegen. Da offenbar:
κ₁ < κ₂ < κ₃ < . . . , ω_{κ_{2ν – 2}} < ω_{κ_{2ν – 1}} < ω_{κ_2ν} , . . .
und diese Zahlen, als Indices, ganze Zahlen sind, so ist:
κ_2ν ≥ 2ν ,
und daher:
ν < κ_2ν ;
wir können daher, und dies ist für das Folgende ausreichend, gewiss sagen:
Dass, wenn ν eine beliebige ganze Zahl ist, die Grösse ω_ν ausserhalb des Intervalls (α^(ν) . . . β^(ν)) liegt.
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[Page 7]

Since the numbers α', α'', α''', . . ., α^(ν), . . . are continually increasing by value while simultaneously being enclosed in the interval [α, β], they have, by a well-known fundamental theorem of the theory of magnitudes [see note 2 below], a limit that we denote by A, so that:
A Templat:Eq Lim α^(ν) for ν Templat:Eq ∞.
The same applies to the numbers β', β'', β''', . . ., β^(ν), . . ., which are continually decreasing and likewise lying in the interval [α, β]. We call their limit B, so that:
B Templat:Eq Lim β^(ν) for ν Templat:Eq ∞.
Obviously, one has:
α^(ν) < A ≤ B < β^(ν).
But it is easy to see that the case A < B can not occur here since otherwise every number ω_ν of our sequence would lie outside of the interval [A, B] by lying outside the interval [α^(ν), β^(ν)]. So our sequence (ω) would not be everywhere dense in the interval [α, β], contrary to the assumption.
Thus, there only remains the case A Templat:Eq B and now it is demonstrated that the number:
η Templat:Eq A Templat:Eq B
does not occur in our sequence (ω).
If it were a member of our sequence, such as the ν^th, then one would have: η Templat:Eq ω_ν.
But the latter equation is not possible for any value of ν because η is in the interior of the interval [α^(ν), β^(ν)], but ω_ν lies outside of it.

[Seite 7]
Da die Zahlen α', α'', α''', . . ., α^(ν), . . . ihrer Grösse nach fortwährend wachsen, dabei jedoch im Intervalle (α . . . β) eingeschlossen sind, so haben sie, nach einem bekannten Fundamentalsatze der Grössenlehre, eine Grenze, die wir mit A bezeichnen, so dass:
A Templat:Eq Lim α^(ν) für ν Templat:Eq ∞.
Ein Gleiches gilt für die Zahlen β', β'', β''', . . ., β^(ν), . . . welche fortwährend abnehmen und dabei ebenfalls im Intervalle (α . . . β) liegen; wir nennen ihre Grenze B, so dass:
B Templat:Eq Lim β^(ν) für ν Templat:Eq ∞.
Man hat offenbar:
α^(ν) < A ≤ B < β^(ν).
Es ist aber leicht zu sehen, dass der Fall A < B hier nicht vorkommen kann; da sonst jede Zahl ω_ν, unserer Reihe ausserhalb des Intervalles (A . . . B) liegen würde, indem ω_ν, ausserhalb des Intervalls (α^(ν) . . . β^(ν)) gelegen ist; unsere Reihe (ω) wäre im Intervall (α . . . β) nicht überalldicht, gegen die Voraussetzung.
Es bleibt daher nur der Fall A Templat:Eq B übrig und es zeigt sich nun, dass die Zahl:
η Templat:Eq A Templat:Eq B
in unserer Reihe (ω) nicht vorkommt.
Denn, würde sie ein Glied unserer Reihe sein, etwa das ν^te, so hätte man: η Templat:Eq ω_ν.
Die letztere Gleichung ist aber für keinen Werth von v möglich, weil η im Innern des Intervalls [α^(ν), β^(ν)], ω_ν aber ausserhalb desselben liegt.
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Note 1: This is the only occurrence of "unserer Reihen" ("our sequences") in the proof. There is only one sequence involved in Cantor's proof and everywhere else "Reihe" ("sequence") is used, so it is most likely a typographical error and should be "unserer Reihe" ("our sequence"), which is how it has been translated.

Note 2: Grössenlehre, which has been translated as "the theory of magnitudes", is a term used by 19th century German mathematicians that refers to the theory of discrete and continuous magnitudes. (Ferreirós 2007, hlm. 41–42, 202.)

[1] Dauben 1993, hlm. 4.

[2] Gray 1994, hlm. 819–821.

[Cantor1874-4] Cantor 1874

[Gray828-5] Gray 1994, hlm. 828.

[Ewald840_841-6] Cantor 1874, hlm. 259.

[7] Cantor 1874, hlm. 259

[8] Cantor 1878, hlm. 242.

[9] Gray 1994, hlm. 820.

[10] Cantor 1874, hlm. 259–260. English translation: Ewald 1996, hlm. 841.

[12] Cantor 1874, hlm. 260–261. English translation: Ewald 1996, hlm. 841–842.

[Ewald842-13] Cantor 1874, hlm. 261.

[16] Gray 1994, hlm. 822.

