**6**

votes

**1**answer

205 views

### Stationary sets in HOD

My questions concern the following quote from “The HOD Dichotomy”, page 8.
"… notice that $\ cof(\omega)\cap\lambda$ belongs to $HOD$ even though it might mean
something else there. Also, ...

**11**

votes

**0**answers

227 views

### Are there additive subgroups of reals of dimension 1 with no subgroups of dimension strictly between 0 and 1?

I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero.
In the 1966 paper "Additive gruppen mit vorgegebener ...

**6**

votes

**1**answer

341 views

### On Consistency of an Existence

Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement:
For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with ...

**7**

votes

**2**answers

244 views

### Cofinality of a $\sigma$-ideal of $\mathbb{R}$

The cofinality of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is [EDIT: cofinal] in $P$, i.e. for every element $p\in P$ ...

**9**

votes

**2**answers

455 views

### Origin of the term “generic” in set theory

In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) ...

**6**

votes

**0**answers

216 views

### A characterization of Mathias reals

Recall that Mathias forcing is the poset $\mathbb{M}$ whose elements are pairs $(s,A)$, where $s$ is a finite subset of $\omega$, $A$ an infinite subset of $\omega$, and $\max(s)<\min(A)$. ...

**7**

votes

**2**answers

364 views

### A “suitably generic” set of Cohen reals without forcing?

I was reading a paper by David Marker, whose main theorem was that if $T$ is a first-order theory which is not small, then $F_2\leq_B \cong_T$. That's not especially relevant to the question at hand, ...

**3**

votes

**1**answer

99 views

### Ultrafilters of weight $\aleph_2$ in Sacks model

It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model ...

**5**

votes

**1**answer

193 views

### Is there a large-cardinal completeness theorem for $L$?

I cannot currently find the original, but if memory serves, Goedel once speculated that there might be a "large-cardinal completeness theorem for $V.$" This theorem would state:
*Theorem. For every ...

**5**

votes

**0**answers

101 views

### Question about the poset of inner models

Definition. Whenever $M$ is a countable model of $\mathrm{ZFC}$ (not necessarily well-founded) write $\mathrm{Inn}(M)$ for the poset of inner models of $M$ ordered by inclusion.
Question. For ...

**6**

votes

**0**answers

175 views

### Co-Heyting Valued Models of Paraconsistent Set Theory

I've been trying to do some forcing arguments in intuitionistic ZF using Heyting valued models where the Heyting algebra I'm using is actually a bi-Heyting algebra (both a Heyting algebra and a ...

**12**

votes

**2**answers

314 views

### Producing no non-constructible reals

The following is stated without proof in Shelah's book "Cardinal arithmetic" (page 276), and is attributed to Uri Abraham:
Suppose that $L[A], L[B]$ have no non-constructible reals and that ...

**2**

votes

**1**answer

212 views

### Partial interpretation of an iteration

Suppose that $\langle\mathbb{P_\alpha,\dot Q_\beta}\mid \beta<\delta,\alpha\leq\delta\rangle$ is a system of iterated forcing.
Let $\dot a$ be a name in $\mathbb P_\delta$, and let $G_\alpha$ be a ...

**11**

votes

**1**answer

349 views

### The independence number

I have been reading about cardinal invariants and I have a question about the independence number $\mathfrak{i}$. In Blass's paper (Combinatorial Characteristics of the Continumm) it is mention that ...

**9**

votes

**1**answer

327 views

### splitting subsets of cardinals

Suppose $\mathbb{P}$ is a separative partial order of uniform density $\kappa$, i.e. for all $p \in \mathbb{P}$, the least size of dense set below $p$ is $\kappa$. Does forcing with $\mathbb{P}$ add ...

**8**

votes

**1**answer

234 views

### Destroying the Mahloness of a cardinal with $\kappa$.c.c. forcing

Question: Is it possible to have a Mahlo cardinal $\kappa$ such that there is a $\kappa$.c.c. forcing that makes it non-Mahlo?
If this is possible then this forcing must change the cofinality of all ...

**4**

votes

**1**answer

214 views

### Choice principle strong enough to defy $V=L$

When using axiom of choice in proofs, people often say that this is non-constructive because AC gives us only proofs of existence, without giving explicit example. However, because in $L$ AC holds, we ...

