# Tagged Questions

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### A Special Pair of Formulas

Consider the first order language $\mathcal{L}=\{\in,\in'\}$ and $\{\in\}$-theory ZFC. Is there a pair $(\varphi (x,y),\psi (x,y))\in \{\in\}-Form\times \{\in'\}-Form$ of ...

**13**

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**1**answer

207 views

### Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?

In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically ...

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**1**answer

166 views

### Questions about Prikry forcing and Cohen forcing

I have two unrelated questions.
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...

**6**

votes

**1**answer

176 views

### Sets of cardinalities of bases without choice

For a vector space $V$, let $BS(V)$ be the set of cardinalities (not necessarily $\aleph$s) of bases of $V$. Of course, in ZFC each $BS(V)$ is a singleton, but supposing the axiom of choice fails, it ...

**7**

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**1**answer

215 views

### Inner model in which every uncountable cardinal is large

The following is known:
$(*)$ If $0^\sharp$ exists, then any uncountable cardinal is is an inaccessible cardinal (and even more) in $L$.
My question is that:
Are there any large cardinal ...

**4**

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**0**answers

136 views

### $\infty$-Borel Determinacy?

An $\infty$-Borel set is a set $X\subseteq\mathbb{R}$ which has an $\infty$-Borel code - a set $r$ coding the construction of $X$ via open sets, complementation, and well-ordered unions (see ...

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219 views

### cardinals below the critical point of a generic embedding

This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension?
To focus on the ...

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370 views

### Different approaches to the multiverse of sets

There are some different approaches to the multiverse of sets, in particular:
1) The approach by Woodin,
2) The approach by Sy Friedman, ...,
3) The approach by Hamkins.
I wonder to know if ...

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302 views

### Applications of SCH outside of set theory

Recall that the Singular Cardinals Hypothesis (SCH) says that if $\kappa$ is a singular cardinal and $2^{cf(\kappa)}<\kappa,$ then $\kappa^{cf(\kappa)}=\kappa^+.$ Clearly it has many applications ...

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310 views

### Does every Aronszajn tree has a Suslin or a Special subtree?

Question: Does every $\omega_1$-Aronszajn tree contains a Suslin sub-tree or a special Aronszajn sub-tree?
Recall that Suslin trees are $\omega_1$-trees (trees of height $\omega_1$, and countable ...

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402 views

### Pathological behavior of Borel sets?

Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the ...

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12 views

### the lexicographix order [migrated]

If it is given ordinals $\alpha$ and $\beta$, the lexicographix order on $\alpha \times \beta$,$\leq_{lex}$ is given by: $(\gamma_0,\delta_0)<_lex(\gamma_1,\delta_1)$ if and only if either ...

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212 views

### 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 ...

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248 views

### The (non-)absoluteness of second-order elementary equivalence

Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, ...

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133 views

### Canonical model for $\neg\mathsf{CH}$ and $\Omega$-logic

Recently I found this book by Woodin. In the introduction of it the author writes the following:
The main result of this book is the identification of a canonical model in which
the Continuum ...

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229 views

### Woodin Cardinals and Inner Models

I have a few questions I have been thinking about that I could definitely use some insights on:
Question 1. Since a Woodin cardinal is a "local" notion, defined with respect to some rank-initial ...

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435 views

### Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...

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**1**answer

272 views

### Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based
on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...

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441 views

### What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?

Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...

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**1**answer

199 views

### Perfect set property implies $\omega_1$ is a limit cardinal in $L$

Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal.
The original proof is in German, and I've been ...

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97 views

### Does the Lévy collapse obey this nice characterization? [duplicate]

This question is related to an issue in my answer to Monroe Eskew's question on the failure of Cantor-Bernstein for the Lévy collapse.
Question. Is the Lévy collapse $\text{Coll}(\omega,\lt\kappa)$ ...

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**1**answer

162 views

### Failure of Cantor-Bernstein for the Levy Collapse

Related to this question, is it possible to give an example of the failure of Cantor-Bernstein for complete embeddings of forcing notions involving the Levy Collapse $Col(\omega,<\kappa)$? Suppose ...

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204 views

### A model of Krivine

In a paper by J.-L. Krivine, Modèles de ZF+AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue [C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A549–A552, ...

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110 views

### How many elementary equivalent models are unifiable by ultrapower?

Definition. A class $\mathcal{C}$ of pairwise elementary equivalent $\mathcal{L}$-structures is unifiable by ultrapower if there is an index set $I$ and an ultrafilter $F$ on it such that $\forall ...

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1k views

### Are there non-diagonal proofs for Cantor's continuum and Godel's incompletness theorems?

There is a formal definition for the notion of a formal proof.
Question 1. Is there any formal definition for the notion of a diagonal formal proof?
Consider the following theorems both proved by ...

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**1**answer

173 views

### Obtaining a lightface pointclass from a boldface one

Define a pointclass to be:
boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and ...

