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11
votes
1answer
303 views

Singular in $V$ regular in $HOD$

Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have: (1) $\kappa$ is singular in $V$ of cofinality $\omega,$ (2) $\kappa$ is regular (and in fact ...
9
votes
1answer
283 views

Two questions about higher Souslin trees

Assume $V=L$ and let $\kappa$ be a Mahlo cardinal. Let $L[G]$ be the generic extension obatined by Mitchell forcing to make $2^{\aleph_0}=\aleph_2=\kappa.$ It is known that in the extension there ...
17
votes
0answers
699 views
+100

Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$. In this question I would like ...
4
votes
1answer
257 views

Is tree property inconsistent with Berkeley cardinals in the absence of Axiom of Choice?

On one hand due to Kunen's inconsistency theorem it is known that within $\sf ZF$, large cardinal axioms beyond Reinhardt cardinal are inconsistent with $\sf AC$. Also some recent results of Bagaria, ...
12
votes
0answers
235 views

The axiom $I_0$ in the absence of $AC$

It is well-known that if $AC$ holds and if $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ is a non-trivial elementary embedding with $crit(j) < \lambda,$ then $\lambda$ has countable cofinality (and in ...
9
votes
0answers
223 views

Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?

The question is: if I assert in ZF that there exists a Reinhardt cardinal, do I really get a theory of higher consistency strength than when I assert in ZFC that there exists an I0 cardinal (the ...
15
votes
1answer
332 views

Namba forcing and semiproperness

This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident. Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...
8
votes
1answer
281 views

How long does the slow inefficient algorithm for computing the product in classical Laver tables take?

Let $(A_{n},*)$ denote the $n$-th classical Laver table. Let $X_{n}$ be the set of all finite sequences of elements from $A_{n}$. Define a function $E_{n}:X_{n}\rightarrow X_{n}$ by letting ...
4
votes
2answers
302 views

Embedding property of weakly compact cardinals

One of the characterizations of $\kappa$ being a weakly compact cardinal is being inaccessible, and for every $\kappa$-model $M$, there is a [$\kappa$-model] $N$ and an elementary embedding $j\colon ...
4
votes
1answer
245 views

Large cardinals without choice?

For any given extension $T$ of ZFC (or perhaps NBGC or something), we can ask whether there is an extension $T'$ of ZF which does not prove AC such that $Con(T) \leftrightarrow Con(T')$ $Con(T) \to ...
6
votes
1answer
340 views

Does the consistency strength hierarchy coincide with the “arithmetic consequence” hierarchy at ZF + Reinhardt?

In these slides (see especially slide 26), Steel emphasizes the phenomenon that for all known "natural" extensions of ZFC, the ordering by consistency strength agrees with the ordering by containment ...
7
votes
1answer
191 views

$\mathsf{AD}_\mathbb{R}$ and Elementary Embeddings

Suppose $\mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R})) + \mathsf{DC}$ holds. (We can use more if it is helpful.) I believe under $\mathsf{AD}_\mathbb{R}$, every $A \subseteq \mathbb{R}$ is ...
2
votes
0answers
115 views

When do wide initial segments ruin admissibility?

Suppose $\alpha$ is admissible and $\beta<\alpha$. We know that $L_\alpha$ is an admissible set (by definition), but adding subsets of $\beta$ to $L_\alpha$ might break admissibility: while set ...
12
votes
0answers
239 views

If $U,D$ are $\kappa$-complete nonprincipal ultrafilters on $\kappa$ and $j_U(U) = j_D(D)$, is $U=D$?

Here, $\, j_U, \, j_D$ are the canonical elementary embeddings induced by $U,D$ respectively. I note that it is consistent with the existence of a measurable that the answer be yes: it is true in the ...
13
votes
1answer
232 views

Consistency strength of $\aleph_2$-Souslin hypothesis

Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis? Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and ...
11
votes
1answer
310 views

Changing cofinalities above supercompact cardinals

Question. Suppose $\kappa$ is a supercompact cardinal and $\lambda > \kappa$ is measurable (or even larger large cardinal if necessary). Is there a set generic extension of the universe in ...
5
votes
1answer
190 views

Do strong embeddings always provide all the ultrafilters that exist?

