forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence ...

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1
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0answers
143 views

The patterns of possibility for nontrivial automorphisms and nontrivial elementary embeddings of the universe

In their paper "The Role of the Foundation Axiom in the Kunen Inconsistency" (arXiv:1311.0814 [Math.LO]), Daghighi, Golshani, Hamkins, and Jerabek show that the patterns of possibility for the ...
-6
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0answers
86 views

Are these answer correct? Simple discrete math problems [closed]

Hi so I just took a test a few hours ago and my teacher posted the test online but he didn't post the answers. I just wanted to know what my score is so I circled all the choices I made. Please tell ...
9
votes
2answers
387 views

Axiom of choice for sets of finite sets

The question I am going to ask is really to satisfy my curiosity, as I am not at all an expert of the subject and do not plan to really work on it. Hence, if you think the question is not suitable for ...
7
votes
1answer
145 views

Characterizing L(R) Cardinals in HOD

We're working in L(R) under AD. We know that $\omega_1$ is the least measurable in HOD, $\Theta$ is the least woodin, $\delta^2_1$ is the least strong to the woodin, etc. My question is about ...
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0answers
98 views

How can I proof that $Set^I \simeq Set /I$? [closed]

I need help to prove this equivalence. Anyone can do an exhaustive explanation about this? Thank you so much
6
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1answer
271 views

Constructing unnatural transformations

In a nutshell, the question is: is it true that any explicit (not involving axiom of choice) pointwise transformation between sufficiently complicated functors is natural almost everywhere? Let $C$ ...
22
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2answers
385 views

Mid point free sets

Given a subset X of unit interval, can we find a subset Y of X of same outer measure as X such that Y does not contain three points of the form x, y and (x+y)/2? I can do this assuming CH but can we ...
5
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1answer
231 views

Logical strength of “choice functions exist for well-ordered families”?

A colleague of mine suggested the following weakening of the axiom of choice: If $\mathscr{F} := \{F_\alpha\}$ is a well-ordered family of non-empty sets (i.e., there is a bijection between ...
0
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0answers
33 views

Calculate optimal path through changing network? [migrated]

Apologies if this question is not suited for this forum. The question extends beyond my knowledge of mathematics and programming, it is quite hard to get my head around it let alone put it in to ...
7
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1answer
182 views

Dedekind-finite arithmetic vs natural numbers arithmetic

It is known that the Dedekind-finite cardinals are closed under addition and multiplication, so one may do arithmetic in them, as opposed to only natural numbers. How much can those two arithmetics ...
5
votes
2answers
249 views

The role of the rigid relation principle ($RR$) in the Kunen inconsistency

Consider the rigid relation ($RR$) principle, i.e. "every set admits a rigid binary relation", that is,"that for every set $A$ there is a binary relation $R$ on $A$ such that the structure $(A,R)$ is ...
1
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1answer
117 views

Does the countable $\sigma$-product of a separable Hilbert space have a first countable topology?

Let $\mathcal{l}^2$ be "the" separable real infinite dimensional hilbert space, e.g. the space of square-summable sequences of real numbers. Let $\Box^{\mathbb{N}}\mathcal{l}^2$ be the countable ...
1
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0answers
167 views

Existence of $\lambda$-transitive linear orders for $\lambda \geq \aleph_0$

A linear order $(L, <)$ is $\lambda$-transitive iff any order-preserving bijection between sets of size $\lambda$ can be extended to an order automorphism of $L$. For $\lambda < \aleph_0$, ...
4
votes
1answer
138 views

Existence of $\kappa$-Suslin trees above a measurable cardinal

We have learned from Joel David Hamkins and Monroe Eskew that: Answers: Having a measurable cardinal $\delta$ we can force a $\kappa$-Suslin tree for many $\kappa$'s above $\delta$. But is the ...
9
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0answers
137 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 ...
3
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1answer
99 views

Infinite non-splittable graphs

Let $G=(V,E)$ be a graph. For $v\in V$ we set $N(v)=\{w\in V:\{v,w\}\in E\}$. We say that $G$ is splittable if there are $S,T\subseteq V$ with $S\cap T=\emptyset$ and $S\cup T = V$ such that for all ...
5
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0answers
159 views

Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?

