**4**

votes

**0**answers

186 views

### random real forcing, independent real

Assume all independent reals that are added by random real forcing. Take enumeration of each independent real. Is the family of all enumerations dominating?

**4**

votes

**2**answers

284 views

### Functor category's objects fail to be a class?

Given two categories, we can form the functor category whose objects are functors. Functors by definition consist of two mappings from in general classes to classes, which makes it fail to be a set. ...

**3**

votes

**4**answers

223 views

### What is the role of absoluteness in existence of a non-trivial self elementary embedding on an inner model?

ِDefinition (1): Call an inner model M "rigid" if there are no non-trivial elementary embeddings $j: M\longrightarrow M$.
It is possible that $L$ is not rigid, while the problem is open for $HOD$. ...

**6**

votes

**1**answer

518 views

### A Hot Betting On HOD

Remark: This question is based on an open question at the end of a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency".
$HOD$ as an inner model of $ZFC$ lies ...

**21**

votes

**0**answers

1k views

### Possible troubles in Shelah's book “Cardinal Arithmetic”

I found some possible troubles in Observation 5.3(7) in the Chapter II of the Shelah's book "Cardinal Arithmetic" (page 86). For convenience, I quote the result and the proof in the book here ...

**3**

votes

**1**answer

278 views

### Is “ultracompact” taken?

Almost-huge cardinals are characterizable in terms of coherent towers of supercompactness measures, with a certain property of the direct limit model (see Kanamori's book). A useful large cardinal ...

**5**

votes

**1**answer

175 views

### Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute?

The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement
...

**11**

votes

**1**answer

627 views

### Can one cover the plane with less than continuum of lines?

I will be working in ZFC, but I am not assuming the Continuum Hypothesis (or Martin's Axiom). I know that it is consistent with ZFC that one can cover the real line with less than continuum of meager ...

**1**

vote

**1**answer

159 views

### Definable families of sets of reals

I apologize for asking the same question twice, since my last question was not really understood and there seems to be a problem preventing me from comment of editing the question.
Is there a model ...

**5**

votes

**1**answer

160 views

### Measurable and definable sets

Is there a model of set theory such that:
AC holds,
Every ordinal definable set is measurable,
Every ordinal definable set of sets of $\mathbb{R}^2$ whose projection on the first (or second axis) ...

**5**

votes

**0**answers

288 views

### Recent application of model theory in set theory by Shelah-Malliaris

Recently one of the oldest open problems in set theory about the cardinal invariants of the continuum (i.e the question of whether $\mathfrak{p}=\mathfrak{t}$) was solved by Shelah and Malliaris (see ...

**6**

votes

**1**answer

205 views

### Adding large sets not containing countable ground model sets

The question is motivated by Toni's question "Approximation of infinite set in generic extension" (see Approximation of infinite set in generic extension).
Before I state the question, let me add ...

**6**

votes

**2**answers

259 views

### Question about Woodin's stationary tower

Suppose that $\delta$ is a Woodin cardinal and that $\kappa$ is the critical point of the generic embedding $j:V\rightarrow M$ after forcing with the stationary tower ($\kappa$ can be $\omega_1$ or ...

**15**

votes

**1**answer

587 views

### Coherent trees: Is this result of Todorcevic correct?

A family of functions $F$ is coherent when for every $f,g \in F$, $\{ x \in dom(f) \cap dom(g) : f(x) \not= g(x) \}$ is finite. A tree on $\omega_1$ is coherent if it is a coherent collection of ...

**4**

votes

**2**answers

237 views

### On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$

The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an ...

**2**

votes

**1**answer

185 views

### Are measures of a measurable cardinal measurable? (Edited and Updated Version)

Update: Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers.
Definition: Let $\kappa$ be a measurable cardinal. Define:
...

**3**

votes

**3**answers

188 views

### Approximation of infinite set in generic extension

Suppose $M$ is a c.t.m and suppose $P$ is $Fn(I,2)$ where $I$ is infinite. Now suppose $G$ is $P$-generic, and $A \in M[G]$ is infinite set.
Is it guaranteed that the exist $B \in M$ such that $B ...

**2**

votes

**1**answer

175 views

### A question about additively indecomposable ordinals

I'm studying some topics about topological games of length $\alpha\geq\omega$, where I came across the following statement about additively indecomposable ordinals (recall that $\alpha$ is additively ...

**2**

votes

**2**answers

209 views

### Transfer of results from one model of set theory to another

Assume we showed that, in a certain transitive model of set theory, we have an isomorphism between two structures $M_1$ and $M_2$. Does the same result still holds in the real world?

**7**

votes

**2**answers

266 views

### Elementary Submodels in Partitions Theorems

I'am reading the paper Elementary Submodels in Infinite Combinatorics from Soukup (http://eprints.renyi.hu/45/1/elementary_submodels_revised.pdf) and there are a lot of proofs using elementary ...

**6**

votes

**0**answers

235 views

### Ultrafilter theorem and translation invariant measures

The usual Vitali construction of a non-Lebesgue measurable set generalizes to a proof that there are no (non-trivial) translation invariant measures on $\mathcal P\mathbb R$.
On the other hand, there ...

**3**

votes

**3**answers

360 views

### A Question on Special Forcings

The usual use of forcing begins with a "countable" and "transitive" ground model of $ZFC$ and reaches to a "countable" and "transitive" generic model of $ZFC$ with the "same ordinals". In a discussion ...

