The large-cardinals tag has no wiki summary.

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### 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 ...

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### The Kunen inconsistency and definable classes

There is a tension between (1) interpreting proper class talk in set theory as talk about first-order formulas and satisfaction; and (2) taking it to be an interesting and non-trivial result that ...

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368 views

### Partition relation, almost a Ramsey cardinal?

Is it consistent with ZFC to have a cardinal $\kappa$ which is not Ramsey and
$\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$ holds for some $n\in \omega$?
The partition relation $\kappa ...

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**1**answer

358 views

### More on Kunen's inconsistency result

I would like to suggest another argument for Kunen's inconsistency result, and I wonder to know if the argument is correct. I am also interested to see, if the proof is correct, which part of the ...

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**1**answer

423 views

### Surreal numbers and large cardinals

This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject.
Part 1 is about foundations. Much of the ...

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**1**answer

256 views

### Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal

Is anything known about the consistency strength of the following statement?
$\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ such that $a \cap ...

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**1**answer

250 views

### Applications of higher-order reflection principles

Let $L_3$ be the third-order language of set theory with identity on the first sort. Variables $x$ are first-order, $y$ are second-order, and $z$ are third-order. In the style of Lévy and Bernays, we ...

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213 views

### Consistency of many Erdos cardinals

Does anyone have a reference for the consistency strength of saying that the Erdos cardinal $\kappa(\alpha)$ exists for all $\alpha$? Or of various weakenings about how far such cardinals extend into ...

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294 views

### Models of Determinacy

Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might look like and if it ...

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**1**answer

150 views

### $\omega$-small and properly small premice.

Let $\mathcal M$ be a premouse.
$\mathcal M$ is said to be $\omega$-small if and only if whenever we have that $\kappa=crit(E)$ for some extender $E$ on the sequence of $\mathcal M$, then $\mathcal ...

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**1**answer

156 views

### Versions of large cardinals with target model in a generic extension

(I ran into this question while thinking about the (Strong) Inner Model Hypothesis: see http://www.jstor.org/stable/4093051 or http://arxiv.org/abs/0711.0680.)
A measurable cardinal is a cardinal ...

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299 views

### A question about Mitchell/Steel Fine Structure and Iteration Trees

In chapter 8 of Mitchell's and Steel's FSIT, they prove a central fine structural result, which basically states that if $\mathcal{M}$ is 1-small, $k$-sound, $k$-iterable premouse then the ...

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543 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 ...

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398 views

### What can the degrees of constructibility be?

If $r, s\in\mathbb{R}$, we say $r$ is constructible relative to $s$ - and write $r\le_cs$ - if $r\in L[s]$. Modding out by the induced equivalence relation $\equiv_c$, we get a partial order, the ...

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294 views

### Homogeneous Namba-like forcing

Let $\kappa \ge \aleph_3$ be a regular cardinal that is countably closed ($\alpha^\omega < \kappa$ for every $\alpha < \kappa$.) I'm mostly interested in the case that $\kappa$ is strongly ...

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264 views

### Elementary Embeddings and Relative Constructibility

Suppose $$j:M\prec N$$ is a non-trivial elementary embedding. Under what conditions on the sets (classes?) $M$ and $N$ (or even the critical point of $j$) does $j$ extend to an elementary embedding ...

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305 views

### if k is weakly inaccessible, then it is the k-th aleph fixed point

A cardinal $\kappa$ is weakly inaccessible iff $\kappa > \omega$, $\kappa$ is regular, and $\forall\lambda<\kappa(\lambda^+<\kappa)$
(here $\lambda^+$ is the successor cardinal)
A cardinal ...

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195 views

### Extending complete filters

Suppose $\kappa$ is a measurable cardinal and let $\mathcal{F}\subset\wp(\kappa)$ be a $\kappa$-complete non-principal filter. Can we extend $\mathcal{F}$ to a $\kappa$-complete ultrafilter?
My ...

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156 views

### $\Delta^1_2$-well ordering vs $\Delta^1_3$

It is a classical result that if $0^{\sharp}$ exists, then there is a model of $ZFC$ in which there is a $\Delta^1_3$ well ordering of reals but no $\Delta^1_2$-well ordering.
My question is: Is ...

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**1**answer

167 views

### Two complementing consequences of supercompactness

I would like to know if the following two consequences of having a supercompact cardinal are orthogonal:
1) On one hand being supercompact is equivalent to being "A ineffable for all A" ...

