# Tagged Questions

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### Radin forcing and large cardinals

Assume $\kappa$ is a $(\kappa+2)$-strong cardinal and let $j: V \to M \simeq Ult(V, E) \supseteq V_{\kappa+2}$ witness this where $E$ is a $(\kappa, \kappa^{++})$-extender. Also let $u$ be the measure ...
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### 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 ...
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### An Extender is a Generalization of an Ultrafilter?

I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here. I've heard that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter ...
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### Intuitive descriptions of some large cardinals

I was trying to formulate intuitive descriptions of some large cardinals. Roughly something equivalent to "A manifold is an object which looks like patches of $R^n$ glued together". Not perfectly ...
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### Are There Mutually Exclusive Large Cardinal Axioms in ZFC?

The various large cardinal axioms are usually described in terms of some roughly linear hierarchy of varying consistency strengths. However, some cardinal axioms potentially contradict one another. As ...
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### Inner models and strongly compact cardinals

The following question is motivated by a result of Magidor that it is consistent that the least strongly compact cardinal is the least measurable cardinal. Question. Assume $\kappa$ is a strongly ...
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### 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 ...
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### 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 ...
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### $\mathsf{AD}_\mathbb{R}$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\vdash$ ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 $A$, ...
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### 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, $LS(L^{2})$ [...
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### 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. ...
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### 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 ...
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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 $\... 1answer 216 views ### Elementary chains in forcing extensions of$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$is a generic ... 1answer 374 views ### Can an ultrapower be undone by forcing? I am not 100% certain this question is appropriate for MO; I may just be missing something obvious. Also, I vaguely recall a similar question being asked here a while ago, but I can't find it; if it ... 2answers 556 views ### Removing large cardinals from an uncountable transitive model The usual way of removing large cardinals from a given model of set theory is to cut off the model below the least large cardinal of interest. But this method may have dramatic effects on the external ... 0answers 169 views ### A question about strongly compact cardinals Is the following equiconsistent with the existence of a strongly compact cardinals: For every$\lambda > \kappa$there exists a$\lambda$-strongly compact embedding$j: V \to M$with the ... 1answer 369 views ### “Largish” cardinals In what follows,$\mathsf{ZCKP}$refers to the subset of$\mathsf{ZFC}$consisting of the axioms of Zermelo set theory with choice and foundation ($\mathsf{ZC}$) plus those of Kripke-Platek set theory ... 1answer 1k views ### Do Measurable Cardinals Exist? (assuming ZFC) In Appendix B of his Uniform Central Limit Theorems (1999), Dudley writes: It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable ... 2answers 413 views ### “Lebesgue-measurable” cardinals and real-closed fields I understand the motivation behind measurable cardinals is to ask the question: "is there any set large enough to admit a non-trivial measure on all of its subsets?" Hence, it's also worthwhile to ... 0answers 191 views ### Adding minimal subsets to$\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.$Question. Is it consistent that there ... 1answer 255 views ### What is known about the large cardinal strength of Shelah's categoricity conjecture? Shelah's categoricity conjecture states that for every Abstract Elementary Class$\mathcal{K}$there is a cardinal$\mu$depending only on$\operatorname{LS}(K)$(i.e. the Löwenheim–Skolem number of$\...
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 ?
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 ...