Let $\kappa$ be a strong cardinal. Then for each $\lambda\geq\kappa$ does there exist a $\mu>\lambda$ such that if $U$ is a $\kappa$-complete ultrafilter on $\lambda$ and $j:V\rightarrow ...
6
votes
1answer
269 views

Are there first order theories of interest to an algebraist or at least a model theorist of large cardinal consistency strength?

I am wondering if there are some first order theories of algebraic structures or structures of interest to model theorists of large cardinal consistency strength or at least unexpectedly high ...
14
votes
1answer
458 views

What is the large cardinal strength of the assertion that every $\kappa$-complete filter on $\kappa$ extends to a $\kappa$-complete ultrafilter?

It is well-known that an uncountable regular cardinal $\kappa$ is strongly compact if and only if every $\kappa$-complete filter on any set extends to a $\kappa$-complete ultrafilter on that set. The ...
10
votes
2answers
560 views

Large cardinal axioms and total recursive functions

Are there known relationships between large cardinal axioms (say Mahlo or Woodin cardinals) and total recursive functions (over the natural numbers) of the type: $ZFC$ + large cardinal axiom $\vdash$ ...
15
votes
2answers
471 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: ...
12
votes
2answers
396 views

tree properties on $\omega_1$ and $\omega_2$

Are the following mutually consistent (relative to large cardinals)? (1) There are no $\omega_2$-Aronszajn trees. (2) There is an $\omega_1$-Kurepa tree. In the models I know of the tree property ...
2
votes
0answers
81 views

Is there a transcendental definable function between algebras of elementary embeddings?

Let $\lambda$ be a cardinal. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $f:V_{\lambda}\rightarrow V_{\lambda}$ is a function and ...
7
votes
1answer
425 views

Taller models of ZFC

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy. Using forcing techniques, at least the ones I know of, one starts from a ...
5
votes
1answer
210 views

A Question on HOD, V and GCH

The theorem 1.1 of the following paper: Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic. states that: Theorem: Assume $V\models ZFC+GCH+~\text{There exists ...
5
votes
1answer
271 views

Statements that Could be Forced by Ultrapowers

Ultrapower of a structure is a very flexible mathematical creature in comparison with the ground structure and its ordinary products. Depending on the nature of ground structure and the good ...
6
votes
0answers
192 views

The Hales-Jewett Theorem for an infinite alphabet

Recall the Hales-Jewett Theorem: HJT: Given a finite alphabet $A$ and some $r \in \mathbb{N}$, there is some $H \in \mathbb{N}$ such that whenever $A^H$, the set of all length-$H$ words from $A$, ...
16
votes
1answer
495 views

What sort of cardinal number is the Löwenheim-Skolem number for second-order logic?

In their paper "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", Magidor and Väänänen make the following statement: "For second order logic, $LS(L^{2})$ ...
10
votes
1answer
386 views

What sort of large cardinal can continuum be?

I have stumbled across a related question asking which large cardinal properties can hold for $\aleph_1$. This question is probably also related, asking in what ways $\aleph_0$ is a "large" cardinal. ...
8
votes
0answers
132 views

Are there always large discrete families of normal measures?

Let $\kappa$ be a measurable cardinal. We give the Stone space of all ultrafilters on $\kappa$ the usual topology, where each $x\subseteq\kappa$ determines a basic open $[x]=\{U;x\in U\}$. The ...
4
votes
0answers
143 views

Two questions about the behavior of the continuum function

The first question asks about the global behavior of the power function in the case of finite gaps. Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let ...
6
votes
1answer
208 views

Elementary chains in forcing extensions of $M_1$

Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic ...
7
votes
1answer
361 views

Can an ultrapower be undone by forcing?