Assume that $V\neq HOD$ and let $\kappa = \min \{\alpha\in On \mid \mathcal{P}(\alpha) \not\subseteq HOD\}$. Clearly, $\kappa$ is a cardinal. Question: Is it consistent that $\kappa = ...
7
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1answer
217 views

Notions of infinity in $\mathsf{ZF}$ without choice

Consider the following statements about a given set $X$ in in $\mathsf{ZF}$: (1) There is $x_0\in X$ such that there is a surjective map $\varphi: X\setminus\{x_0\}\to X$. (2) There is an injective ...
6
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0answers
148 views

A generalization of SOCA

Roughly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of the unordered pairs over an uncountable separable metric space you can always find an uncountable homogeneous subset ...
6
votes
2answers
150 views

Reference for proof that consistency of $\omega_1$-Erdos cardinal implies Con(Chang's Conjecture)

What is a good source for Silver's proof (or a more modern version) that Con($\exists \omega_1$-Erdos cardinal) implies Con(Chang's Conjecture)? Silver's original proof seems to have never been ...
0
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0answers
48 views

Number of “small” subsets to a “large” set [migrated]

For the following we assume the axiom of choice. Let $X$ be a set of cardinality $l$ for some infinite cardinal number $l$, and let $p(X)$ be the number of subsets of $X$ that have cardinality ...
1
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0answers
164 views

In set theory, is there a name for a function which maps the empty set to zero and all the others to one? [closed]

I would like to avoid inventing something which might be standard. Thus, I'am asking if there is a name for a function which is defined as $f$: Let $S$ be any set, then $f(S)=0$ if $S$ is empty and ...
13
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3answers
1k views

Applications of set theory in physics

In the introduction of the paper "Links between physics and set theory", the following quote of Eris Chric is stated: "Set theory perhaps is too important to be left just to ...
3
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0answers
134 views

A property of the Frechet filter and every ultrafilter

(Joint question with Piotr Szewczak.) Definitions and notation. By filter we mean a filter on $\omega$ containing the cofinite sets at least. For a filter $\mathcal{F}$, let ...
11
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1answer
234 views

A classic cardinal characteristic of the continuum in disguise?

We believe the answer to the following question, that is relevant to a joint research project with Piotr Szewczak, should be known. We would appreciate any help or pointer. Needed definitions may be ...
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votes
0answers
15 views

Approximate point spectrum is complement of set of points of regular type [migrated]

I have a question concerning the approximate point spectrum of a closed linear operator. I need to show that the approximate point spectrum is the complement of the set of points of regular type, ...
8
votes
3answers
345 views

Freiling's Axiom of Symmetry Concretized

Freiling's Axiom of Symmetry says that for any function $f:[0,1]\to \mathcal{P}([0,1])$ such that for every $x\in [0,1]$ we have $|f(x)|=\aleph_0$, then there exist $y,z\in [0,1]$ such that $z\notin ...
13
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4answers
1k views

Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?

In a conversation where it came up that the Pythagoreans probably found an enumeration of the rational numbers I erroneously remarked that Georg Cantor found a natural bijection from $\mathbb{N}$ to ...
3
votes
0answers
161 views

What algebraic identities does the iteration of forcing operation satisfy?

Let $G$ be the set of all formulas $\phi(x)$ in the language of such that $ZFC\vdash\exists x\phi(x)$ exists, $ZFC\vdash\phi(x)\rightarrow``x\,\textrm{is a complete Boolean algebra}"$, ...
4
votes
1answer
198 views

$\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication

A. S. Daghighi, M. Golshani, J. D. Hamkins, and E. Jeřábek proved in "The foundation axiom and elementary self-embeddings of the universe" that, working in ZFGC$^{\text{−f}}$+BAFA, there are ...
8
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2answers
294 views

For a partition of $\mathbb{R}$ into countably infinite sets, must there be an almost-disjoint family of $2^{\frak c}$ many selectors?

My question arises from a construction I gave in my recent answer to a question of Alexander Pruss concerning large families of independent non-measurable sets of reals. In that argument, using the ...
4
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0answers
204 views

Is there some absoluteness between the Boolean valued universe $V^{B}$ and $V$?

It is well known that if $\phi$ is a $\Delta_{1}$-formula and $x_{1},..,x_{n}$ in $V$ and $V[G]$ is a forcing extension, then $V\models\phi(x_{1},...,x_{n})$ if and only if ...
8
votes
2answers
231 views

Images of $\{0,1\}^\kappa$

Is there a compact topological space $(X,\tau)$ such that for no cardinal $\kappa$ there is a surjective continuous map $e:\{0,1\}^\kappa \to X$? (We assume that $\{0,1\}$ is endowed with the ...
0
votes
2answers
114 views

Surjectivity from union of a set system to the set system

Let $\mathcal{A}$ be a non-empty systems of non-empty sets such that there is an injective map $f:\bigcup \mathcal{A}\to \mathcal{A}$ such that $a\in f(a)$ for all $a\in\bigcup\mathcal{A}$. Assuming ...
1
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0answers
121 views

Surjective marriages

Let $M, W\neq \emptyset$ be sets and $K\subseteq M\times W$. We say that $(M, W, K)$ has a marriage if there is an injective function $f:M\to W$ such that $f\subseteq K$. If $(M,W, K)$ has a ...
8
votes
1answer
332 views

If $\kappa$ is weakly inaccessible and $A\subset\kappa$, can $L[A]$ violate $\kappa^{\lt\kappa}=\kappa$?

