# set-theory's questions - English 1answer

3.115 set-theory questions.

### 2 Example of a Baire Class $1$ function $f$ satisfying $\omega\cdot n<\beta(f)\leq \omega\cdot (n+1)$ for some natural number $n\geq 1.$

Definitions: Let $X$ be a Polish space (separable completely metrizable topological space). A function $f:X\to\mathbb{R}$ is Baire Class $1$ if it is a pointwlise limit of a sequence of continuous ...

### 5 Negation of CH implied by lots of special subtrees?

In the following, I focus on trees of height $\omega_1$: if there exists a nonspecial tree any of whose $\aleph_1$-subtrees is special, must CH fail? Some neither consistent nor coherent thoughts: ...

### 3 Determinacy interchanging the roles of both players

3 answers, 554 views set-theory co.combinatorics game-theory
Let me refer to Jech's "Set Theory" Chap. 33 Determinacy: "With each subset A of $\omega^\omega$ we associate the following game $G_A$, played by two players I and II. First I chooses a natural ...

### 11 Can we have two different set theories extending Z and yet be bi-interpretable?

Here, it is mentioned that different set theories extending ZF are never bi-interpretable. Where different means "not theoretically equivalent",i.e. there must be a theorem that one has and the other ...

### 9 The Axiom of Determinacy and the Banach-Mazur game

The Wikipedia article on the Axiom of Determinacy (AD) claims: Equivalent to the axiom of determinacy is the statement that for every subspace X of the real numbers, the Banach–Mazur game BM(X) is ...

### 9 ZF + “every Suslin set of reals is ${\bf \Sigma}^1_2$”

What is known about the theory ($\ast$) ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"? By "reals" I mean elements of the Baire space $\omega^\omega$. For a cardinal $\kappa$, a set of ...

### 7 Is every set smaller than a regular cardinal, constructively?

Constructively, my only interest in regular cardinals is in terms of the "$\Sigma$-universes" they generate. By a $\Sigma$-universe, I mean a collection of triples $(X,Y,f: X \to Y)$ closed under base ...

### 11 Does Foundation increase the strength of second-order logic?

0 answers, 220 views set-theory lo.logic
Thinking about the recent threads on structural consequences of the Axiom of Foundation (AF) over ZF-AF, I've been trying to find some conservativity result which explains why AF doesn't seem to have ...

### 5 Ultrainfinitism, or a step beyond the transfinite

Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the TRANSFINITE. $\aleph_0, \aleph_1,\aleph_2\dots$ the lists ...

### -2 Can ${\cal P}({\frak{c}})$ be embedded into $\text{Sub}(\text{Sym}(\omega))$? [on hold]

Let $\text{Sym}(\omega)$ be the group of bijections $f:\omega\to\omega$ with composition. For any group $G$ let $\text{Sub}(G)$ be the complete lattice of the subgroups of $G$, ordered by set ...

### 8 Spaces without maximal homogeneous subspaces

A homogeneous space $(X,\tau)$ is a topological space such that for all $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x)=y$. As a previous question implies, the union of an ...

### 8 Is the measurable space $(\omega_1,\mathcal{P}(\omega_1))$ separable?

Here $\omega_1$ is the first uncountable ordinal, and $\mathcal{P}(\omega_1)$ denotes the power set of $\omega_1$. Separable means countably generated as a $\sigma$-algebra.

### 9 Does Easton forcing preserve measurable cardinals?

The question is in the title. For Easton's theorem see Wikipedia. Loosely speaking we can use forcing to manipulate the powerset function on regular cardinals as much as we like given we satisfy the ...

### 5 What is the consistency strength of this theory?

Language: first-order logic Primitives: $=, S, \in$ (the first denotes identity, the second denotes “is a successor of”, and the third denotes membership relation). Axioms: those of identity ...

### Borel class of multivalued mapping

I'm researching Borel class of mapping $G: 2^Y \mapsto 2^X$, where $X,Y -$ compact metric spaces, $f: X \mapsto Y$ $-$ 1st class Borel mapping and $G(A)=cl(f^{-1}(A))$ for $A \in 2^Y$. My idea of ...

### 47 The Logic of Buddha: A Formal Approach

Buddhist logic is a branch of Indian logic (see also Nyaya), one of the three original traditions of logic, alongside the Greek and the Chinese logic. It seems Buddha himself used some of the features ...

### 9 Embeddings of linear orders in $\wp(\omega)/Fin$ under Martin's axiom

We know that, under MA, every linear order $(X,\le)$ with $|X|<\mathfrak c$ embedds in $\wp(\omega)/Fin$. Does this hold for linear orders with cardinality $\mathfrak c$?

