**4**

votes

**1**answer

226 views

### Transcendence degree of the surreals over the subfield generated by the ordinals

Consider the Grothendieck ring $K[\Omega]$ of the semiring $\Omega$ of all ordinals under the operations of natural sum and product. Its quotient field $K(\Omega)$ is naturally a subfield of the ...

**7**

votes

**2**answers

348 views

### When can we reach a real by forcing?

I'm sure this is well-known, but: suppose I have a non-constructible real $r\in V-L$. Under what conditions is there a poset $\mathbb{P}\in L$ and a $G$ which is $\mathbb{P}$-generic over $L$, such ...

**4**

votes

**1**answer

153 views

### Simplest non-constructible set of integers compatible with the nonexistence of $0^\sharp$?

What is the simplest non-constructible set of integers (say, in the analytical hierarchy) that is compatible with the nonexistence of $0^\sharp$? In particular, can there still be a non-constructible ...

**4**

votes

**0**answers

121 views

### Is it provable in $\mathsf{ZF}$ that there is a group structure on any set $X$? [duplicate]

Given a set $X$ is it provable in $\mathsf{ZF}$ that there is a binary operation $\ast: X\times X\to X$ such that $(X,\ast)$ is a group?

**3**

votes

**1**answer

132 views

### How many pairwise non-homeomorphic compact, zero-dimensional topologies are there on $\mathbb{N}$?

To make the question more precise:
We call a topological space $(X,\tau)$ zero-dimensional if for $x\neq y \in X$ there is a clopen set $U\subseteq X$ with $x\in U, y\notin U$.
Let $\mathcal{C}$ be ...

**8**

votes

**2**answers

172 views

### Elementary embeddings with the same critical point

Question: Is it consistent (relative to the existence of large cardinals) that there is an elementary embedding $j\colon V\to M$ (where $M$ is transitive model) that factors as $j = j_n \circ k_n$ for ...

**3**

votes

**1**answer

126 views

### A question on recursion in Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection

$\Sigma_{3}KP\omega$ be Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection. What strengthening of Barwise's Definition by $\Sigma$ Recursion (Theorem 6.4 on ...

**15**

votes

**1**answer

765 views

### Is it possible to define higher cardinal arithmetics

In number theory there are several operators like addition, multiplication and exponentiation defined from $\omega\times\omega$ to $\omega$. Each of them is defined as an ...

**10**

votes

**2**answers

449 views

### Can a parent and child node have the same type in a well-founded digraph tree?

$\newcommand\toward{\rightharpoonup}$It would help me to
understand something in a current research project if someone
could provide an example of directed graph $\langle
G,\toward\rangle$ with the ...

**9**

votes

**2**answers

273 views

### When was Bounded Zermelo set theory first formulated?

Bounded Zermelo set theory, and many variants named for MacLane in some way, are used in equiconsistency proofs for Simple Theory of Types plus infinity, and for the Elementary Theory of the Category ...

**1**

vote

**0**answers

100 views

### is the set of algebraic numbers equivalent to rays of R-dense projections? [closed]

ok, the title is made of words I made up for lack of vocabulary. so let's define the words I used in the title:
$\mathbb{R}$-dense is a set dense in $\mathbb{R}$, rays of such a set means I take ...

**4**

votes

**0**answers

190 views

### A question on the size of an admissible ordinal

Let $\mathbf{L}_{\varsigma}$ be the level of ordinal $\varsigma$ of Gödel's constructible universe $\mathbf{L}$. Let $\Sigma_{3}$-KP be Kripke-Platek set theory with infinity and ...

**-5**

votes

**1**answer

184 views

### An axiomatic system with a set of constants that form a complete ordered field [closed]

I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...

**1**

vote

**1**answer

220 views

### At what level of the analytic hierarchy do Cohen reals lie?

In his doctoral thesis titled "Three models of ordinal computability", Benjamin Seyfferth proved the following theorems:
i) A set $\mathtt A$ of reals is Ordinal Turing Machine-enumerable if and only ...

**5**

votes

**2**answers

381 views

### Gorelic's Forcing for large Lindelöf spaces with points $G_\delta$

I am trying to understand a step for proving that there exists large Hausdorff Lindelöf Spaces with points $G_\delta$ using forcing. I am following Isaac Gorelic's "The Baire Category And Forcing ...

**5**

votes

**0**answers

70 views

### Large discrete subspaces in spaces of separately continuous functions

For topological spaces $X,Y,Z$ let $SC_p(X\times Y,Z)$ be the space of separately continuous functions $f:X\times Y\to Z$ endowed with the topology of pointwise convergence.
It is easy to see that ...

**7**

votes

**2**answers

316 views

### Possible cardinality and weight of an ordered field

Is it true (in ZFC) that for any regular infinite cardinal $\kappa$ there exists an ordered field of weight $\kappa$ and cardinality $2^\kappa$ (or at least $>\kappa$)?
The field of real numbers ...

