**5**

votes

**1**answer

165 views

### 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 ...

**5**

votes

**0**answers

30 views

### What is the smallest density of a metrizable space without countable separation?

A Tychonoff space $X$ is defined to have countable separation if some (equivalently, any) compactification $bX$ of $X$ contains a countable family $\mathcal U$ of open sets such that for any points $x\...

**8**

votes

**2**answers

584 views

### Difference between Laver's and Mathias's forcing

Mathias forcing consists of conditions of the form $(s,A)$ where $s$ is a finite subset of $\omega$, $A$ is an infinite subset of $\omega$ such that $\max(s) < \min(A)$ and $(s,A)\leq (t,B)$ iff $s\...

**-5**

votes

**0**answers

28 views

### Cardinality of infinite sets' subtraction [on hold]

$A = \mathbb{N}$; $B = \{\frac{p}{q} : p, q \in \mathbb{Z}\setminus\{0\}\}$. Find the cardinality of $P(A\setminus B)$.
We know that the cardinality of $\mathbb{N}$ is $\aleph_0$, so as the ...

**5**

votes

**1**answer

188 views

### Why relative consistency results by forcing arguments are provable in finitistic metatheory

It is claimed in many textbooks that relative consistency results, such as $\text{Con}(\text{ZFC})\rightarrow\text{Con}(\text{ZFC}+2^{\aleph_0}\geq\aleph_2)$, are provable in the finitistic metatheory....

**3**

votes

**1**answer

105 views

### $2M = M$ and its subset

I have some question concerning arithmetic of cardinal in ZF.
Write $ X = Y$ if there is a bijection between them.
Let $M$ be a set such that $2M = M$.
Can I show, in ZF, that any infinite subset $X$ ...

**5**

votes

**1**answer

134 views

### Showing that $\alpha$ isn't a cardinal in $J_{\alpha+1}^{\vec E}$ for a fine extender sequence $\vec E$

In [FSIT] and [OIMT] it is claimed that there is a surjection from $P(\kappa)\cap J^{\vec E}_{\nu(E_\alpha)}\times[\nu(E_\alpha)]^{<\omega}$ onto $\alpha$, and that this surjection lies in $J_{\...

**1**

vote

**1**answer

146 views

### Well-ordering of power set of omega

Assuming initially the background set theory ZF, what is the exact status of the existence of a well-ordering on the power set of $\omega$? How much needs to be added to guarantee this?

**4**

votes

**2**answers

241 views

### Critical points in $ZF$ without Choice

Recall the definition of critical point for set theory:
A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to ...

**7**

votes

**1**answer

225 views

### 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 ...

**2**

votes

**0**answers

119 views

### Is there a model of ZF+ACC where transfer fails for the definable hyperreals?

A decade ago Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go ...

**7**

votes

**1**answer

185 views

### Definability using rudimentary function

Denote by RUD the set of all rudimentary functions, together with the function that takes any set to its transitive closure.
Assume that I know that a binary relation $R$ is definable by some ...

**2**

votes

**1**answer

171 views

### Transfer with minimal choice

Let FUF postulate the existence of a Free UltraFilter on $\mathbb{N}$ and ACC the axiom of countable choice. Consider the superstructure on $\mathbb{R}$ and its inclusion in the bounded ultrapower. ...

**-1**

votes

**0**answers

12 views

### Continuous function on compact topological space [migrated]

I came across the following statement.
Let $X$ a uncountable set, $p \notin X$ and $X^* = X \cup \{p\}$. Let $$\mathcal O := \{O \subseteq X^* \mid O \subseteq X \text{ or } p \in O \text{ and } X \...

**13**

votes

**1**answer

805 views

### 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 ...

**8**

votes

**1**answer

134 views

### On a strengthening of strong measure zero

Recall that a set of $X$ of reals has strong measure zero (SMZ) if for every sequence $\{\epsilon_n:n<\omega\}$ of positive real numbers, there is a sequence $\{I_n:n<\omega\}$ of intervals such ...

**11**

votes

**3**answers

2k views

### Has Dedekind's proof of existence of infinite sets been analyzed by historians?

This pdf by David Joyce notes that in paragraph 66 of his famous essay, Dedekind claims to prove the existence of an infinite set.
The proof exploits the assumption that there exists a set $S$ of all ...