### 20 When will the real numbers be Borel?

1 answers, 672 views set-theory lo.logic forcing
In set theory Borel sets are important, but we don't actually care about the sets. We can about the Borel codes. Namely, the algorithm to generate a given Borel set starting with the basic open sets (...

### 6 Absoluteness of well-orderability

The property of well-orderability is upward absolute for transitive models of ZF: by Replacement in the smaller class, specifically Mostowski collapse, this is equivalent to the upward absoluteness of ...

### -3 What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to capture? [closed]

This question has been moved to philosophy.stackexchange.com I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by ...

### $\text{Max}\big(\text{Sub}(\text{Sym}(\omega))\setminus \{\text{Sym}(\omega)\}\big)$

If $G$ is any group, then by $\text{Sub}(G)$ we denote the collection of all subgroups, ordered by $\subseteq$. If $(P,\leq)$ is a partially ordered set we let $\text{Max}(P)$ and the set of maximal ...

### 3 The Rise and Fall of Dictators & How it Depends on Our Choice [closed]

This question is loosely inspired by the following paper of Shelah on Arrow property in which he answered a question of Gil Kalai affirmatively. Shelah, Saharon, On the Arrow property. Adv. in Appl. ...

### Embedding $(\mathbb{R},+)$ into $(\text{Sym}(\omega),\circ)$ [migrated]

0 answers, 70 views gr.group-theory set-theory
Is there an injective group homomorphism from $(\mathbb{R},+)$ into $(\text{Sym}(\omega),\circ)$, where $\text{Sym}$ denotes the set of all bijections $f:\omega\to\omega$? If not, is there a group ...

### 8 Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$

We say that $f:\mathbb{R}\to\mathbb{R}$ is of Baire Class $1$ if it is a pointwise limit of a sequence of continuous functions. One can generalize the definition above by taking pointwise limit of ...

### 11 Cops, Robbers and Cardinals: The Infinite Manhunt

Cops & Robbers is a certain pursuit-evasion game between two players, Alice and Bob. Alice is in charge of the Justice Bureau, which controls one or more law enforcement officers, the cops. Bob ...

### 19 Short proof of $\frak p=t$

1 answers, 1.248 views reference-request set-theory
It is known for a while now that $\frak p=t$, as a result of Malliaris-Shelah. The original paper draws from model theoretic methods. I've heard rumors that there was a proof which was purely set ...

### 2 Short proof of $\mathfrak{p}=\mathfrak{t}$ by Juris Steprans [duplicate]

2 answers, 166 views reference-request set-theory
I have just read this question Short proof of $\frak p=t$. The link present in the answer about the proof given by Steprans doesn't work anymore. Since I don't have enough reputation neither to talk ...

### Large complete minors of $\mathbb{Z}^\omega$

1 answers, 103 views graph-theory set-theory graph-minors
Let $x,y\in \mathbb{Z}^\omega$ and let $x,y\in\mathbb{Z}^\omega$ form an edge if there is $i\in\omega$ such that $|x_i - y_i|=1$ and $x_k = y_k$ for all $k\in \omega\setminus\{i\}$. $K_\omega$, the ...

### 10 Bijection from $\mathbb{R}$ to $\mathbb{R^2}$

Importantly, I am looking for a constructive proof (which does not rely on the Cantor–Bernstein–Schroeder theorem). Motivated by this discussion.

### 3 Does $\text{Sym}(\omega)$ have $2^{\aleph_0}$ pairwise non-isomorphic subgroups?

2 answers, 273 views gr.group-theory set-theory
Let $\text{Sym}(\omega)$ denote the set of all bijections $f:\omega\to\omega$ together with composition as group operation. Does $\text{Sym}(\omega)$ have $2^{\aleph_0}$ pairwise non-isomorphic ...

### 29 What is the dimension of the mathematical universe?

Forcing construction in set theory leads to a new understanding of the mathematical (multi)universe by providing a machinery through which one can construct new models of the universe from the ...

### 7 Reference Request: Existence of Ordinal Rank Theory?

Notations: Recall that $\omega_1$ is the first uncountable ordinal. Let $X$ be a Polish space (completely metrizable and separable) and $F(X)$ be the collection of all real-valued functions on $X.$ ...

### 9 How many iterations of inner models/generic extensions are sufficient?

0 answers, 202 views set-theory forcing
Let $M=M_0$ be a ctm of ZF. If $(M_0, \ldots, M_n)$ is a sequence of ctms of ZF, where for all $i,$ either $M_{i+1}$ is an inner model of $M_i$ or a generic extension of $M_i,$ then call $M_n$ an $n$-...

### 3 Selective ultrafilter on $\omega$ is normal. Clear proof

2 answers, 205 views set-theory ultrafilters
In this question I had asked about proof of the property of selective ultrafilter. As was answered, the proof is trivial if we know that ultrafilter is selective iff it is Ramsey ultrafilter. The ...