The large-cardinals tag has no wiki summary.

**28**

votes

**7**answers

5k views

### Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...

**25**

votes

**3**answers

936 views

### Latest stand of core model theory?

What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004:
...

**22**

votes

**1**answer

609 views

### Does an existence of large cardinals have implications in number theory or combinatorics?

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...

**21**

votes

**4**answers

835 views

### What is the definition of a large cardinal axiom?

In different books one can find different implicit definitions for a large cardinal axiom.
My question is that which one of these definitions are more popular or standard amongst set theorists?
Any ...

**19**

votes

**2**answers

772 views

### Can there be an embedding j:V to L, from the set-theoretic universe V to the constructible universe L, when V not= L?

Main Question. Can there be an embedding $j:V\to L$ of the
set-theoretic universe $V$ to the constructible universe $L$, if
$V\neq L$?
By embedding here, I mean merely a proper class isomorphism from
...

**19**

votes

**0**answers

660 views

### Supercompact and Reinhardt cardinals without choice

A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it:
Definition. A cardinal $\kappa$ is supercompact if for all ordinals ...

**17**

votes

**2**answers

2k views

### Recent claim that inaccessibles are inconsistent with ZF

Here it is mentioned that someone claims to have proven that there are no weakly inaccessibles in ZF.
Question 1: What reasons are there to believe that weakly inaccessibles exist?
Question(s) 2: ...

**17**

votes

**0**answers

466 views

### categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result?
Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...

**16**

votes

**6**answers

2k views

### Reasons to believe Vopenka's principle/huge cardinals are consistent

There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...

**16**

votes

**2**answers

687 views

### Woodin's unpublished proof of the global failure of GCH

An unpublished result of Woodin says the following:
Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$
In the paper "The ...

**16**

votes

**1**answer

681 views

### Logical endofunctors of Set?

What set-theoretic assumptions are necessary and sufficient to ensure the existence of a nontrivial (i.e. not isomorphic to the identity) endofunctor of the category Set which is logical (i.e. ...

**15**

votes

**3**answers

1k views

### Any paradoxical theorems arising from large cardinal axioms?

If we accept the axiom of choice, we take the responsibility of living in a world in which, e.g., a ball in Euclidean 3-space is equiscindable to two isometric copies of itself (Banach-Tarski). So we ...

**15**

votes

**2**answers

2k views

### Large cardinal axioms and Grothendieck universes

A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of ...

**15**

votes

**1**answer

819 views

### Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...

**14**

votes

**2**answers

2k views

### Should there be a true model of set theory?

As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...

**14**

votes

**4**answers

1k views

### A New Continuum Hypothesis (Revised Version)

Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$.
What happens after exponentiation?
We have the following equation: $2^{N_n}=N_{2^{n}}$.
(Which says: For all finite cardinal $n$ ...

**14**

votes

**2**answers

953 views

### Why does inner model theory needs so much descriptive set theory (and vice versa)?

I am curious about how much descriptive set theory is involved in inner model theory.
For instance Shoenfield's absoluteness result is based on the construction of the Shoenfield tree which ...

**14**

votes

**2**answers

410 views

### Singularizing forcing of “small” cardinality?

Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size ...

**14**

votes

**2**answers

665 views

### Questions about $\aleph_1-$closed forcing notions

"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:
1) $GCH$ fails ...

**14**

votes

**3**answers

1k views

### What are the known implications of “There exists a Reinhardt cardinal” in the theory “ZF + j”?

This is, alas, in large part a series of questions on unpublished work of Hugh Woodin; it's also quite frivolous if Reinhardt cardinals turn out inconsistent.
Definitions:
Call $\kappa$ an ...

**13**

votes

**2**answers

606 views

### Can measures be added by forcing?

The LÃ©vy-Solovay theorem says that small forcings do not create measures. J.D. Hamkins has generalized this to a larger class of forcings called gap forcings. I would assume this cannot be ...

**13**

votes

**1**answer

788 views

### Devlin's “Constructibility” as a resource

It is fairly well-known among set-theorists that Keith Devlin's 1984 book "Constructibility" has flaws in its initial development of fine structure theory. (See Lee Stanley's review 1 of the text for ...

**13**

votes

**0**answers

359 views

### How to prove projective determinacy (PD) from I0?

Martin and Steel (in 1987?) showed that if there are infinite many Woodin cardinals then every projective set of reals is determined (PD).
However, it is mentioned in many texts that in 1983/1984 ...

**12**

votes

**3**answers

331 views

### Is Prikry forcing minimal?

Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb ...

**12**

votes

**1**answer

810 views

### Large cardinal axiom: everything that happen once must happen an unbounded number of times

I remember reading something about a large cardinal axiom saying something like
If some cardinal $\kappa$ has some property $P$, then there should be a proper class of cardinals with the property ...

**12**

votes

**0**answers

225 views

### capturing small sets in small factors

Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known:
...

**11**

votes

**4**answers

671 views

### Is there a least-fixed-point formulation of inaccessible cardinals?

The infinity axiom can be formulated by defining a function $S$ as
$$S(N) = \{0\} \cup \{n+1\\ |\\ n \in N\}$$
(FWIW, I'm assuming the von Neumann ordinals.) The axiom is then
$$\exists I . I = ...

**11**

votes

**4**answers

516 views

### Proving ZFC results using large cardinals

There are many $ZFC$ results that their proof uses forcing: The idea is that we force the statement to be true, and then using absoluteness (or other reasons) we conclude that the result is true in ...