**6**

votes

**1**answer

226 views

### Characterization of intermediate submodels of generic extensions

Question 1: Suppose that $V[G]$ is a set generic extension of $V$ by some forcing notion $P\in V,$ and suppose that $W$ is a model of $ZFC, V \subset W \subset V[G].$ Can we find a forcing notion ...

**2**

votes

**0**answers

317 views

### Topological proof of a result in Logic

I proved the result below using logic. My questions:
Can this theorem be proved by purely topological means?
Do you know any theorems that either can be used to prove the same result, or which give ...

**4**

votes

**1**answer

221 views

### Circulant matrix with integer entries and determinant 1 or -1

CONJECTURE
Let $A= (c_0,c_1,\ldots,c_n)$ be a circulant matrix, i.e if $(c_0,c_1,\ldots,c_n)$ is the first column of $A$ then the $i$th column of $A$ is obtained by applying the permutation ...

**9**

votes

**2**answers

823 views

### Is there any research on set theory without extensionality axiom?

In practice (say, in computer science), collections with many "labels" ("identities"), or collections which occur in many copies, are more frequently used than sets. Such collections do not satisfy ...

**4**

votes

**0**answers

120 views

### Has Quine's set theory NF been proved consistent (relative to ZF) [duplicate]

This question is an updated repetition of mathoverflow.net question The status of 'the consistency of NF relative to ZF' which I asked about a year ago. At that time the answer was that "the ...

**7**

votes

**1**answer

353 views

### Demuth's theorem in set theory

I am quite sure the following fact must have been known for set theorists, though I could not find it anywhere.
If $r$ is random over $L$ and $x\in L[r]\setminus L$, then there must be some real ...

**11**

votes

**0**answers

301 views

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

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 ...

**5**

votes

**2**answers

833 views

### Why can't mathematics be formalised in terms of classes rather than sets? [closed]

I've always been curious about the seeming compulsion to found mathematics upon sets, be it ZF(C) or some other system. Of course, there are other approaches these days like category theory and type ...

**0**

votes

**1**answer

158 views

### Possible no standard use of replacement axiom

The idea is to build in ZFC using replacement, a set REPLACEMENT(x ∈ A: TERM(x))
from a set and a term in the same way the set {x ∈ A: FORMULA(x)} is built
using specification from a set and a ...

**1**

vote

**1**answer

254 views

### Is there an analogue of Shoenfield's absoluteness theorem, but for $\mathrm{On}$?

From wikipedia:
Shoenfield's absoluteness theorem shows that $\Pi^1_2$ and $\Sigma^1_2$
sentences in the analytical hierarchy are absolute between a model $V$
of ZF and the constructible ...

**3**

votes

**1**answer

566 views

### Is there a “large powerset axiom” so extreme that it disproves the existence of strongly inaccessible cardinals?

If $\kappa$ is a strongly compact cardinal, then the singular cardinal hypothesis holds above $\kappa$. Hence the existence of large cardinals at the level of "strongly compact" or above is ...

**21**

votes

**1**answer

599 views

### Is there a model of ZF set theory with a set that does not inject into the cardinals?

Question. Is there is a model of ZF set theory with a set $X$ that does not inject into the cardinals?
I use the term "cardinal" here in the ZF sense, so they are not necessarily well-orderable.
To ...

**5**

votes

**1**answer

318 views

### A categorical characterization of ordinal numbers

It's rather easy to notice that the operation of join of categories reproduces the ordinal sum once restricted to act on (iso classes of) well-ordered set; it's rather easy to see that $\alpha\star ...

**1**

vote

**1**answer

222 views

### A conjecture about certain relations

In my research the following problem appeared (and if it is true, this solves positively several my conjectures):
Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set ...

**2**

votes

**1**answer

147 views

### Intermediate Extensions Determined by Reals

Does there exist a forcing $P$ which adds a generic real in the sense that $V[G] = V[x]$ for some $x \in ({}^\omega\omega)^{V[G]}$, and for all reals $y \in ({}^\omega\omega)^{V[G]}$, if $V[y] \neq ...

**6**

votes

**2**answers

226 views

### Only admissibles start gaps in clockable ordinals

This is a question about ITTM model introduced by Hamkins et al. In this paper it is proven that no admissible ordinal is clockable, so it either starts or lies within a gap in clockable ordinals. I ...

