Articles of large cardinals

Proving that the existence of strongly inaccessible cardinals is independent from ZFC?

ZFC can’t prove that strongly inaccessible cardinals exist, or else it would prove that a model of itself exists and hence $Con(ZFC)$. So this leaves us with two options: ZFC proves there are no strongly inaccessible cardinals The existence of strongly inaccessible cardinals is independent from ZFC I’ve heard, however, that you can’t actually prove […]

Universe cardinals and models for ZFC

I’m reading through Joel David Hamkins’ set theory lecture notes. On page 14, on the subject of inaccessible cardinals and submodels of ZFC in $V$, he defines a universe cardinal to be a cardinal $\gamma$ such that $V_\gamma$ models ZFC. For example, if an inaccessible cardinal $\kappa$ exists, then $\kappa$ is a universe cardinal. But […]

On the number of countable models of complete theories of models of ZFC

This question already has an answer here: Number of Non-isomorphic models of Set Theory 2 answers

Measurable cardinal existence condition

We know that a cardinal $k > \omega$ is measurable if there is a measure function $\mu :2^k\mapsto \{0, 1\}$ that satisfies the following 3 conditions: 1.$&ltk$ -additive: for every set I of indices with card(I) < k, and for every family of pairwise disjoint sets $z_i$, where $i\in I$, we have $\displaystyle \mu(\bigcup_{i\in I} […]

Consistency strength of weakly inaccessibles without $\mathsf{GCH}$

Actually, the thing is to bypass $V=L$. So, is there a way to prove that (if consistent) $\mathsf{ZFC}$ can’t prove that there exists a weakly inaccessible without first showing that $\mathsf{GCH}$ is relatively consistent? Obviously, if we can show that $\mathsf{ZFC}$ doesn’t prove the consistency of a weakly inaccessible is much better. The only really […]

Why isn't second-order ZFC categorical?

This builds off of (and reuses some of the examples from) this question about failures of categoricity not related to size. Second-order $\mathsf{ZFC}$ isn’t categorical, but it’s almost-categorical, meaning $\mathcal{M} = \langle D, \in \rangle$ is a model of second-order $\mathsf{ZFC}$ iff there is an (strongly) inaccessible cardinal $\kappa$ such that $\mathcal{M}$ is isomorphic to […]

$V_k$ being a model of ZFC whenever $k$ is strongly inaccessible

ZFC implies that the $V_k$ is a model of ZFC whenever $k$ is strongly inaccessible.. So if $k$ is weakly inaccessible, it can’t be a model of ZFC? Why is it like this? And ZF implies that the Godel universe $L_k$ is a model of ZFC whenever $k$ is weakly inaccessible. Again, the same question.

Forcing Classes Into Sets

I am still studying the topics in forcing and did not yet study much about forcing with a class of conditions. I know from Jech’s Set Theory that you can force that the class of ordinals in the world will be countable in the generic extension, which means that you can take a proper class […]

What are large cardinals for?

I’ve heard large cardinals talked about, and I (think I) understand a little about how you define them, but I don’t understand why you would bother. What’s the simplest proof or whatever that requires the use of large cardinals? Is there some branch of mathematics that makes particularly heavy use of them?

A question regarding the status of CH in the Gitik model

Consider models of ZF+”Every uncountable cardinal is singular” (eg. Moti Gitik: “All uncountable cardinals can be singular”, Israel journal of Mathematics, 35(1-2): 61-88, 1980). How should CH be formulated in such models? Can CH be false in such models and if so, how many intermediate cardinalities between $\omega$ and $2^{\omega}$ can there possibly be since […]