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5
votes
3answers
188 views

In $L$, does there exist a definable non-principal ultrafilter on $\mathbb{N}$

The axiom of constructibility $V=L$ leads to some very interesting consequences, one of which is that it becomes possible to give explicit constructions of some of the "weird" results of AC. For ...
8
votes
1answer
206 views

Genericity by names

If $P$ is a notion of forcing in $M$, then $G$ is a $P$-generic filter over $M$ if $G\subseteq P$ is a filter, and for every $D\in M$ which is a dense subset of $P$, $G\cap D\neq\varnothing$. ...
8
votes
1answer
221 views

Preservation of some stationary sets by sufficiently closed forcing

The following statement can be proven using elementary submodels and sufficiently generic conditions: "If $S \subseteq cof(<\kappa) \cap \kappa^+$ is stationary, and $\kappa^{<\kappa} =\kappa$, ...
1
vote
0answers
108 views

Dedekind reals in heyting valued models

Let $V^{H}$ be a Heyting valued model of intuitionistic set theory. What conditions does $H$ have to satisfy in order for the following claim to hold? (where $\| \phi(u) \| \in H$ is the truth value ...
12
votes
1answer
289 views

Families of pairwise incomparable subsets of the integers

Certain maximal objects whose existence follows from Zorn's Lemma have received some set-theoretic attention. Examples are maximal independent families and maximal almost disjoint families. There is ...
3
votes
0answers
158 views

Strong measure zero sets and selection principles

A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...
13
votes
2answers
273 views

Can a Suslin line be 2-entangled?

A Suslin line is a linear order $L$ which is dense with no endpoints, complete, and ccc but not separable. I'm wondering what kind of order-preserving maps there are from $L$ into $L$. Specifically, ...
3
votes
1answer
113 views

Cardinality of the set of Boolean subalgbras of the lattice of projections on a Hilbert space.

A simple question I've managed to gey myself quite confused about. Given a Hilbert space H, what do we know about the cardinality of (a) the set P(H) of projection operators onto H (equivalently, ...
7
votes
0answers
270 views

“Hard” separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...
21
votes
1answer
372 views

When does the choice of the generic matter?

It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases ...
7
votes
0answers
106 views

Extending a Ramsey filter

We take filter to mean filter on $\omega$ containing all cofinite sets. We say a filter $F$ is Ramsey if ZERO does not have a winning strategy in the following infinite game between the two players ...
18
votes
2answers
2k views

Where are Georg Cantor's Original Manuscripts?

Georg Cantor is famous for introducing transfinite numbers and set theory. A main part of his mathematical point of view about this new type of "numbers" and this new "realm of mathematics" cannot be ...
8
votes
1answer
263 views

What is known about equiconsistency of PFA and existence of supercompact cardinals?

Question: What is the last status of known partial results and new approaches on the problem of equiconsistency of PFA and existence of supercompact cardinals?
4
votes
1answer
190 views

Increasing and Descending Chains of Inner Models for Measurable Cardinals

Notation: For each measurable cardinal $\kappa$ and a non-trivial $\kappa$-additive two-valued measure $\mu$ on it let $M_{\kappa,\mu}$ be the corresponding inner model. Question: Assuming suitable ...
4
votes
0answers
130 views

What is the meaning of restricting a Boolean value to a subalgebra?

$\require{AMScd}$ I am studying the proof that the ordering principle does not imply the axiom of choice in Jech's book "The Axiom of Choice" (Section 5.5). Let $P$ be the set of finite partial ...
5
votes
1answer
162 views

Order Types and Replacement Schema

Using Replacement Schema we can prove any well-ordering is isomorphic to an ordinal number. Q: Is the following consistent? $ZFC-Rep+\neg Rep+\text{Any well-ordering is isomorphic to an ordinal ...
10
votes
0answers
193 views

Analytic contraction of the Stone-Cech compactification of $\mathbb C$

Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$. Do the meromorphic functions separate the points of $S$? ...
25
votes
3answers
936 views

Latest stand of core model theory?

