# Tagged Questions

**6**

votes

**0**answers

99 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
...

**1**

vote

**1**answer

149 views

### Completing class-sized Fields

Let's say that an ordered Field is a class (proper or not) which satisfies the axioms of ordered fields. We work in NBG set theory with global choice.
Let's say that an ordered Field is real closed ...

**12**

votes

**0**answers

252 views

### The axiom $I_0$ in the absence of $AC$

It is well-known that if $AC$ holds and if $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ is a non-trivial elementary embedding with $crit(j) < \lambda,$
then $\lambda$ has countable cofinality (and in ...

**4**

votes

**2**answers

305 views

### Embedding property of weakly compact cardinals

One of the characterizations of $\kappa$ being a weakly compact cardinal is being inaccessible, and for every $\kappa$-model $M$, there is a [$\kappa$-model] $N$ and an elementary embedding $j\colon ...

**-1**

votes

**1**answer

97 views

### Subgraphs of $\mathbb{R}^2$ in the Hadwiger-Nelson problem

In the setting of the Hadwiger-Nelson problem, two points of $\mathbb{R}^2$ form an edge if and only if their distance is $1$. The resulting graph $G$ has chromatic number $\chi(G)\in \{4,5,6,7\}$ and ...

**6**

votes

**1**answer

242 views

### moving up a consequence of PFA

The Proper Forcing Axiom (PFA) implies that every forcing which adds a subset of $\omega_1$ either adds a real or collapses $\omega_2$. Is it consistent that every forcing which adds a subset of ...

**6**

votes

**0**answers

170 views

### Nonexistence of generic objects over $L(\mathbb{R})$

A well known result (stated and credited to Todorcevic in "Semiselective Coideals", by Farah, Mathematika, 1997, but with antecedents going back to Mathias) says that, under the appropriate large ...

**4**

votes

**2**answers

154 views

### Can one satisfaction class code another?

Let $M$ be a model of ${\sf ZFC}$. A satisfaction class $S$ for $M$ is subset of $M$'s ordered pairs which satisfies in $M$ the standard Tarskian compositional axioms. E.g.:
$M\vDash \forall \phi, ...

**9**

votes

**0**answers

231 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 ...

**4**

votes

**1**answer

248 views

### Large cardinals without choice?

For any given extension $T$ of ZFC (or perhaps NBGC or something), we can ask whether there is an extension $T'$ of ZF which does not prove AC such that
$Con(T) \leftrightarrow Con(T')$
$Con(T) \to ...

**4**

votes

**2**answers

221 views

### Is injectivity of $2^{(\ldots)}$ weaker than $\mathsf{GCH}$? [duplicate]

The following statement cannot be proven in $\mathsf{ZFC}$:
(S) : If $A, B$ are sets with $|A| < |B|$, then $2^{|A|} = |{\cal P}(A)| < |{\cal P}(B)| = 2^{|B|}$.
Obviously, ...

**4**

votes

**1**answer

201 views

### Does “Every infinite set is splittable” imply $\mathsf{AC}$? [duplicate]

We say an infinite set $X$ is splittable if there are $X_1, X_2\subseteq X$ with $X_1\cap X_2 = \emptyset$, $X_1\cup X_2 = X$ and there are bijections $\varphi:X_1\to X_2$ and $\psi:X_1\to X$.
Does ...

**5**

votes

**1**answer

267 views

### Cardinality of connected Hausdorff topologies

Let $X$ be an infinite set and let $C(X)$ denote the collection of connected Hausdorff topologies on $X$. Suppose $N\subseteq C(X)$ has the property that whenever $\tau\neq\sigma \in N$ then ...

**6**

votes

**1**answer

342 views

### Does the consistency strength hierarchy coincide with the “arithmetic consequence” hierarchy at ZF + Reinhardt?

In these slides (see especially slide 26), Steel emphasizes the phenomenon that for all known "natural" extensions of ZFC, the ordering by consistency strength agrees with the ordering by containment ...

**7**

votes

**1**answer

193 views

### $\mathsf{AD}_\mathbb{R}$ and Elementary Embeddings

Suppose $\mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R})) + \mathsf{DC}$ holds. (We can use more if it is helpful.)
I believe under $\mathsf{AD}_\mathbb{R}$, every $A \subseteq \mathbb{R}$ is ...

**3**

votes

**1**answer

176 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 ...