[17] Havil 2012, hlm. 208–209.

[18] Havil 2012, hlm. 209.

[19] LeVeque 1956, hlm. 154–155.

[20] LeVeque 1956, hlm. 174.

[21] Weisstein 2003, hlm. 541.

[24] Arkhangel'skii & Fedorchuk 1990, hlm. 16.

[27] Noether & Cavaillès 1937, hlm. 12–13. English translation: Gray 1994, hlm. 827; Ewald 1996, hlm. 844.

[Noether18-28] Noether & Cavaillès 1937, hlm. 18. English translation: Ewald 1996, hlm. 848.

[29] Noether & Cavaillès 1937, hlm. 13. English translation: Gray 1994, hlm. 827.

[Dec7letter-30] ^ ^a ^b ^c ^d ^e ^f ^g Noether & Cavaillès 1937, hlm. 14–15. English translation: Ewald 1996, hlm. 845–846.

[31] Gray 1994, hlm. 827

[32] Dauben 1979, hlm. 51.

[33] Noether & Cavaillès 1937, hlm. 19. English translation: Ewald 1996, hlm. 849.

[34] Ewald 1996, hlm. 843.

[35] Noether & Cavaillès 1937, hlm. 16. English translation: Gray 1994, hlm. 827.

[36] Perron 1921, hlm. 162.

[Kanamori4-38] Kanamori 2012, hlm. 4.

[39] Gray 1994, hlm. 827–828.

[40] Perron 1921, hlm. 162

[42] Fraenkel 1930, hlm. 237. English translation: Gray 1994, hlm. 823.

[43] Kaplansky 1972, hlm. 25.

[45] Gray 1994, hlm. 829–830.

[46] Gray 1994, hlm. 821–824.

[48] Bell 1937, hlm. 568–569; Hardy & Wright 1938, hlm. 159 (6th ed., pp. 205–206); Birkhoff & Mac Lane 1941, hlm. 392, (5th ed., pp. 436–437); Spivak 1967, hlm. 369–370 (4th ed., pp. 448–449).

[50] Dasgupta 2014, hlm. 107; Sheppard 2014, hlm. 131–132.

[51] Jarvis 2014, hlm. 18; Chowdhary 2015, hlm. 19; Stewart 2015, hlm. 285; Stewart & Tall 2015, hlm. 333.

[52] Birkhoff & Mac Lane 1941, hlm. 392, (5th ed., pp. 436–437).

[53] Burton 1995, hlm. 595.

[54] Dauben 1979, hlm. 69.

[55] Gray 1994, hlm. 824.

[Ferreiros184-57] Ferreirós 2007, hlm. 184.

[58] Noether & Cavaillès 1937, hlm. 12–16. English translation: Ewald 1996, hlm. 843–846.

[59] Dauben 1979, hlm. 67.

[Noether16_17-60] Noether & Cavaillès 1937, hlm. 16–17. English translation: Ewald 1996, hlm. 847.

[61] Grattan-Guinness 1971, hlm. 124.

[62] Dauben 1979, hlm. 67, 308–309.

[63] Ferreirós 2007, hlm. 184–185, 245.

[64] Ferreirós 2007, hlm. 185

[65] Ferreirós 2007, hlm. 177.

[66] Dauben 1979, hlm. 67–68.

[67] Ferreirós 2007, hlm. 183.

[68] Ferreirós 2007, hlm. 185.

[69] Ferreirós 2007, hlm. 109–111, 172–174.

[70] Ferreirós 1993, hlm. 349–350.

[71] Noether & Cavaillès 1937, hlm. 12–13. English translation: Ewald 1996, hlm. 844–845.

[72] Noether & Cavaillès 1937, hlm. 13. English translation: Ewald 1996, hlm. 845.

[73] Ferreirós 2007, hlm. 179.

[74] Noether & Cavaillès 1937, hlm. 14–16, 19. English translation: Ewald 1996, hlm. 845–847, 849.

[75] Ferreirós 1993, hlm. 358–359.

[76] Ferreirós 1993, hlm. 350.

[78] Cantor 1878, hlm. 245–254.

[79] Cantor 1879, hlm. 4.

[81] Ferreirós 2007, hlm. 267–273.

[82] Ferreirós 2007, hlm. xvi, 320–321, 324.

[83] Cantor 1878, hlm. 243.

[84] Hawkins 1970, hlm. 103–106, 127.

[85] Hawkins 1970, hlm. 118, 120–124, 127.

[86] Ferreirós 2007, hlm. 362–363.

[87] Cohen 1963, hlm. 1143–1144.

[1]

[2]

[A]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[B]

[10]

[C]

[11]

[D]

[12]

[13]

[14]

[15]

[16]

[17]

[E]

[F]

[18]

[G]

[proof 1]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[H]

[29]

[30]

[31]

[I]

[32]

[33]

[J]

[34]

[35]

[K]

[36]

[L]

[37]

[38]

[39]

[M]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

[N]

[63]

[O]

[65]

[66]

[67]

[68]

[69]

[70]

[71]

[28]

[40]

[41]

[42]

[64]