**-4**

votes

**0**answers

42 views

### Let Z be the set of integers, and consider the function f : Z → Z defined f(x) = 2x. Which of the following is correct? [on hold]

a) f is invertible
b) f is injective
c) f is bijective
d) f is surjective
I put b, though apparently that's wrong.
Thanks :)

**0**

votes

**0**answers

8 views

### LYX: how to mark power set? [migrated]

This is my first time here and I did my best to figure out if such a question is in place here. If I missed something I apologize.
My questions is how can I mark a power set in the lyx document ...

**12**

votes

**1**answer

435 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 ...

**9**

votes

**2**answers

368 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

243 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

94 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

174 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

160 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

182 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

**0**answers

168 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 ...

**4**

votes

**0**answers

63 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 ...

**6**

votes

**1**answer

219 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

364 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

276 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 ...

**2**

votes

**1**answer

140 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 ...

**2**

votes

**1**answer

120 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

322 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

973 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

252 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

277 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

149 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

358 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

231 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

153 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

128 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

194 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

220 views

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

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

**7**

votes

**1**answer

260 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 ...

**6**

votes

**0**answers

211 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 ...

**4**

votes

**1**answer

578 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

449 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

200 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

101 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

1k 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

224 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

149 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

245 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

196 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

193 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 ...

**5**

votes

**1**answer

176 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

185 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 ...

**4**

votes

**1**answer

318 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

187 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

157 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

200 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 ...

**3**

votes

**1**answer

150 views

### Does existence of $\omega_1$ subset of reals imply $\omega_1$ choice for subsets of reals?

Suppose there exists a subset of $\Bbb R$ which has cardinality $\omega_1$. Is it then necessarilly true that for every collection of $\omega_1$ subsets of $\Bbb R$ there exists a choice function?
I ...

**3**

votes

**1**answer

165 views

### Is Every New Real in the Silver Extension a Silver Generic Real?

Let $\mathbb{V}$ denote Prikry-Silver forcing. That is, $\mathbb{V}$ is forcing with partial functions $\omega \rightarrow 2$ with coinfinite domain or forcing with uniform trees.
Let $\dot x$ ...

**6**

votes

**1**answer

322 views

### Explicit examples of undetermined games

Suppose we have a game between two players in which they take alternating turns. The game can have finite length, length $\omega$ or any transfinite number of steps (however, I'm not concerning games ...

**10**

votes

**1**answer

218 views

### On the definition of the $\alpha$-iterable cardinals

I am reading the paper Ramsey-like cardinals II by Victoria Gitman and Philip Welch (Journal of Symbolic Logic, vol. 76, no. 2. pp. 541-560, 2011) and maybe I am missing something.
According to the ...

**9**

votes

**4**answers

481 views

### Boolean Valued Models of PA

O.K, a massively naive question. I've never really studied any non-standard models of PA before. I was just wondering if there's ever been any attempt to use the kind of Boolean valued model theory ...