**6**

votes

**0**answers

119 views

### Forcing in GBC, the ctm approach

There is a nice, detailed survey about forcing in GBC in the appendix of the dissertation of Jonas Reitz. At page 115 the author wrote: " If $ \Gamma $
is a finite collection of sentences forced by $ ...

**6**

votes

**1**answer

192 views

### Theorem of Bukovsky characterizing ground models

It was mentioned in a talk that Bukovsky proved the following are equivalent for inner models $M \subseteq V$:
(1) There is a partial order $\mathbb P \in M$ and a $\mathbb P$-generic filter $G \in ...

**3**

votes

**1**answer

189 views

### “Lexicographic” ordering on ${\cal P}(\omega)$

For $A\neq B\in {\cal P}(\omega)$ we set $$\mu(A,B) = \min\big((A\setminus B)\cup (B\setminus A)\big).$$ We define $A < B$ if and only if $A \neq B$, and
$A = B\cap \mu(A,B)$ (that is $A$ is an ...

**5**

votes

**1**answer

108 views

### Can There be Rudin-Keisler Immediate Sucessors?

There are several well-studied orderings on the set $\omega^*$ of ultrafilters on the natural numbers. Three popular ones are $\le_i$ for $i = 1,2,3$. We define $\mathcal U \le_i \mathcal V$ to mean ...

**8**

votes

**1**answer

209 views

### Must $L_\alpha$ be correct about well-foundedness?

If $R \in L_\alpha$ is a binary relation so that $L_\alpha$ thinks $R$ is well-founded, must $R$ truly be well-founded? (Edit) That is, if $L_\alpha$ thinks that every nonempty subset of the domain of ...

**8**

votes

**1**answer

264 views

### Interpreting Robinson arithmetic in a very weak set theory

It is known that adjunctive set theory interprets Robinson arithmetic, and that extensionality is not needed for that. (Montagna and Mancini, "A minimal predicative set theory", Notre Dame Journal of ...

**4**

votes

**2**answers

222 views

### Least inner model of ZF without power set axiom

I'm interested in the existence and properties of an analogue version of $L$ for models of ZF$^-$ (ZF without the power set axiom), which for simplicity I'll call $L^-$. By "analogue" I mean the least ...

**35**

votes

**1**answer

726 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, $f(S)\subset Y$ ...

**23**

votes

**0**answers

444 views

### Can one divide by the cardinal of an amorphous set?

This question arose in a discussion with Peter Doyle.
It is provable in ZF that one can divide by any positive finite cardinal $k$: if $X \times \{1,\ldots,k\} \simeq Y \times \{1,\ldots,k\}$ then $X ...

**9**

votes

**1**answer

284 views

### Two questions about higher Souslin trees

Assume $V=L$ and let $\kappa$ be a Mahlo cardinal. Let $L[G]$ be the generic extension obatined by Mitchell forcing to make $2^{\aleph_0}=\aleph_2=\kappa.$
It is known that in the extension there ...

**6**

votes

**1**answer

192 views

### Fat stationary sets

Recall a stationary subset $S$ of a regular cardinal $\kappa$ is fat when for every $\alpha < \kappa$, and every club $C$, there is a closed set of order type $\alpha$ contained in $S \cap C$. It ...

**5**

votes

**1**answer

172 views

### Cofinality of countable ordinals in ZF, and in toposes

A countable limit ordinal $\kappa$ has cofinality $\omega$. One proves this in ZF, say, using the usual trick for representing $\kappa$ as a countable set of reals having closed convex span $[0,1]$ ...

**9**

votes

**2**answers

220 views

### Limits of rearranged sequences along ultrafilters

Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in ...

**2**

votes

**0**answers

115 views

### When do wide initial segments ruin admissibility?

Suppose $\alpha$ is admissible and $\beta<\alpha$. We know that $L_\alpha$ is an admissible set (by definition), but adding subsets of $\beta$ to $L_\alpha$ might break admissibility: while set ...

**15**

votes

**2**answers

611 views

### $\mathfrak{ufo}$: An unidentified combinatorial cardinal characteristic of the continuum?

An ultrafilter ornament is a chain of free filters on $\mathbb{N}$ that are not ultrafilters, whose union is an ultrafilter.
Let $\mathfrak{ufo}$ be the minimal cardinality of
an ultrafilter ...

**12**

votes

**0**answers

240 views

### If $U,D$ are $\kappa$-complete nonprincipal ultrafilters on $\kappa$ and $j_U(U) = j_D(D)$, is $U=D$?

Here, $\, j_U, \, j_D$ are the canonical elementary embeddings induced by $U,D$ respectively.
I note that it is consistent with the existence of a measurable that the answer be yes: it is true in the ...