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735 views

### Is it ever a good idea to use Keisler-Shelah to show elementary equivalence?

The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem ...

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**1**answer

182 views

### A ZFC construction to get a proper extension which is a $\omega_1$-model

In $V$, let me call a set theory structure A is a $\omega_1$ model if the $\omega_1$ of $A$ is the same as the $\omega_1$ in $V$ (up to isomorphism). The question I would like to ask is the following: ...

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165 views

### Skolem Hulls in $H_{\omega_2}$

I put this on stack exchange over a week ago with no answer, so let's try here.
Consider a model of the form $\mathfrak{A} = (H_{\omega_2}, \in, \lhd, f_0, f_1, ...)$, some expansion of ...

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220 views

### Vaught's conjecture for partial orders

In
``Steel, John R. On Vaught's conjecture. Cabal Seminar 76–77, pp. 193–208''
the following is proved:
Theorem. Let $\phi\in L_{\omega_1,\omega}.$ If every model of $\phi$ is a tree, then $\phi$ has ...

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**2**answers

265 views

### Forcing for Arbitrary First Order Theories

Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of ...

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**1**answer

123 views

### Intermediate submodels which do not satisfy AC

The following is known:
Theorem. Suppose $V[G]$ is a generic extension of $V$ by a set forcing, and let $N$ be a model of $ZFC$ with $V\subseteq N\subseteq V[G].$ Then $N$ is a generic extension of ...

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149 views

### How many Dedekind-finite sets can $\mathbb{R}$ be partitioned into?

Building off Asaf Karagila's answer to my previous question (Can $\mathbb{R}$ be partitioned into dedekind-finite sets?) on partitioning $\mathbb{R}$ into strictly Dedekind-finite sets:
(1) What ...

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150 views

### Algebra generated by a tree [Edit] [closed]

Suppose that $(T,\leq)$ is a partially ordered set, we say $T$ is a tree* if for every $i\in T$, $\{s: s\in T, s\leq t\}$ is a well-founded chain.
What I need to know is: Can the algebra ...

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819 views

### Weak forms of the Axiom of Choice

Let $n\geq 2$ be a natural number and consider the following:
$AC(n)$: For each family $\{X_i\}_{i \in I}$ of $n$-element sets the product $\prod_{i\in I}X_i$ is non-empty.
Is it known that for ...

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**1**answer

222 views

### Can $\mathbb{R}$ be partitioned into dedekind-finite sets?

Assuming $ZF$ itself is consistent, it is consistent that there are sets $D$ which are infinite but cannot be placed in bijection with any of their proper subsets; such sets are called "strictly ...

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180 views

### Climbing quickly up $L$

This question is motivated by Joel David Hamkins' answer to Godel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem), in which he shows that, if we replace ...

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### 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 ...

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275 views

### Who proved “sets in every generic are already in the ground model?”

Suppose $\mathbb{P}$ is a notion of forcing in the ground model $V$, and $X$ is a set which is in $V[G]$ for every $\mathbb{P}$-generic filter $G$. Then $X\in V$ already, by a fairly simple (if ...

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845 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 ...

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559 views

### PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)

We know $PFA$ implies $2^{\aleph_0}=\aleph_2$.
Q1. What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum ...

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228 views

### Are superstrong stronger than strongly compact cardinals? (or vice versa)

In the last part of Kanamori's excellent "The Higher Infinite" there is a small diagram about the strength and consistency strength of some major large cardinal axioms.
Below supercompact cardinals ...

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207 views

### Genericity by names

If $P$ is a notion of forcing in $M$, then $G$ is a $P$-generic filter over $M$ if $G\subseteq P$ is a filter, and for every $D\in M$ which is a dense subset of $P$, $G\cap D\neq\varnothing$.
...

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108 views

### Dedekind reals in heyting valued models

Let $V^{H}$ be a Heyting valued model of intuitionistic set theory. What conditions does $H$ have to satisfy in order for the following claim to hold? (where $\| \phi(u) \| \in H$ is the truth value ...

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273 views

### “Hard” separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...

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373 views

### When does the choice of the generic matter?

It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases ...

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942 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:
...

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458 views

### Can we force the existence of this function?

I want to see if it is possible to force the existence of a function
$F:\aleph_2 \times \aleph_2\rightarrow \aleph_1$ such that:
a) $F(a,b)=F(b,a)$, for all $a,b\in \aleph_2$ and
b) for all ...

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90 views

### Stability of analytic Zariski structures

Noetherian Zariski structures are introduced by Hrushovski and Zilber.(1996)
An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield.
...

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218 views

### Is “approximate categoricity” absolute?

Let $T$ be a countable first-order theory, and assume that $T$ has exactly one atomic model up to isomorphism in every uncountable cardinality. (By "atomic" I mean a model which omits the ...