**8**

votes

**4**answers

709 views

### Is it inconsistent for a model of set theory to contain its own first order theory?

I am wondering if it is inconsistent to have a model of set theory V such that V contains an $A\subset \omega$ that codes its first order theory.I.e. for all $\{\underline\epsilon\}$-sentences $\phi$, ...

**1**

vote

**0**answers

142 views

### Can Dedekind's 'proof' of the existence of infinite sets be properly formulated and carried out in positive set theory?

This question is related to Mikhail Katz's recent mathoverflow question, "Has Dedekind's proof of the existence of existence of infinite sets been analyzed by historians?". Dedekind's 'proof' seems (...

**6**

votes

**1**answer

308 views

### Finding non convex functions satisfying a weak form of convexity, without the axiom of choice

If a real-valued function $f$ over reals satisfies $$ (1) \; \; \; f({x+y\over2})\le {f(x)+f(y)\over2}, $$and it is continuous, then it is not hard to see that $f$ is indeed convex. On the other hand,...

**1**

vote

**0**answers

136 views

### Full epsilon-induction and bounded epsilon-induction

epsilon-induction is the scheme: $\forall x(\forall y\in x\varphi (y)\rightarrow \varphi (x))\rightarrow \forall x\varphi (x)$.
Let "bounded epsilon-induction" be the above scheme, but only for ...

**10**

votes

**1**answer

312 views

### When are generic models not too wild?

This is a question related to ideas raised in http://arxiv.org/abs/1410.1224 and http://arxiv.org/pdf/1405.7456.pdf. Basically, the idea is the following:
Suppose I have a first-order theory $T$. ...

**9**

votes

**1**answer

348 views

### Real-valued measurable cardinals

A cardinal $\kappa$ is real-valued measurable if there is a probability measure on the $\sigma$-algebra of all subsets of $\kappa$ which is zero on singletons and additive on disjoint families of ...

**-2**

votes

**1**answer

210 views

### Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable? [closed]

Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable?
A professor said it to me a long time ago, but I don't have any references.
...

**10**

votes

**2**answers

822 views

### Witness to a failure of Fubini/Tonelli

Is it provable in ZFC that there is a subset of the plane all of whose vertical cross sections have Lebesgue measure zero and all of whose horizontal cross sections are complements of sets of Lebesgue ...

**17**

votes

**2**answers

1k views

### Analogues of Primitive Recursive Functions

Let $\mathbf{A}$ be an admissible set (possibly with urelements). I am wondering if there is some good notion of "primitive recursive arithmetic" relative to $\mathbf{A}$. More precisely, I would like ...

**1**

vote

**1**answer

150 views

### Is there any infinite set which is Dedekind finite and weakly Dedekind finite?

Is there any infinite set which is Dedekind finite and weakly Dedekind finite?
If $X$ is a weakly Dedekind finite amorphous set, can we show that $\mathcal P(X)$, the power set of $X$, is also weakly ...

**5**

votes

**1**answer

228 views

### Is this weak form of $V=L$ (in)consistent with large cardinals?

I have been considering a (definability-free) weak form of the constructibility axiom, which is intended to capture the coarse structure of the constructible hierarchy. This means that this weak form ...

**1**

vote

**1**answer

217 views

### Sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$

Consider sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$. Are there sets of Vitali's type in both $L$ and $V \backslash L$? If so, is there any way one can distinguish ...

**5**

votes

**1**answer

145 views

### Martin-Solovay Tree of Weakly Homogeneous Tree under $\mathsf{AD}_\mathbb{R}$

A tree $T$ on $\omega \times \lambda$ is weakly homogeneous if there is a countable set $\sigma$ of countably complete measures on ${}^{<\omega}\lambda$ so that $x \in p[T]$ if and only if there is ...

**41**

votes

**1**answer

1k views

### Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...

**5**

votes

**1**answer

208 views

### A variant of Freiling's Axiom of Symmetry and a weak form of the Continuum Hypothesis in models where all sets of reals are Lebesgue measurable

Consider the following variant of Freiling's Axiom of Symmetry, $\mathsf{AS}$, which will be denoted $A_{< 2^{\aleph_0}}$:
given any function $f$ from $\mathbb{R}$ into the families of of subsets ...

