Pengguna:Dedhert.Jr/Daftar sifat kardinal besar

Halaman berikut ini mencakup sebuah daftar kardinal dengan sifat kardinal besar. Ini disusun dengan kasar dalam rangka konsistensi kuat dari aksioma menyatakan keberadaan kardinal dengan sifat yang diberikan. Keberadaan bilangan kardinal $\kappa$ dari sebuah tipe yang diberikan menyiratkan keberadaan kardinal hampir tipe yang didaftarkan di atas, dan untuk deskripsi kardinal paling terdaftar $\varphi$ mengenai kekuatan konsistensi lebih rendah, $V_{\kappa }$ memenuhi "adanya sebuah kelas takterbatas mengenai kardinal $\varphi$".

Tabel berikut mengurutkan bilangan kardinal berdasarkan tingkat konsistensinya. Jika konsistensinya sama kuat, kriteria selanjutnya yang dipakai adalah ukuran/besaran kardinal. Dalam beberapa kasus (seperti kardinal kompak kuat) kekuatan kekonsistenan yang tepat tidak diketahui dan tabelnya menggunakan penebak terbaik saat ini.

• "Small" cardinals: 0, 1, 2, ..., $\aleph _{0},\aleph _{1}$,..., $\kappa =\aleph _{\kappa }$, ... (see Aleph number)
• worldly cardinals
• weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals
• weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals.
• reflecting cardinals
• weakly compact (= Π11-indescribable), Πmn-indescribable, totally indescribable cardinals
• λ-unfoldable, unfoldable cardinals, ν-indescribable cardinals and λ-shrewd, shrewd cardinals (not clear how these relate to each other).
• ethereal cardinals, subtle cardinals
• almost ineffable, ineffable, n-ineffable, totally ineffable cardinals
• remarkable cardinals
• α-Erdős cardinals (for countable α), 0^# (not a cardinal), γ-iterable, γ-Erdős cardinals (for uncountable γ)
• almost Ramsey, Jónsson, Rowbottom, Ramsey, ineffably Ramsey, completely Ramsey, strongly Ramsey, super Ramsey cardinals
• measurable cardinals, 0^†
• λ-strong, strong cardinals, tall cardinals
• Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin cardinals
• superstrong cardinals (=1-superstrong; for n-superstrong for n≥2 see further down.)
• subcompact, strongly compact (Woodin< strongly compact≤supercompact), supercompact, hypercompact cardinals
• η-extendible, extendible cardinals
• Vopěnka cardinals, Shelah for supercompactness, high jump cardinals
• n-superstrong (n≥2), n-almost huge, n-super almost huge, n-huge, n-superhuge cardinals (1-huge=huge, etc.)
• Wholeness axiom, rank-into-rank (Axioms I3, I2, I1, and I0)

The following even stronger large cardinal properties are not consistent with the axiom of choice, but their existence has not yet been refuted in ZF alone (that is, without use of the axiom of choice).

• Reinhardt cardinal, Berkeley cardinal

References

• Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2. 
• Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (edisi ke-2nd). Springer. ISBN 3-540-00384-3. 
• Kanamori, Akihiro; Magidor, M. (1978). "The evolution of large cardinal axioms in set theory". Higher Set Theory. Lecture Notes in Mathematics. 669 (typescript). Springer Berlin / Heidelberg. hlm. 99–275. doi:10.1007/BFb0103104. ISBN 978-3-540-08926-1. 
• Solovay, Robert M.; Reinhardt, William N.; Kanamori, Akihiro (1978). "Strong axioms of infinity and elementary embeddings" (PDF). Annals of Mathematical Logic. 13 (1): 73–116. doi:10.1016/0003-4843(78)90031-1 . 

External links

• Cantor's attic
• some diagrams of large cardinal properties