What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004: ...
10
votes
0answers
308 views

Intermediate submodels without Boolean algebras

My question is motivated by the following well-known fact regarding intermediate submodels of generic extensions. I would like to know if it can be proven using posets without the need for Boolean ...
5
votes
1answer
179 views

A question about the first Cohen model

Consider the first Cohen model, i.e. let $M$ be a countable transitive model of ZFC + $V=L$, let $\mathbb P$ be the poset consisting of finite partial functions from $\omega\times\omega$ to $2$, let ...
11
votes
2answers
456 views

Can we force the existence of this function?

I want to see if it is possible to force the existence of a function $F:\aleph_2 \times \aleph_2\rightarrow \aleph_1$ such that: a) $F(a,b)=F(b,a)$, for all $a,b\in \aleph_2$ and b) for all ...
10
votes
1answer
362 views

How strong is the iterated consistency of ZFC?

Let $T_0$ be $\mathsf{ZFC}$ and, for $n\in\omega$, set $T_{n +1}=T_{n}+\mathrm{Con}(T_{n})$. Question 1: Is there a natural number $n$ such that $T_{n}$ is equiconsistent with $\mathsf{ZFC}+$ ...
4
votes
1answer
144 views

How can I collapse all cardinals of ground model except one of them?

Let $\kappa$ be an uncountable cardinal of a c.t.m $M$ of $ZFC$. Is there a generic extension of $M$ like $M[G]$ such that all uncountable cardinals of ground model collapse except $\kappa$?
8
votes
1answer
169 views

A question about the Ordinal Definable elements of Power Sets

Assume that we are working in a set theory T, formalized in the language of ZFC, whose axioms---in addition to those of ZFC---include also the negation of "V=OD". For any set X, let CARD(X) denote ...
2
votes
0answers
90 views

Stability of analytic Zariski structures

Noetherian Zariski structures are introduced by Hrushovski and Zilber.(1996) An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield. ...
10
votes
1answer
218 views

Is “approximate categoricity” absolute?

Let $T$ be a countable first-order theory, and assume that $T$ has exactly one atomic model up to isomorphism in every uncountable cardinality. (By "atomic" I mean a model which omits the ...
0
votes
0answers
161 views

Ask for recommendation for the books about rough set theory

I am a graduate student of Shanxi Normal University in China and I'm very interested in the rough set theory.However,I cannot find some good books about this set.So,I want to get some information by ...
12
votes
0answers
330 views

Hausdorff gaps and $\mathfrak{p}=\mathfrak{t}$

Recently Malliaris and Shelah proved that $\mathfrak{p}=\mathfrak{t}$ (see: http://math.uchicago.edu/~mem/Malliaris-Shelah-CST.pdf). Their results are far more general, however for the specify context ...
5
votes
0answers
317 views

Does the following result hold in the full Solovay model?

In Solovay paper, "A model of set theory in which every set of reals is Lebesgue measurable", he cites in the end a theorem 4.1 (generalization of results), which says, roughly that in ...
6
votes
1answer
222 views

Recursive ordinals and the minimal standard model of ZF

Does the minimal standard model of ZF contain all recursive ordinals or is it limited (probably by the proof theoretic ordinal of ZF as I suspect but cannot prove)? Paul J. Cohen's definition of the ...
0
votes
0answers
72 views

Uncountable models of set theory [duplicate]

Assume we have a countable transitive model of set theory which satisfies some set of first-order axioms $T$. Is it possible to get from this a model $\mathcal{M}$ of set theory which satisfies the ...
4
votes
1answer
182 views

Bad subforcings of nice forcing notions

Let $\mathbb{P}$ and $\mathbb{Q}$ be two forcing notions. Recall that we say $\mathbb{Q}$ is a subforcing of $\mathbb{P}$ if there exists a regular embedding $\mathbb{Q} \to \text{r.o.}(\mathbb{P}).$ ...
10
votes
1answer
315 views

V=HOD & The Height of the Large Cardinal Tree

As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are ...
1
vote
1answer
148 views

Some questions about functions of Ordinal Numbers

Does there exist a strictly increasing and continuous function of ordinal numbers whose smallest critical number (i.e. fixed point) is the smallest non-constructive ordinal number (in the sense of ...
9
votes
1answer
203 views

Can “syntactic forcing” add ordinals?

