Large cardinals are such cardinals whose existence cannot be proved within ZFC, and requires stronger axioms to be added to ZFC, they are often used to measure the consistency strength of a certain statement in the language of set theory.

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Can any large cardinal axiom be reached by a recursive sequence of unbounded sequences and fixed points?

The lowest large cardinals can be seen as a simple recursive rule that defines an unbounded sequence, then defines a fixed point of the sequence, then an unbounded sequence of fixed points, a fixed ...
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Existence of a probability measure under which a subset of $[0,1]$ cannot be approximated by a Borel set

Is there some probability measure, $p$, on $\left(\left[0,1\right],2^{\left[0,1\right]}\right)$ relative to which some $D\subseteq\left[0,1\right]$ cannot be approximated by a Borel set, i.e. if $E$ ...
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388 views

On the large cardinals foundations of categories

(This was cross-posted to MathOverflow.) It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when ...
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Are large cardinals bi-interperable with type theory?

"Sets in Types, Types in Sets" establishes that the calculus of constructions is bi-interpretable with ZFC + infinitely many inaccessibles. Are there any more results like this higher up the large ...
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134 views

Is the expressive power of infinitary logic language $L(\infty,\infty)$ larger, the same or smaller than that of ZFC+large cardinal axioms?

In a previous question I learned that the power of statements of the form $\Pi_m^n$ or $\Sigma_m^n$ for arbitrary positive $m$ and $n$, is smaller than that of ZFC. For instance, the GCH cannot be ...
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What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom?

Consider the first-order theory obtained by adjoining to ZFC (or, better, Mac Lane set theory) the Grothendieck–Verdier (or Bourbaki?) universe axiom: For each set $x$ there exists a Grothendieck ...
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80 views

A reference that forcing with a poset of cardinality less than $\kappa$ preserves supercompact cardinals

I'm looking for a reference for the fact that if $\kappa$ is supercompact and $Q$ is a poset of cardinality less than $\kappa$, then $V^Q\models \check{\kappa} \ \text{is supercompact}$. Apparently ...
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Is the class of all ordinals independent of set theory?

I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. ...
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193 views

Weakly Compact Cardinals and the Extension Property

From $\textit{The Higher Infinite}$ by Kanamori, in his proof (page 39 - 41) that $\kappa$ is weakly compact if and only if $\kappa$ satisfies the Extension Property, the proof does something that I ...
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Are all large cardinal axioms expressible in terms of elementary embeddings?

An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ ...
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184 views

Did large cardinals exist before 1963?

I'm curious to know the history of the interaction between large cardinals and traveling to (creating) universes through forcing. The question arose because I understand that Peano Arithmatists ...
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Strongly inaccessible cardinals and certain kinds of universes

Write $\mathrm{tc}(*)$ for transitive closure. For all cardinals $\kappa$, let $H(\kappa)$ denote the collection of all sets $X$ such that $|X|<\kappa$, and For all $x \in \mathrm{tc}(X)$, it ...
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Does asserting the existence of large cardinals allow one to prove the existence of new subsets of $\mathbb{R}$?

ZFC proves the existence of certain subsets of $\mathbb{R}$. Suppose ZFC is consistent, and we adjoin a large cardinal axiom to ZFC, obtaining ZFC'. Assume ZFC' is also consistent. It it possible to ...
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142 views

How is Mahlo cardinal used?

I would like to know how Mahlo cardinals are used - as such examples may help me understand why they were created and so on.
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115 views

$V_k$ 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 ...
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78 views

Does TG prove that ZFC2 has a model?

Does TG (Tarski-Grothendieck Set Theory) prove that ZFC2 (second-order ZFC) has a model? Does it at least prove the consistency of ZFC2?
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141 views

Effective Well ordering of reals

Is there an effective (constructive) well order on reals ? I know several questions were already asked on this topic, and the answers were very good to this well known problem. My question is more ...
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Ultrapowers by extenders of potential premice

I have a problem with an argument in Fine structure and iteration trees by Mitchell and Steel. Let $E$ be a $(\kappa, \lambda)$-extender. Let $\dot E^{\mathcal{M}}$ the a unary predicate with is ...
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Which are “big theorems” of descriptive set theory?