In some current work, my co-authors and I had wanted in a certain argument to appeal to $\kappa^{\lt\kappa}=\kappa$ in $L[A]$, in a situation where $A\subset\kappa$ and $\kappa$ was weakly ...
6
votes
1answer
163 views

Surjective (strong) reducibility of Borel equivalence relations

Suppose $E$ and $F$ are Borel equivalence relations on Polish spaces $X$, $Y$, resp. Say that $E$ is surjectively Borel reducible to $F$ iff there is a Borel surjection $f:X \to Y$ such that $xEy$ iff ...
4
votes
1answer
311 views

Inaccessible cardinal and $\Sigma_1$ reflection

A theorem of A. Levy says that, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1}V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ formulas. Where ...
7
votes
2answers
487 views

Independence of the countable axiom of choice

How does one proove that the Countable axiom of choice is not provable in ZF?Is there any brief proof?Does the Independence of the countable axiom of choice implies the independence of the axiom of ...
4
votes
1answer
124 views

$\mathfrak{p}=\mathfrak{b}=\mathfrak{a}=\aleph_1$ and $\mathfrak{d}=\mathfrak{c}=\kappa$

Asumme tha in $M$, $CH$ holds and $\kappa>\aleph_0$ and $\kappa^{\aleph_0}=\kappa$. Let $K$ be $Fn(\kappa,2)$-generic over $M$. Question: Then we can say in $M[K]$ that: $(i)$ ...
3
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1answer
200 views

Hamkins infinite time Turing machines: dovetailing ordinal time

It is claimed in the Hamkins and Lewis founding article "Infinite time Turing machines" (proof of the gap existence theorem 3.4) that for $\omega$ steps of a computation of a machine performing a ...
5
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116 views

$\kappa$-impediments (according to Shelah, Nash-Williams, Aharoni)

Let $\Gamma = (M, W, K)$ be a bipartite graph, that is $M, W$ are sets and $K\subseteq M\times W$. If there is an injective function $f:M\to W$ such that $f\subseteq K$ we say $f$ is an espousal and ...
2
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0answers
279 views

A query on how to climb inaccessibles in £

I am investigating to what extent extensions of the librationist property or set theory £ may support relative inaccessible sets; see Librationist Closures of the Paradoxes and Elements of ...
2
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1answer
107 views

“Namba forcing adds reals” independent of $ZFC + \neg CH$?

I know that, in the presence of $CH$, Namba forcing does not add reals. But when $CH$ fails, is it consistent that it still does not add reals?
14
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2answers
507 views

Scott-Solovay unpublished paper on ``Boolean valued models of set theory''

I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on ``Boolean valued models of set theory'' is given as a main reference, with many references to the ...
6
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3answers
240 views

Borel cross section

It is known from metric space topology that a closed equivalence relation on a Polish space has either countably many or $\mathfrak{c}$ many equivalence classes. A short elementary proof is given in ...
3
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3answers
836 views

Reverse Skolem's paradox

By using the Löwenheim–Skolem theorem & Mostowski collapse, in every model $V$ of $ZF+Con(ZF)$ there is a countable transitive set $M$ such that $(M,\in_M) \models ZF$. Is the following "converse" ...
17
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2answers
1k views

Philosophical arguments in defense (or against) large cardinals

The question is essentially what is asked in the title. I split it into two parts (A) (Arguments supporting the existence of large cardinals) What are the main philosophical arguments in defense ...
5
votes
1answer
196 views

Are Sharps in Countable Models Really Sharps?

Suppose $M$ is a countable transitive model of $\mathsf{ZFC}$ (maybe more). Suppose $x, y \in {}^\omega\omega \cap M$ and $M \models y = x^\sharp$. Is it true (possible with additional assumptions), ...
8
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2answers
226 views

Is there a version of Miller forcing “guided by” an ultrafilter?

It’s well known that Mathias forcing factors as a two-step iteration $P(\omega)/\mathrm{fin}\ast \mathbb{M}_{\dot U}$, where $\mathbb{M}_{\dot U}$ is Mathias forcing guided by the generic ultrafilter ...