I am not 100% certain this question is appropriate for MO; I may just be missing something obvious. Also, I vaguely recall a similar question being asked here a while ago, but I can't find it; if it ...
18
votes
2answers
540 views

Removing large cardinals from an uncountable transitive model

The usual way of removing large cardinals from a given model of set theory is to cut off the model below the least large cardinal of interest. But this method may have dramatic effects on the external ...
8
votes
0answers
165 views

A question about strongly compact cardinals

Is the following equiconsistent with the existence of a strongly compact cardinals: For every $\lambda > \kappa$ there exists a $\lambda$-strongly compact embedding $j: V \to M$ with the ...
10
votes
1answer
339 views

“Largish” cardinals

In what follows, $\mathsf{ZCKP}$ refers to the subset of $\mathsf{ZFC}$ consisting of the axioms of Zermelo set theory with choice and foundation ($\mathsf{ZC}$) plus those of Kripke-Platek set theory ...
8
votes
1answer
1k views

Do Measurable Cardinals Exist? (assuming ZFC)

In Appendix B of his Uniform Central Limit Theorems (1999), Dudley writes: It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable ...
6
votes
2answers
404 views

“Lebesgue-measurable” cardinals and real-closed fields

I understand the motivation behind measurable cardinals is to ask the question: "is there any set large enough to admit a non-trivial measure on all of its subsets?" Hence, it's also worthwhile to ...
6
votes
0answers
163 views

“Clubiness” of projective sets of ordinals

I'm sure this is just my google-fu failing me, but: what are sufficient, non-overkill large cardinal axioms which guarantee "Every (boldface) $\Pi^1_n$ set of (real codes for) countable ordinals ...
6
votes
0answers
188 views

Adding minimal subsets to $\aleph_\omega$

Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all $\alpha < \kappa.$ Question. Is it consistent that there ...
6
votes
1answer
251 views

What is known about the large cardinal strength of Shelah's categoricity conjecture?

Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of ...
6
votes
1answer
223 views

Mahlo cardinal and hyper k-inaccessible cardinal

It is known that every Mahlo cardinal $\kappa$ is hyper $\kappa$-inaccessible. It the converse true, namely: every cadinal $\kappa$ which is hyper $\kappa$-inaccessible is a Mahlo cardinal ?
3
votes
1answer
237 views

How can the critical point of an elementary embedding be omega_1?

I've seen an example of an elementary embedding such that $\omega_1$ is the critical point. I was wondering what's wrong with the following proof that this cannot be: Let $\phi(x_1,x_2)$ be the ...
6
votes
0answers
334 views

Is there any elementary embedding characterization for $\Pi_{1}^{1}$ - reflecting cardinals?

Similar to one of the characterizations of weakly compact cardinals, a $\Pi_{1}^{1}$ - reflecting cardinal is defined as follows: A cardinal $\kappa$ is $\Pi_{1}^{1}$ - reflecting if $\kappa$ is ...
6
votes
1answer
586 views

Replacing Axiom of Choice with Axiom of Countable Choice

Many people find ACC more intuitive than AC ("Pick something from the first set, then something from the second set, then...) and it also doesn't lead to "controversial consequences" (See for eg: ...
8
votes
1answer
262 views

Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$?

Is it consistent that there exists an inaccessible cardinal $\lambda$ and a forcing extension $V[G]$ so that $$V[G]\models\text{There is some non-trivial elementary embedding ...
0
votes
0answers
17 views

In what sense is the function that maps $\alpha$ to the least $\alpha$-Erdős cardinal fast-growing

If we let $\alpha$ be an infinite limit ordinal and denote by $\kappa(\alpha)$ the least $\alpha$-Erdős cardinal, there should be some sense in which $\kappa(\alpha+\omega)$ is much larger than ...
6
votes
0answers
166 views

PCF conjecture and fixed points of the $\aleph$-function

Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|pcf(a)| \geq \aleph_1.$ See his papers Short extenders forcings I and Short ...
7
votes
1answer
337 views

Is every ordinal potentially definable?

It is easy to see that, if $V\models\alpha>\omega_1^{CK}$, then $\alpha$ is not recursive in any forcing extension of $V$. The argument goes as follows: The relation "$\Phi_e=r$" is $\Pi^0_2$. ...
5
votes
0answers
155 views

Can one take roots of rank-into-rank embeddings infinitely many times?

If $\lambda$ is a cardinal, then let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $j,k\in\mathcal{E}_{\lambda}$, then define ...