Question: If one were to fully understand 10 theorems in DST, or 15,20,25,30 theorems, which ones would be the most important to understand in order to work towards an understanding of descriptive set ...
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Relation between inacessible cardinals and CH

I know that the two statements “There exists an inaccessible cardinal” and “Continuum Hypothesis” are both independent of ZFC. Now, are those two statements independent of each other? That older ...
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107 views

Intuition behind extenders

What's the best way to think about extenders? For instance if we have a normal $\kappa$-complete ultrafilter on $\kappa$, call it $D$, $M$ the ultrapower given by $D$, and if we look at $j: V \to M$, ...
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Kunen's result and $L(\mathbb{R})$

Recall Kunen's result: If $j:V \to M$ is a nontrivial elementary embedding then $V \ne M$. Assume $AD$ is true in $L(\mathbb{R})$. Let $j : L(\mathbb{R}) \to M$ be a nontrivial elementary embedding. ...
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$\kappa$-complete ultrafilter and bounded subsets of $\kappa$

If $U$ is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$ then every bounded subset of $\kappa$ has measure $0$. This is because if we had $X \in U$ with $X$ having size less than $\kappa$, ...
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How many weak/strong limit cardinals exist under different assumptions?

I am trying to hastily answer the question in the title because I need it for another problem. I have been consulting multiple sources and am confused about the standard meanings of the following ...
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Shelah Cardinals and Modern Set Theory

Do Shelah cardinals play an essential role in any modern set theory results or was the concept basically made obsolete by Woodin cardinals?
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Which infinite cardinals can be defined using partition relations?

Several types of infinite cardinals are easily defined in terms of partition relations. For instance, if $\kappa > \omega$ then $\kappa$ is weakly compact if $\kappa$ satisfies ...
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An inequality in a proof of Kunen's Inconsistency

This may be a silly question, but I keep coming back to it. Let $j:V\prec M$ be a non-trivial elementary embedding with $M$ a transitive class and $\kappa$ the critical point of $j$. Define the ...
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What's the latest on Laver tables?

A couple of years ago, I was astonished and delighted to learn about Laver tables, a sequence (indexed on $n$) of Cayley-like tables for a binary operation $\star$ on numbers $i,j\leq 2^n$ that ...
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example of weakly inaccessible cardinal that is not a strongly inaccessible cardinal

As I study through inaccessible cardinals, I find many examples that show some cardinal being both weakly inaccessible and strongly inaccessible. So, can anyone show me the example of weakly ...
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167 views

consistency of large cardinal axiom

It is known that if ZFC is consistent, ZFC+"no such large cardinal exists" is consistent. Then, is ZFC+"such large cardinal exists" known to be consistent? This would imply that proving large cardinal ...
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143 views

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.$<k$ -additive: for every set I of ...
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IF $\kappa$ is weakly compact, then $\kappa$ has the tree property.

Let $\kappa$ be weakly compact, and let $(T, <_T)$ be a tree of height $\kappa$ such that each level of $T$ has size $< \kappa$. Assume $T = \kappa$. We extend the partial ordering $<_T$ of ...
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Why is Kunen inconsistency at the top of Cantor's upper attic?

Motivation: I have reproduced part of page 396 and 397 from Handbook of Mathematical Logic below: So if we start with a concept of number and play the game of naming the largest one, does Kunen ...
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Inaccessible cardinals and well-founded models of ZFC

This was a problem in an exam I took last semester, but I never got the chance to ask my professor how to solve it. Here goes: Let $\kappa$ be an inaccessible cardinal. Then $V_\kappa$ is a ...
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What is the largest cardinal which can inject into $\mathbb{R}$ in ZF?

