Proper class of inaccessibles and 1-inaccessible cardinals
Dec 30, 2022 · A 1 -inaccessible is "at least as expensive" as a proper class of inaccessibles in the following sense: we can prove (in a very weak base theory) that if ZFC + "There is a 1 …
Hyper-inaccessible cardinals - Mathematics Stack Exchange
It's in fact trivial from the fact that inaccessible cardinals are regular. κ κ is 1 1 -inaccessible if and only if. κ κ is a limit of inaccessible cardinals if and only if. κ κ has κ κ inaccessible below it if …
elementary set theory - About the definition of inaccessible …
Apr 7, 2020 · It is consistent that every weakly inaccessible cardinal is inaccessible: If we assume GCH, then every limit is a strong limit, since λ <κ λ <κ implies 2λ =λ+ <κ. 2 λ = λ + <κ. …
Inaccessible cardinal in standard transitive models of ZFC
May 16, 2024 · Inaccessible cardinal in standard transitive models of ZFC. For a standard model of ZFC N, ∈ N, ∈ (where N N is a transitive set), if κ ∈ N κ ∈ N is an inaccessible cardinal in the …
Inaccessible cardinals and Andrew Wiles's proof - MathOverflow
A notable exception is the most celebrated result of recent years, the proof of Fermat's last theorem by the British mathematician Andrew Wiles in 1994 [...] To complete his proof, Wiles …
logic - Inaccessible cardinals and consistency of ZFC
May 18, 2024 · For κ κ an inaccessible cardinal, Vκ ⊨ ZFC V κ ⊨ Z F C. This makes sense as a statement of set theory and is provable in ZFC. Your first statement is almost the statement …
Is it possible that |$\\mathbb R$| is an inaccessible cardinal?
Sep 4, 2017 · Since this forcing is c.c.c., it preserves cardinals and cofinalities, so $\kappa$ remains a weakly inaccessible cardinal; but now $2^{\aleph_0}=\kappa$. (Actually, all that's …
set theory - Worldly vs Inaccessible Cardinals, why different ...
Dec 13, 2017 · A worldly cardinal κ κ is defined by Vκ ⊨ ZFC V κ ⊨ Z F C . An inaccessible cardinal ι ι is defined in such a way that Vι V ι is a Grothendieck universe and so provides a …
set theory - Properties of Inaccessible Cardinals - Mathematics …
Jan 9, 2017 · κ κ is strongly inaccessible if it is a regular strong limit cardinal, that is, if κ κ is weakly inaccessible and 2α <κ 2 α <κ for all α <κ α <κ. Assuming the Generalized Continuum …
ZFC + "There exists an inaccessible cardinal" proves Con (ZFC)
The key step here is the construction of a truth predicate over Vκ (since the latter is only a set-sized structure); this enables us to directly talk about the truth values of arbitrary sentences in …