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6
votes
1answer
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
0answers
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
1answer
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
2answers
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
2answers
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
2answers
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
2answers
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
1answer
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 ...