Kunen, in paragraph VII.9 of his book talks about forcing "via syntactical models", where the we do not use set models of ZFC. Still, the functional $x \mapsto \check x$ can be defined as usual and ...
5
votes
1answer
148 views

extending elementary embeddings

Suppose $j : M \to N$ is an elementary embeddings between transitive models of ZFC. Everyone knows that if $G$ is $\mathbb{P}$-generic over $M$, $H$ is $j(\mathbb{P})$-generic over $N$, and $j[G] ...
6
votes
1answer
362 views

Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities?

Is there some suitable generalization to the notion of Baire property for topological spaces of arbitrary cardinalities which satisfies the following condition: The meager sets are sets which are ...
3
votes
1answer
132 views

Outer Definability of a Class

Definition: Let $C$ be a class of sets and $\mathcal{L}$ a first order relational language. We say $C$ is "outer definable" by $\mathcal{L}$ if there is a first order theory $T$ and for each $n_{R}$ - ...
5
votes
1answer
151 views

Intermediate submodels and the continuum hypothesis

Let $V$ be a model of $ZFC+GCH$ and let $V[G]$ be a generic extension of $V$ in which $CH$ fails. Question 1. Is there a model $W$ such that: 1) $V \subseteq W \subseteq V[G],$ 2) $W\models CH,$ ...
8
votes
1answer
333 views

Can there be only one (uncountable transitive model of ZFC)?

It is an immediate consequence of Cohen's forcing that if there is one countable transitive model of $\sf ZFC$, then there are many of them. Even if all of these models are of the same height, there ...
1
vote
0answers
142 views

Has the Ramified Theory of Types been applied to Predicative Set Theories?

Questions of predicativity are well-studied in the context of arithmetic. We have a base theory, first-order Peano arithmetic. Some people, like Edward Nelson (in chapter 1 of his book) and Charles ...
5
votes
2answers
395 views

How complicated is the formula expressing that a set is non-measurable?

The question is exactly stated by the title, i.e. how complicated is the formula $\psi(x)$ (in the language of set theory) expressing that a given set of reals $x$ is non-measurable? A second ...
5
votes
1answer
173 views

Definability in HOD

Let $\mathfrak{M}$ be a countable transitive model of set theory, and consider HOD (the hereditarily ordinal definable elements of $\mathfrak{M}$). Let $x$ be an object $x \in HOD$. So $x$ is ...
25
votes
2answers
867 views

Does Fermat's last theorem hold in the ordinals?

My question is whether there are no nontrivial solutions in the ordinals of the equations arising in Fermat's last theorem $$x^n+y^n=z^n$$ where $n\gt 2$, and where we use the natural ordinal ...
6
votes
2answers
405 views

The Theory of Transfinite Diophantine Equations [closed]

The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for ...
3
votes
1answer
124 views

The type of nondefinable elements-2

Consider a countable transitive model $\mathfrak{M}$ of set theory. Let $X$ be a definable collection of sets of reals. My question is: is the type of nondefinable elements in $X$ is definable over ...
4
votes
4answers
601 views

Prime numbers and limit ordinals

As a set, i.e. as a von Neumann ordinal, the $\omega$-th limit ordinal $\omega^2$ is fairly complex and not so easy to visualize (for the novice). But as an explicit well-ordering of $\mathbb{N}$, ...
2
votes
1answer
211 views

$(\kappa,\lambda)$ - Minimal Models & Stronger Version of Rowbottom's Theorem

Definition 1: Let $M$ be a $\mathcal{L}$ - structure and $A\subseteq Dom(M)$. Define: $Def_{A}(M):=\{X\subseteq Dom(M)~|~\exists n\in \omega~~\exists \varphi (x,y_1,...,y_n)\in ...
1
vote
1answer
243 views

$(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$

The notion of minimality in model theory is related to the existence of a gap in the size of definable subsets of a model. Now consider the following generalization: Definition 1: Let $M$ be a ...
2
votes
1answer
540 views

Subsets of Real Numbers (Edited & Revised Version)

Question 1: Is it consistent with $\text{ZF}$ that only countable subsets of $\mathbb{R}$ are well-orderable? Question 2: Is it consistent that for some $\lambda$, $\aleph_0 < \lambda < ...