**5**

votes

**2**answers

246 views

### Is it consistent that $\frak{d} < 2^{\aleph_0}$?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. We write $f <^* g$ if there is $N\in\omega$ such that $f(n) < g(n)$ for all $n>N$. A set $D\subseteq \omega^\omega$ is ...

**12**

votes

**4**answers

1k views

### Sierpinski's construction of a non-measurable set

In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to ...

**2**

votes

**3**answers

819 views

### Where do Set Theory and Number Theory meet together?

As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...

**12**

votes

**3**answers

699 views

### Complete resolutions of GCH

Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph_\alpha} = \aleph_{F(\alpha)}$ for all ...

**9**

votes

**0**answers

165 views

### Preserving Jonsson cardinals

I am (still) interested in trying to characterize and describe forcings that preserve Jonsson cardinals. A cardinal $\kappa$ is a Jonsson cardinal if there is no Jonsson algebra on $\kappa$, i.e. ...

**4**

votes

**1**answer

183 views

### Is there a “natural” bijection between models of $ZFC$ and $ZF\neg C$?

I'm quite confident that there is the same number of models of $ZFC$ and of $ZF\neg C$, which means that there would exist a bijection between the former and the latter, by definition. However, I was ...

**3**

votes

**1**answer

117 views

### Classify set theories whose transitive models sharing the same sets of ordinals are equal

This question is a follow-up from my recent question, Classifying set theories whose standard models sharing the same ordinals are equal
Let's say that a (recursively axiomatizable) set theory $T$ ...

**3**

votes

**1**answer

175 views

### A question on models of set theory and Lebesgue measure

In a question and an answer at MO, Joel David Hamkins showed that (if ZFC is consistent) there are models of ZFC in which $V\neq HOD$ and every $\Sigma_2$-definable set has a definable member.
Let ...

**12**

votes

**2**answers

287 views

### Classifying set theories whose standard models sharing the same ordinals are equal

Let's say that a (recursively axiomatizable) set theory $T$ extending ZF is "ordinal-categorical" if, whenever $M$ and $N$ are standard models of $T$ sharing the same ordinals, one has $M = N$. For ...

**11**

votes

**1**answer

553 views

### Is forcing computable?

By results similar to Tennenbaum's theorem we know that there exist no computable models of $ZF$. But suppose we are given, as a sort of oracle, access to some model of $ZF$ (e.g. we can make oracle ...

**30**

votes

**1**answer

783 views

### Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations,
where we can use forcing to prove the existence of finite objects with some ...

**0**

votes

**1**answer

212 views

### A question regarding $ZFC^{-}$

Given $ZFC^{-}$, that is, ZFC-Powerset+Collection+Separation, is there a set of alternative axioms $X$ (other than the trivial one, namely, {Powerset}) that, when added to $ZFC^{-}$, allow one to ...

**9**

votes

**1**answer

226 views

### Strongest large cardinal axiom compatible with $V = L$?

What is the strongest known natural large cardinal axiom compatible with $V = L$ (strongest in the sense that it implies all known "small" large cardinal axioms, where a large cardinal axiom is said ...

**0**

votes

**1**answer

123 views

### Lowering from filters to ultrafilters for an infinitary relation

Let $U$ be a set. Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation on $U$ (that is $f$ is a set of functions $N\rightarrow U$).
I denote $\mathcal{L}\in \upuparrows f ...

**12**

votes

**1**answer

682 views

### Lots of combinatorial interpretations of Catalan numbers

During a lecture I gave on Catalan numbers, I pointed out that that it
is possible to give a continuum number of combinatorial
interpretations of these numbers. See the solution to (f$^5$) on
page 54 ...

**-2**

votes

**1**answer

159 views

### Expressing a value related to an infinitary relation through ultrafilters

Let $U$ be a set. I denote $\mathfrak{A}$ the lattice of filters on $U$ ordered reverse to set theoretic inclusion of filters. I denote $\bigvee$ and $\bigwedge$ correspondingly the supremum and ...

**6**

votes

**1**answer

132 views

### Consistency of the collection axiom scheme compared to replacement

Set theory ZFC- is ZFC without power set, but with replacement. It does not imply the collection axiom scheme, as discussed in http://jdh.hamkins.org/what-is-the-theory-zfc-without-power-set/
Does ...

**2**

votes

**0**answers

69 views

### A question on recursion and transfinite recursion in extensions of KP

Is the $\Sigma_{n}$-recursion supported by $\Sigma_{n}KP=KP+\Sigma_{n}$-separation + $\Sigma_{n}$-collection equivalent with $\Sigma_{n}$ transfinite recursion? If not, how do these notions differ?

**2**

votes

**1**answer

111 views

### Interweaving two indexed families of filters

Conjecture
Let $U$ be an (infinite) set. Let $f$ be an $N$-ary (where $N$ is an
arbitrary index set) relation on $U$ (that is a set of functions $N
\rightarrow U$).
Let $\mathcal{L}_0$, ...