**10**

votes

**2**answers

400 views

### Number of paths through infinite trees with given “growth rates”

(Preface: This may be a naive or easy question for experts....)
Consider an infinite tree, rooted on the left, where each node has two children; the number of nodes at each level (distance from the ...

**3**

votes

**1**answer

324 views

### Are there sets which are computable in one model, but uncomputable in another?

Suppose we have two models of set theory, $U$ and $V$ which have the same $\Bbb N$. Is it possible that there is a set $A\subseteq\Bbb N$ such that, in $U$, this set is computable, i.e. there is a ...

**3**

votes

**1**answer

156 views

### Uniformizing a relation on ordered sets

Suppose $A$ and $B$ are (complete) ordered sets. Suppose $R\subseteq A\times B$, and
$f(a)=\inf\{b : (a,b)\in R\}$
$g(b)=\inf\{a : (a,b)\in R\}$
then what can we call $f$ and $g$? Perhaps there is ...

**3**

votes

**1**answer

127 views

### About the Caratheodory class.

Let $X$ a set, and $\mathcal{P}(X)$ the class of its subset's.
Let $\mathcal{A}\subset \mathcal{P}(X)$, we call a map $L: \mathcal{P}(X)\to[0, \infty]$ $\mathcal{A}$-regular if for any $S\subset X$ ...

**8**

votes

**2**answers

367 views

### centeredness in forcing iterations

Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$, and $B$ is $\kappa$-centered. Let $G \subset A$ be a generic ultrafilter. Is $B/G$ $\kappa$-centered in $V[G]$?
Naively, we ...

**11**

votes

**3**answers

1k views

### Difference between ZFC and ZF+GCH

I hear that the axiom of choice (AC) derives from
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH are independent of
Zermelo–Fraenkel set theory(ZF).
So, I'm just ...

**8**

votes

**1**answer

265 views

### A question about three forcings

Let $P_1$ be the finite support iteration of random forcings of length $\omega$. Let $P_2 = \text{Random} \times \text{Random}$ and $P_3 = \text{Random} \times \text{Cohen}$. Are $P_i, P_j$, for $1 ...

**3**

votes

**2**answers

292 views

### A Question related to the Formula Hierarchy

Let large Latin symbols as $X$ and $Y$ denote sets of natural numbers and small symbols as $n$ and $n´$ denote natural numbers and small Greek letters stand for formulas.
Suppose $\alpha$ is ...

**4**

votes

**1**answer

161 views

### almost huge embeddings and stationary correctness

Suppose $\kappa$ is an almost huge cardinal. Using the characterization of Theorem 24.11 in Kanamori's book, one can show that if $j : V \to M$ is an embedding derived from an almost-huge tower of ...

**10**

votes

**1**answer

393 views

### Is there an $L$ like inner model for $\sf Z$?

Godel proved the consistency of the axiom of choice with the axioms of $\sf ZF$ by showing that given any model of $\sf ZF$, there is a definable class which satisfies $\sf ZFC$.
The proof uses a lot ...

**9**

votes

**0**answers

264 views

### Reinhardt cardinals and iterability

Work in $ZF$. Let $j:V\to V$ be a non-trivial elementary embedding which is iterable, so that we can iterate it and form models $M_\alpha, \alpha\in ON,$ with $M_0=V,$ and elementary embeddings ...

**7**

votes

**0**answers

178 views

### Countable choice in $L(\mathbb{R}^*_G)$

Let $\lambda$ be a singular strong limit cardinal and let $G \subset \text{Col}(\omega,\mathord{<}\lambda)$ be a $V$-generic filter. Let $\mathbb{R}^*_G = \bigcup_{\alpha < \lambda} ...

**7**

votes

**0**answers

149 views

### Does $\mathsf{fReR}_0$ prove the existence of the cartesian product of two sets

$\mathsf{fReR}_0$ is the set-theoretical system whose axioms consist of:
(1) Axiom of extensionality: $\forall z\in x\ (z\in y)\wedge\forall z\in y\ (z\in x)\rightarrow x=y$
(2) Axiom of empty set: ...

**1**

vote

**1**answer

202 views

### Can (how) one distinguish germs of continuous functions by a countable set of params?

Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point.
So in both cases we see ...

**3**

votes

**2**answers

234 views

### A question on Gandy-Jensen system and the rudimentary functions

Let $\mathrm{R}_0,\cdots,\mathrm{R}_8$ be the following functions:
$\mathrm{R}_0(x,y)=\{x,y\}$
$\mathrm{R}_1(x,y)=x-y$
$\mathrm{R}_2(x)=\bigcup x$
$\mathrm{R}_3(x,y)=x\times y$
...

**8**

votes

**1**answer

277 views

### On $V$-decisive and weakly homogeneous forcings

Suppose that $\Bbb P$ is a forcing in $V$, we say that $\Bbb P$ is $V$-decisive if whenever $\varphi(x_1,\ldots,x_n)$ is a statement in the language of forcing, and $u_1,\ldots,u_n\in V$ then $1_{\Bbb ...