**3**

votes

**1**answer

242 views

### Representation of meager sets in Cohen extensions

Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set $...

**2**

votes

**6**answers

2k views

### Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...

**4**

votes

**0**answers

183 views

### Unbounded towers and combinatorial cardinal characteristics of the continuum

Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations.
This question assumes familiarity with combinatorial cardinal characteristics of ...

**3**

votes

**1**answer

159 views

### Reference request: The consistency of a tall tower in $\mathbb{N}^\mathbb{N}$

A $\kappa$-tower in $\mathbb{N}^\mathbb{N}$ is a sequence
$\langle a_\alpha : \alpha<\kappa\rangle$ in $\mathbb{N}^\mathbb{N}$
that is $\le^*$-increasing with $\alpha$
and has no $\le^*$-upper ...

**2**

votes

**1**answer

163 views

### Boolean algebras and free filters generated by chains

Suppose $\mathbf{B}$ is a complete Boolean algebra with an infinite domain $B$. Suppose $\mathbf{B}$ is atomic (i.e. every element is the supremum of some set of atoms). This algebra contains the co-...

**7**

votes

**0**answers

129 views

### Examples of analytic $\mathcal{I}$-mad families

If $\mathcal{I}$ is an ideal (proper and containing the finite sets) on $\omega$, call a family of subsets $\mathcal{A}\subseteq[\omega]^\omega$ $\mathcal{I}$-almost disjoint if for all distinct $A,B\...

**8**

votes

**0**answers

199 views

### 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 ...

**5**

votes

**1**answer

300 views

### When is there an unbounded tower in $[\mathbb{N}]^\infty$?

(Edit: I'm splitting the question, leaving here only what is answered by Ashutosh, and moving the rest to another question.)
This question assumes familiarity with combinatorial cardinal ...

**10**

votes

**1**answer

270 views

### Precipitous ideals and GCH

It is well known that ZFC + "There is a measurable cardinal" is equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$." Is ZFC + "There is a measurable $\kappa$ such that $2^\kappa &...

**14**

votes

**1**answer

473 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

**3**answers

672 views

### Singularizing forcing of “small” cardinality?

Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size $\...

**10**

votes

**1**answer

378 views

### A question on subsets of $\omega_1$

Is there a family $\{A_\alpha:\alpha<2^{\omega_1}\}\subset [\omega_1]^{\omega_1}$ in ZFC such that for each countable set $I\subset 2^{\omega_1}$ and $\alpha\in 2^{\omega_1}\setminus I$ we have
$$...

**7**

votes

**1**answer

208 views

### What is the height (or depth) of $[\mathbb{N}]^\infty$?

(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)
Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$,
partially ordered by $\...

**4**

votes

**1**answer

221 views

### Is it consistent that the gaps between cardinals $\kappa$ and $2^\kappa$ “get larger and larger”?

Is the following statement consistent in $\mathsf{ZFC}$?
For every ordinal $\beta$ there is an ordinal $\lambda_0$ such that for all ordinals $\lambda\geq\lambda_0$ we have $2^{\aleph_{\...

**19**

votes

**2**answers

1k views

### What is the largest Laver table which has been computed?

Richard Laver proved that there is a unique binary operation $*$ on $\{1,\ldots,2^n\}$ which satisfies $$a*1 \equiv a+1 \mod 2^n$$
$$a* (b* c) = (a* b) * (a * c).$$
This is the $n$th Laver table $(A_n,...

**11**

votes

**1**answer

402 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 ...

**-8**

votes

**1**answer

362 views

### Missing Axiom: There are no other axioms. Leads to a proof of CH within ZFC [closed]

I have come to the conclusion that there is often an implied axiom: There are no other axioms. Failure to explicitly state this axiom and to consider its consequences can result in the misleading ...

**0**

votes

**1**answer

161 views

### Can epsilon-induction be derived from the transitive closure operator?

I was wondering (and could not seem to prove or disprove) if epsilon-induction could be derived from the transitive closure operator for binary relations, if we do not have the Foundation Axiom.
The ...