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274 views

### Relation between $\neg \square(\kappa)$ and the tree property at $\kappa$.

If $\kappa$ is an inaccessible cardinal then the tree property at $\kappa$ is equivalent to weak compactness of $\kappa$, which implies that $\square(\kappa)$ fails---that is, that every coherent ...

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626 views

### Can measures be added by forcing?

The Lévy-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 ...

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308 views

### A question about the comparability of large cardinals.

Are there any examples of two large cardinal axioms $AX$ and $AY$, in the language of first order $ZFC$, which satisfy the following conditions.
Each of them defines a unique cardinal number - ...

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341 views

### large cardinal tree properties as properties of sheaves

As follows from this talk Large Properties for Small Cardinals, p.7,p.4 http://www2.dm.unito.it/paginepersonali/viale/SEMINARS-TORINO/Fontanella-Torino-19.1.2012.pdf, the definitions of weakly compact ...

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### Internal large cardinal embeddings. How deep can we go?

Suppose, in V, that $\kappa$ is a large cardinal, say $\kappa$ is supercompact, and $j:V \to M$ is an ultrapower embedding witnessing the large cardinal properties of $\kappa$. From here, we can take ...

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### 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 ...

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### a partial order not dense iff a measurable exists

For $\kappa>0$ a regular cardinal, let $Ht_\kappa$ denote the following partial quasi-order:
(i) elements(objects) of $Ht_\kappa$ are classes X of sets of size $\kappa$
with the property that,
...

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**1**answer

361 views

### Can $\omega_1$ be supercompact?

Is "ZF + $\omega_1$ is supercompact" consistent relative to "ZFC + there is a supercompact cardinal"?
In particular, if $\delta$ is supercompact, does it remain so in $V(\mathbb{R} \cap V[G])$ where ...

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### 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 ...

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### Measures that are not OD

Is anything known about the consistency strength of the statement:
"There is a normal measure (on a cardinal) that is not ordinal-definable"?
In particular, is it consistent relative to the ...

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### 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
...

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### Large cardinals without the ambient set theory?

In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan
Talk about cardinals without the
(ambient) ...

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### ULTRAINFINITISM, or a step beyond the transfinite

Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the TRANSFINITE.
$\aleph_0, \aleph_1,\aleph_2\dots$
the lists ...

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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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### 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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300 views

### Vopenka's Theorem on L(A) and Large Cardinals from Weaker Assumptions?

I know that sometime ago Vopenka proved this:
Theorem: Assume there is a strongly compact cardinal. Then for any set $A$, $V \neq L(A)$.
Can we get by with a consistency-wise strictly weaker ...

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### 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 ...

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### What can we learn about an elementary embedding from the image of the ordinals?

If $j : V \rightarrow M$ is an elementary embedding, what can we learn in $M$ from $j''ORD$? That is, what is $M[j''ORD]$?
In particular,
Is it $M[j''ORD]$ equal to all of $V$?
If not, do we ...

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243 views

### On Successive Regular Cardinals With No Ladders

Definition: Let $\kappa$ be an $\aleph$ cardinal, we say that $\langle f_\alpha\colon\alpha\to\kappa\mid\alpha<\kappa^+\rangle$ is a ladder if every $f_\alpha$ is injective.
Equivalently this is ...

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546 views

### Set Cardinality Game - Can a player with numbers in R win over a player with numbers in N as each of them in one turn has to present a new number?

Let there be 2 players, p$\mathbb{N}$ and p$\mathbb{R}$. They are playing the Set Cardinality Game where p$\mathbb{N}$ has to present some number n that has not been used in the game so far by either ...

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### 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.
...

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### 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 ...

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### GCH+ Kurepa Families

I have a couple of questions about known theorems for GCH+Kurepa families.
Definition first: Let $\kappa$ be a infinite cardinal. A $\kappa^+$ Kurepa family is a family $F$ of subsets of $\kappa^+$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### End extensions of models which do not preserve axioms

Assuming the axiom of choice there is a neat way defining inaccessible cardinals as uncountable, regular, strong limit cardinals.
Without the axiom of choice we have several notions of ...

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### Why “adding” a single extender cannot give an L-like inner model for say, a strong cardinal?

The constructible universe $L$ is too thin for large cardinals greater than measurable. To build $L$-like inner models for large cardinal, it is natural to think about "adding" the evidences into the ...

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### Some questions from the paper “Forcing the failure of CH by adding a real” by Shelah and Woodin

1-Let $P=Add(\omega_1, \kappa)$, and let $D$ be a dense open subset of $P$. Then there is a dense subset $S$ of $D$ such that for every $f \in S$ and any $g \in P$, if $domf=domg$ and the set $\{ ...

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### 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 ...