**7**

votes

**1**answer

303 views

### Canonical functions in set theory and their applications

Given regular cardinal $\kappa>\omega,$ we can define the canonical functions $f_\alpha: \kappa\to \kappa,$ for $\alpha<\kappa^+.$
Some of their properties are presented in Chapter 22 of the ...

**5**

votes

**1**answer

611 views

### About the hypothesis of Zorn's lemma

The proofs I know of Zorn's lemma give the following refinement:
Let $(X,<)$ be a partially ordered set such that every well-ordered
subset of $X$ has an upperbound. Then $X$ has a maximal ...

**10**

votes

**3**answers

486 views

### The continuum hypothesis for packing shapes without overlapping

Consider the finite cross $C$ (=union of line segments $\overline{(0, -1)(0, 1)}$ and $\overline{(-1, 0)(1, 0)}$) and the unit half-circle $H$. It is easy to see that we may pack continuum-many ...

**6**

votes

**1**answer

274 views

### Can the Cohen forcing collapse cardinals?

Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...

**2**

votes

**2**answers

105 views

### proof that “small” sets in an extension by iterated forcing already appear in an earlier stage

In Kunen's book (introduction to independence proofs, ) the following lemma is proved (chapter 8, lemma 5.14):
Assume that in M, $\alpha$ is a limit ordinal,
$( ( \mathbb{P}_\xi : \xi \leq \alpha) , ...

**13**

votes

**4**answers

2k views

### Is it possible to formulate the axiom of choice as the existence of a survival strategy?

Consider the following situation:
There is an infinite set $G$ of giraffes.
A lion comes and announces a set $C$ of all possible colours and an infinite cardinal $\kappa$.
The hungry lion ...

**6**

votes

**1**answer

227 views

### non(Meager) in Random times Random extension

Suppose the least size of a non meager set of reals is $\kappa$. Is it still $\kappa$ after forcing with Random $\times$ Random?

**4**

votes

**0**answers

169 views

### Recursively Pointed Sacks Forcing and Preserving $\omega_1$

Let $\mathbb{P}$ denote recursively pointed Sacks forcing. This is forcing with recursively pointed perfect trees ordered by inclusion. A tree $T \subseteq {}^{<\omega}2$ is recursively pointed if ...

**12**

votes

**1**answer

268 views

### Discrete subsets in the topology of pointwise convergence vs. metrisability

While reading Arkhangel'skii's Topological function spaces, I encountered an unexpected application of Martin's Axiom. This is Theorem II.5.20:
Assume $\mathsf{MA}+\neg \mathsf{CH}$. Let $X$ be a ...

**3**

votes

**1**answer

207 views

### How to change the successor of a singular with a Woodin?

I'm looking for references on how to change the successor of a singular cardinal from "more or less" minimal assumptions. If possible, then without adding bounded subsets to the singular either.
In ...

**3**

votes

**1**answer

212 views

### Consequences of ZFC+“$|x|\lt|y| \rightarrow |2^x|\lt|2^y|$”

This question stems from http://mathoverflow.net/a/6594/22332 and really is summarized in the title: Has there been any research on the power of $|x|\lt|y| \rightarrow |2^x|\lt|2^y|$? It seems like it ...

**6**

votes

**1**answer

191 views

### Consistency of Weak Diamond with a Weak Version of Martin's Axiom

If $S \subset \omega_1$ is stationary, then the weak diamond principle $\Phi(S)$ states that for any $F: 2^{<\omega_1} \to 2$, there is a $g: \omega_1 \to 2$ such that for all $f: \omega_1 \to 2$, ...

**4**

votes

**1**answer

196 views

### Another question on Borel sets and projections

Let $A$ be a (bounded) Borel set in $R^n$. Then we know that its projection $A_1$ on $R^{n-1}$ does not have to be Borel. But does $A_1$ have the following property?
Let $\mu$ be a given ...

**5**

votes

**1**answer

393 views

### Forcing is intuitionistic

The main idea of why it´s necessary a generic filter $G$ to extend a countable transitive $\epsilon$-interpretation (not necessarily a model) $M$ is given by the condition (for which $G$ being a ...

**8**

votes

**1**answer

218 views

### Erdős cardinals and $0^\sharp$

It is well-known that if the Erdős cardinal $\kappa(\omega_1)$ exists, then $0^\sharp$ exists, but what if $\kappa(\lambda)$ exists for a limit ordinal $\omega_1^L\leq \lambda<\omega_1$? Does this ...

**0**

votes

**1**answer

176 views

### A Question Regarding the Powerset Size Axiom

Consider the the Powerset Size Axiom, that is, the following:
(PSA) ($\forall$x,y) |x|$\lt$|y|$\Rightarrow$$2^{|x|}$$\lt$$2^{|y|}$.
Does there exist a class $\mathscr M$ of models of ZFC such that ...

**4**

votes

**1**answer

229 views

### Iterated ultrapowers of L

If there exists a measurable cardinal, we can generate a sequence of iterated ultrapowers $\{Ult_U^\alpha(V)\}_{\alpha\in ON}$. If $0^\sharp$ exists, i.e. if there exists an elementary embedding ...