**5**

votes

**1**answer

224 views

### Iteration of random reals

Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections ...

**3**

votes

**1**answer

88 views

### Levels of L resembling each other, take 2

(Everything below is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to ...

**3**

votes

**1**answer

87 views

### Generic sections of non-null sets are non-null

Consider the Cantor space $\mathcal C={}^\omega2$ with the usual product measure, and let $r$ be a random real (over a transitive model $V$ of ZFC). Let $B\subset \mathcal C^V\times\mathcal C^V$ a ...

**6**

votes

**1**answer

204 views

### Fine structure question: when do levels of $L$ look “a lot” like each other?

(Everything is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\},$$ ...

**2**

votes

**0**answers

305 views

### First few research papers [closed]

I was planning on posting this on academia.stackexchange, but I want an answer from mathematicians who've dealt with a similar issue when they were beginning graduate students. If this site doesn't ...

**5**

votes

**1**answer

214 views

### A button for individual reals

Hamkins introduced the notion of a "button" in forcing. This is a set-theoretic statement that can be forced, and can never be made false by further forcing. An example is $V \not= L$. Another ...

**5**

votes

**1**answer

238 views

### References for higher descriptive set theory surveys

A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and ...

**6**

votes

**3**answers

372 views

### When does the generalized Cantor space embed in a $\kappa$-compact space

The generalized Cantor space is the space $2^\kappa$, with basic open sets
$$
[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},
$$
for $\sigma\in 2^{<\kappa}$.
A space is $\kappa$-compact if ...

**8**

votes

**1**answer

284 views

### How long does the slow inefficient algorithm for computing the product in classical Laver tables take?

Let $(A_{n},*)$ denote the $n$-th classical Laver table. Let
$X_{n}$ be the set of all finite sequences of elements from $A_{n}$.
Define a function $E_{n}:X_{n}\rightarrow X_{n}$ by letting
...

**3**

votes

**0**answers

161 views

### Borel equivalence relations in models of determinacy

The following appears as fact 3.1 in the slides from Hjorth's 2010 Tarski Lectures. Assume ${L(\mathbb R)} \models \mathrm{AD}$.
Fact 3.1 For E and F Borel equivalence relations one has
$$E ...

**15**

votes

**1**answer

470 views

### Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?

Background. The field of Borel equivalence relation theory
provides a robust, unifying theory that organizes most of the
classification problems of classical mathematics into a hierarchy,
allowing us ...

**2**

votes

**1**answer

146 views

### Non-completeness of the Borel-Lebesgue measure and countable choice

Is it possible to prove the non-completeness of the Borel-Lebesgue measure on $\mathbb{R}$ (restricted to the Borel $\sigma$-algebra) without the full axiom of choice, but still with Countable Choice ...

**5**

votes

**0**answers

234 views

### Equivalence of the Banach-Tarski paradox

I am working on the Banach-Tarski paradox and the fact that the Hahn-Banach theorem implies that paradox. The proof involves the equivalence of the Hahn-Banach theorem and the fact that for every ...

**10**

votes

**0**answers

222 views

### A variant of strong ideals, is it consistent?

Is it consistent relative to large cardinals that there is a precipitous ideal on $\omega_1$ forcing a generic elementary embedding $j : V \to M \subseteq V[G]$, such that $j(\omega_1) = \omega_n^V$ ...

**11**

votes

**1**answer

310 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 ...

**1**

vote

**1**answer

126 views

### Tileable subsets of $\mathbb{Z}\times\mathbb{Z}$

For $t\in \mathbb{Z}\times\mathbb{Z}$ and $A\subseteq\mathbb{Z}\times\mathbb{Z}$ we set $t+A :=\{t+a: a\in A\}$.
Call $A\subseteq\mathbb{Z}\times\mathbb{Z}$ tileable if there is ...

**11**

votes

**1**answer

322 views

### Singular in $V$ regular in $HOD$

Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have:
(1) $\kappa$ is singular in $V$ of cofinality $\omega,$
(2) $\kappa$ is regular (and in fact ...

**12**

votes

**1**answer

470 views

### What is the motivation behind inner model theory?

Inner model theory aims to construct canonical inner models which captures as much of V as possible, which now is formulated more concretely as to build (fine structural) mice that contain many large ...