**3**

votes

**2**answers

336 views

### Is it consistent with ZFC that $\mathrm{dv}(\kappa) = \kappa$ for all infinite cardinal numbers $\kappa$?

Whenever $\kappa$ is an infinite cardinal number, write $L(\kappa)$ for the powerset of $\kappa$ ordered lexicographically. (Where the "$L$" stands for linear order.) Furthermore, write $B(\kappa)$ ...

**7**

votes

**1**answer

373 views

### Existence of infinite groups that are too reluctant to be topological

With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...

**2**

votes

**1**answer

225 views

### Absolutness of $\Pi_1^1$ statements

Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe.
But it is also known that $\Pi_1^1$ statements are ...

**21**

votes

**2**answers

1k views

### Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...

**5**

votes

**0**answers

358 views

### Last Status of Feferman's Conjecture on Indefinite Value of Continuum

The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...

**5**

votes

**1**answer

274 views

### Ultraproduct of Forcing Extensions & Forcing Extension of Ultraproduct

Notation:
$M[{\mathbb{P}}:G]$ denotes the forcing extension of $M$ by $\mathbb{P}$-generic filter $G$.
$\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$ denotes the ultraproduct of models using the ...

**5**

votes

**2**answers

171 views

### absorption of strategically closed posets

It is a (folklore?) fact that if $\kappa$ is a regular cardinal, and $\mathbb{P}$ is a $\kappa$-closed poset such that $\Vdash_\mathbb{P} |\mathbb{P}| = \kappa$, then $\mathbb{P}$ is equivalent to ...

**2**

votes

**1**answer

254 views

### Consistency of: “The continuum function is injective, and for all infinite cardinals $\kappa$ we have that $2^\kappa$ is weakly inaccessible.”

I asked here about "large powerset axioms" and to my delight, learned that such axioms are being taken seriously. I've been toying with them ever since. My favourite is: "The continuum function is ...

**5**

votes

**1**answer

191 views

### Scott sentence in models of set theory

Let $\mathfrak{M}$ be a countable transitive model of set theory.
Let $L$ be some countable language and $A$ be a countable (in $\mathfrak{M}$) $L$-structure.
My question is:
In $\mathfrak{M}$ can ...

**4**

votes

**1**answer

123 views

### Class theory with support for self-application of class functions?

To every natural number $n$, we can assign its Church numeral $\underline{n}.$ A formal definition would be:
$\underline{0}(f)=\mathrm{id}_{\mathrm{dom}(f)}$
$\underline{n+1}(f) = ...

**19**

votes

**1**answer

636 views

### Why isn't this a computable description of the ordinal of ZF?

In a previous MO question, I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...

**7**

votes

**1**answer

160 views

### Preservation of ultrafilters by Sacks forcing

It is well-known that, if $p$ is a Ramsey (selective) ultrafilter on $\omega$, then after adding a Sacks real $p$ remains an ultrafilter (well, it's really the upwards closure of $p$ the one that's an ...

**6**

votes

**0**answers

174 views

### Foundation scheme for $\Sigma_{n+1}$-formulas

I have trouble working out a proof in the second part of
Jean-Pierre Ressayre and Alex Wilkie. Modèles non standard en arithmétique et théorie des ensembles. Publications ...

**10**

votes

**0**answers

189 views

### Namba forcing and semiproperness

This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident.
Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...

**5**

votes

**2**answers

456 views

### A question about “small” uncountable cardinal numbers

If $X$ denotes a set, let $C(X)$ denote its cardinal number and let $P(X)$ denote its power set. There is a school of thought which considers any set having the cardinal number ...

**5**

votes

**2**answers

264 views

### For which cardinal numbers $\kappa$ is it consistent with ZFC that $\kappa^{\mathrm{cf}(\kappa)} < \kappa^\kappa$?

ZFC proves that $\kappa^{\mathrm{cf}(\kappa)} \leq \kappa^\kappa$ for all infinite cardinal numbers $\kappa$. Further, it is consistent with ZFC that we always have equality (e.g. assume GCH).
...

**0**

votes

**1**answer

81 views

### Question about the consistency of assuming (via axiom) that $\kappa < \nu$ for certain pairs of cardinal numbers provably satisfying $\kappa \leq \nu$

Call an ordered pair of formulae $\langle P(\kappa,\tilde{a}), Q(\nu,\tilde{a})\rangle$ in the language of $\{\in\}$ unproblematic iff
ZFC proves that for all $\tilde{a}$ and all cardinal numbers ...