**11**

votes

**2**answers

788 views

### What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?

Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that's unclear, in this ...

**11**

votes

**1**answer

167 views

### extending elementary embeddings from initial segments of V to all of V

Suppose $j:V_\lambda \rightarrow V_\eta$ is (elementary and) cofinal. Can $j$ be extended to all of $V$?
(Subsidiary question: What conditions are there on an ultrafilter/extender/whatever so that ...

**11**

votes

**1**answer

511 views

### Can Vopenka's principle be violated definably?

One form of Vopenka's principle (a large cardinal axiom) states that no locally presentable category contains a full subcategory which is large (= a proper class) and discrete (= contains no ...

**11**

votes

**1**answer

408 views

### Generic Extensions and $L(V_{\lambda+1})$

Suppose $\lambda$ is a strong limit cardinal of cofinality $\omega$ and for $A$ a transitive set, define $L(A)$ in the usual fashion by setting
$$L_0(A)=A;$$
$$L_{\alpha+1}(A) = L_\alpha (A)\cup ...

**11**

votes

**0**answers

368 views

### Some questions about $0^{\sharp}$ and forcing over $L$

1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is ...

**10**

votes

**2**answers

514 views

### Kunen's inconsistency result

A well-known result of Kunen says that there is no non-trivial elementary embedding $j: V \rightarrow V.$ There are several proofs of this theorem (see Kanamori, The higher infinite). I wonder to know ...

**10**

votes

**1**answer

362 views

### How strong is the iterated consistency of ZFC?

Let $T_0$ be $\mathsf{ZFC}$ and, for $n\in\omega$, set $T_{n +1}=T_{n}+\mathrm{Con}(T_{n})$.
Question 1: Is there a natural number $n$ such that $T_{n}$ is equiconsistent with $\mathsf{ZFC}+$ ...

**10**

votes

**1**answer

217 views

### Consistency strength of projective determinacy (PD)

Let PD stand for projective determinacy, and consider the two claims:
(1) For each n=1,2,..., Con(ZFC+PD) implies Con(ZFC + there are n Woodin cardinals)
(2) Con(ZFC+PD) implies Con(ZFC + there are ...

**10**

votes

**2**answers

174 views

### What is the strength of chains of 1-extendibles?

Let $X$ be a collection of cardinals such that if $\kappa,\lambda\in X$ and $\kappa<\lambda$, then there is a non-trivial elementary embedding $j:V_{\kappa+1} \to V_{\lambda+1}$ with $crit(j) = ...

**10**

votes

**1**answer

290 views

### Is the inclusion version of Kunen inconsistency theorem true?

The relations $\in$ and $\subsetneq$ seem so similar in some sense. For example they are equal on ordinal numbers. So there is a natural question about their possible similar behaviors on the ...

**10**

votes

**3**answers

910 views

### What sort of large cardinal can $\aleph_1$ be without the axiom of choice?

Assuming the axiom of choice it is very easy to see that $\aleph_1$ is a regular Joe of a successor cardinal. It is not very large in any way except the fact that it is the first uncountable cardinal.
...

**10**

votes

**1**answer

315 views

### V=HOD & The Height of the Large Cardinal Tree

As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are ...

**10**

votes

**2**answers

500 views

### What are Moschovakis cardinals?

The question is exactly that of the title: what are Moschovakis cardinals?
Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency ...

**10**

votes

**2**answers

249 views

### Singular successors without large cardinals

Assuming the axiom of choice we have that successor cardinals are regular. However as one of the first examples of uses of forcing show, it is consistent relative to $\sf ZF$ that $\omega_1$ is ...

**10**

votes

**1**answer

379 views

### Forcing mildly over a worldly cardinal.

A cardinal $\theta$ is worldly if $V_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$, but is there a ...

**10**

votes

**1**answer

280 views

### Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals?

Background
I am interested in elementary embeddings from a model of set theory into itself. One way of producing such elementary embeddings is when the model is generated by indiscernibles; this idea ...

**10**

votes

**1**answer

345 views

### Consistency strengths related to the perfect set property

I want a model of $\mathrm{MA}_{\sigma\mathrm{-centered}}+\neg\mathrm{CH}$ in which every set of reals in $L(\mathbb{R})$ has the perfect set property. In terms of consistency strength, it is known ...

**10**

votes

**1**answer

449 views

### Why does the generalised Galvin-Prikry Theorem only hold at Ramsey cardinals?

The Galvin-Prikry theorem says that Borel sets are Ramsey. This means that for every Borel set $S\subseteq[\omega]^\omega$, there is an $A\in\[\omega]^\omega$ such that either $[A]^\omega \subseteq S$ ...

**10**

votes

**0**answers

195 views

### Absoluteness of “$\kappa$-homogeneously Suslin” for sets of reals

What is known about the absoluteness, or lack thereof, of the notion of "$\kappa$-homogeneously Suslin" for sets of reals?
For example, if $A$ is $\kappa$-homogeneously Suslin and $\lambda > ...

**9**

votes

**2**answers

576 views

### Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?

I was recently told that the following (due to M. Viale) is a nice theorem:
Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper ...

**9**

votes

**2**answers

1k views

### Large cardinals

Looking at the chart of cardinals in Kanamori's book, one realizes that all large cardinals are implied by stronger ones and imply weaker ones. For instance measurable implies Jonsson which implies ...

**9**

votes

**3**answers

733 views

### How elementary can we go?

It is a theorem of A. Levy, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1} V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ sentences.
One ...