This question takes place in ZF. Assume some mild large cardinals; then it is consistent (in fact, it follows from AD, the consistency of which follows from mild large cardinals) that there are very ...
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“Nice” well-orderings of the reals

I have a question which I believe could be easily resolved if I happened to look at the right source - hence my asking it here as opposed to at MathOverflow. I've tried googling it, but I haven't been ...
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288 views

Can forcing push the continuum above a weakly inacessible cardinal?

There is a famous quote of Paul Cohen which reads $\lt\lt$ A point of view which the author feels may eventually come to be accepted is that the continuum hypothesis is obviously false ... The ...
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Is there an axiom scheme exhausting all types of Mahlo cardinals?

Is there an axiom scheme exhausting all types of Mahlo cardinals? Mahlo cardinals may be considered as the first stage in the following construction : let $C_{0,0}$ be the class of all inacessible ...
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Generalizing $0^\#$

Background and motivation: The following theorem is due to Silver: If there exists a Ramsey cardinal then: For every $\aleph_0 < \kappa < \lambda$, $L_{\kappa}$ is an elementary ...
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Good Reference for Large Cardinals/Homotopy Theory

I'm planning on attending a conference in Barcelona in September called "Large Cardinal Methods in Homotopy Theory" and want to try to be as prepared as I possibly can. Are there good references for ...
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316 views

Consistency of $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$

It is known that if $\operatorname{cf}(\omega_1)=\operatorname{cf}(\omega_2)=\omega$ then $0^\#$ exists. I am familiar with the Feferman-Levy model in which $\omega_1$ is singular, which has the same ...
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The real cofinality of singular cardinals in $L$ under $0^\#$

Suppose that $0^\#$ exists, is there a relatively simple way to show that for any ordinal $\lambda$, if $\lambda$ is a singular cardinal in $L$ then its real cofinality is $\omega$?
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$A^\#$ and inner models

For a set of ordinals $A$ we say that $A^\#$ exists if there exists a closed and unbounded class of indiscernibles, $I\subseteq\operatorname{Ord}$, for $L[A]$. Formally, if such class exists we define ...
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Equivalent ways to describe the Mitchell order

For a measurable cardinal $\kappa$, we define an ordering over $\kappa$-complete ultrafilters as follows: Suppose $W,U$ are both $\kappa$-complete free ultrafilters over $\kappa$, we say that $U\lhd ...
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Silver indiscernibles and definable injections

This is a follow up to my previous question on Silver indiscernibles. Background: Suppose that $0^\#$ exists, $\alpha<\lambda$ are limit ordinals, $i_\alpha$ is the $\alpha$th Silver ...
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Silver indiscernibles and constructibility

We know that if $0^\#$ exists then it's not in $L$. For an infinite ordinal $\alpha$, denote by $I_\alpha$ the initial segment of length $\alpha$ of Silver indiscernibles. Question: For which ...
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When does $V=L$ becomes inconsistent?

In a wonderful course I'm taking with Magidor we are finishing the proof of the Covering Theorem for $L$. The theorem, in a nutshell, says that $V$ is very close to being $L$ if and only if $0^\#$ ...
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Basic constructibility question

I'm currently reading J. D. Hamkins' paper "Unfoldable cardinals and the GCH," and I've run across a comment that I think I ought to find trivial, but I don't. On page 1187, he says that ...
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Strong cardinals and reflection

I'm new to all this large cardinal thing and I have trouble in proving the following: If $\kappa$ is a $\gamma$-strong cardinal, for some large enough $\gamma$, then $\kappa$ is ...
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How do I explicitly see that the Ultrapower map is the identity below its critical point?

I apologize in advance for how basic this question is... Let $j:V\rightarrow V/U$ be the ultrapower map where U is an ultrafilter on a set S, and $j(x)=[c_x]$. Now, let $f\in j(0)$. Then $f$ is ...