**2**

votes

**0**answers

80 views

### An analogue of CH for proper classes

Working in NBG set theory, with AC but without Global Choice, we ask for two proper classes A and B such that A strictly injects in B and B strictly injects in P(A); so
Question: In NBG set theory, is ...

**7**

votes

**1**answer

236 views

### Are there known ways to posit definable global choice in ZF without positing V=L?

I need a global choice function defined by a formula in (a fragment of) ZF. There is no harm in assuming V=L for my purposes. But I wonder if there are any familiar alternative ways to get this?
...

**10**

votes

**2**answers

319 views

### Semiproper but not proper

Assume V=L. Is there a semi-proper notion of forcing that is not proper?
Namba forcing isn't semi-proper in L, and Prikry forcing isn't even available there.

**9**

votes

**0**answers

168 views

### countable OD sets in the Solovay model

It is known that the following is true in the Solovay model (SM) for ZFC: any countable OD (ordinal-definable) set $X$ of reals necessarily consists of OD elements. What about countable OD sets of ...

**10**

votes

**3**answers

213 views

### First-order definable bijection between $P(On)$ (or $No$) and $V$? (Is this equivalent to $V = HOD$?)

It is known that locally one can ``code'' any set in the von Neumann universe $V$ by a set of ordinals. But can one do this globally? In other words, is there a first-order definable bijection ...

**3**

votes

**1**answer

142 views

### CCC Forcing and $\omega_1$ conditions

I have a question about the proof of the Lemma 7.2 in the paper
I. Juhász, P. Koszmider and L. Soukup,
A first countable, initially $\omega_{1}$-compact but non-compact space,
Topology and its ...

**11**

votes

**1**answer

354 views

### What are the current views on consistency of Reinhardt cardinals without AC?

It's well known that Reinhardt cardinals are inconsistent, provided that we have access to axiom of choice, but, as far as I know, we are clueless about this when we don't assume choice. For me, the ...

**5**

votes

**2**answers

425 views

### When do we have a bijection between a proper class A and its power set class P(A)?

We work in the set theory NBG with the axiom of (local choice but without global (class) choice. For every class A P(A) is the class of all sets x included in the class A.
We know that P(A) is a set ...

**3**

votes

**1**answer

134 views

### Formal systems needed to formalize relative independence results

We know that Con(ZF) implies Con(ZFC+GCH), Con(ZF+neg(AC)) and Con(ZFC+neg(CH)). But what are some weak theories in which these relative independence results are provable? In particular, are they ...

**3**

votes

**0**answers

356 views

### Can there be ordinals larger than those contained in Ord, and if so, can they be used to extend the constructible universe $L$?

Can there be ordinals larger than those contained in Ord, the class of all ordinals,and if so, can these ordinals be used to extend the constructible universe $L$?
In a simplified form, my question ...

**16**

votes

**2**answers

1k views

### Is non-existence of the hyperreals consistent with ZF?

I know that it is possible to construct the hyperreal number system in ZFC by using the axiom of choice to obtain a non-principal ultrafilter. Would the non-existence of a set of hyperreals be ...

**2**

votes

**1**answer

137 views

### A question regarding forcing extensions

Can one, for an infinite set A in ZFC, use forcing to add so many generic subsets of A as to make the collection of all subsets of A a proper class? Consider now a model $M$ of ZFC and use $Add( , ...

**1**

vote

**2**answers

137 views

### A Question Regarding Defining Generic Extensions of ZF and ZFC in Morse-Kelly Set Theory

It is known that Morse-Kelly (MK) set theory forms a metatheory for ZFC. For example:
MK proves Con(ZFC). In fact, Joel David Hamkins claims in his blog post "Kelly-Morse set theory implies Con(ZFC) ...

**2**

votes

**2**answers

132 views

### Order in bijective-equivalent collections of proper classes in set-theory

We work in the set theory NBG (with local choice, but not global choice), because if there is global choice, every proper class is well-ordered, so that every proper class is bijective with the class ...

**5**

votes

**1**answer

253 views

### Injection of the proper class of ordinals in every proper class

Is it possible to prove in the set theory NBG (with local choice but without global choice) that the proper class of ordinals injects in every proper class ?

**2**

votes

**1**answer

151 views

### Injection of every proper class in the ordinal class

Is it possible in the set theory NBG (with local choice but without global choice) that every proper class injects in the proper class of ordinals ?

**3**

votes

**0**answers

110 views

### Cores of infinite graphs

Let $\kappa$ be a cardinal and let $\textrm{Grph}(\kappa)$ be the set of graphs $G = (V,E)$ such that $V \subseteq \kappa$ and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$.
We ...

**7**

votes

**0**answers

265 views

### Two questions about universally measurable sets

I have two questions about universally measurable sets:
(1) Is there a universally measurable set of reals which does not have the Baire property?
(2) Is there a universally measurable set of reals ...

**4**

votes

**1**answer

232 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

353 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

154 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 ...

**5**

votes

**0**answers

123 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

138 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

176 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

127 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

798 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

279 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

101 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

193 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

191 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 ...