**5**

votes

**1**answer

135 views

### About non-stationary sets of $\omega_1$

Suppose $A$ is a non stationary set of $\omega_1$. Define by induction the following sequence of sets:\
$A_0 = A$
$A_{\alpha+1} = A_{\alpha}'$ [$X'$ is the subset of $X$, of all points the are ...

**8**

votes

**1**answer

284 views

### Nontrivially nontrivial automorphisms of $P(\omega_1)/fin$

Velickovic proved (Theorem 4.1 of OCA and automorphisms of $\mathcal{P}(\omega)/\mathrm{fin}$) that, assuming OCA and $\rm MA_{\aleph_1}$, every (Boolean algebra) automorphism of ...

**23**

votes

**3**answers

729 views

### Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?

This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal.
Question 1. Does every set of reals contain a measure-zero subset
of the same ...

**10**

votes

**1**answer

290 views

### Is the inclusion version of Kunen inconsistency theorem true?

The relations $\in$ and $\subsetneq$ seem so similar in some sense. For example they are equal on ordinal numbers. So there is a natural question about their possible similar behaviors on the ...

**5**

votes

**1**answer

160 views

### what kind of ordinal is the degree of strongness of a partially strong cardinal (Edited and revised)

For an infinite cardinal $\kappa$ and an ordinal $\lambda>\kappa,$ $\kappa$ is called $\lambda-$strong, if there is a non-trivial elementary embedding $j: V \rightarrow M$ with $crit(j)=\kappa$ ...

**3**

votes

**2**answers

404 views

### Are descriptive and ontological notions of equality equal? [closed]

Let $a$ and $b$ are two "objects". What is the meaning of $a=b$? This is one of the deepest problems of philosophy and logic because one needs a complete information about ...

**9**

votes

**1**answer

297 views

### Does existence of a proper class model imply the consistency?

The fundamental theorem of model theory says that:
Theorem: A first order theory is consistent if and only if it has a model.
In the above theorem we assume that the domain of any model is a ...

**5**

votes

**2**answers

255 views

### Can we force with Fraisse filters to solve Vaught's conjecture?

Around the classic Fraisse amalgamation theorem in model theory we have the following notions:
Definition (1): If $M$ be an $\mathcal{L}$-structure then define:
$age(M):=\lbrace N~|~N~\text{is ...

**2**

votes

**1**answer

191 views

### What is the order type of $L$ with Godel's well ordering?

In some sense $Ord$ is a "proper class" ordinal. Unfortunately the notion of a proper class ordinal is not a straight forward generalization of the notion of "set" ordinals because the proper classes ...

**2**

votes

**0**answers

146 views

### Is there a non-trivial consistency preserving transformation?

In set theory "equiconsistency" (and not "consistency") of the theories is the main part of researches. So we usually try to construct a new model using a given one. In the ...

**6**

votes

**1**answer

206 views

### the choice of representing formulas and Gödel's second incompleteness theorem

In Rautenberg's book (A Concise Introduction to Mathematical Logic, Universitext, Springer 2006), Gödel's second incompleteness theorem is stated:
Theorem 3.2 (Second incompleteness theorem). PA ...

**6**

votes

**1**answer

165 views

### What are the Possible Large Cardinals of $L[X]$?

I've been doing some basic reading in inner model theory, and I'm at the point where I've seen the definition of things like Martin-Steel and Mitchell-Steel inner models. I am wondering about the ...

**10**

votes

**1**answer

257 views

### Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?

Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense ...

**11**

votes

**1**answer

360 views

### Open coloring axiom vs. CH

Is there a simple, direct proof that the open coloring axiom contradicts CH (straight from the definitions, no machinery allowed)? The separable metric spaces version of OCA, if that helps.
Edit: ...

**11**

votes

**3**answers

602 views

### Is the class of n-dimensional manifolds essentially small?

Question: Consider the proper class of all $n$-dimensional smooth manifolds. If we take the equivalence classes where two manifolds are identified if there exists a diffeomorphism between them, is ...

**1**

vote

**0**answers

148 views

### A question on definable categories

One way to define a category set-theoretically might be to give four $\in$-formulas (not sets!)
$$\begin{array}{rl}
\mathsf{O}(X)&\text{(“$X$ is an object”)}\\
\mathsf{M}(X,Y,z)&\text{(“$z$ ...

**15**

votes

**0**answers

368 views

### Souslin trees on the first inaccessible cardinal

This may be well-known or simply deducible from the existing theorems, but I didn't find an answer in my set theory books:
Is there a model of $ZFC$ in which there are no $\kappa$-Souslin trees where ...

**6**

votes

**0**answers

157 views

### Proving equivalence of a tree-based version of Countable Choice for families of finite sets

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5:
Each of the following statements imply those beneath it.
The countable union of ...

**14**

votes

**1**answer

469 views

### Does “Higher Infinite” have a volume II?

Kanamori in the introduction of his famous book "The Higher Infinite" says that his book is the first volume of a complete book and the second volume is about large cardinals and forcing. I saw ...

**3**

votes

**1**answer

125 views

### Is there a maximal (or maximal Tychonoff) non normal space?

Is there a maximal (or maximal Tychonoff) non normal space? In "A Problem of Set-Teoretic Topology" the existence of a maximal Tychonoff space is asserted. Also there exists a perfectly normal maximal ...

**13**

votes

**3**answers

708 views

### Where is the end of universe?

In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...

**6**

votes

**1**answer

159 views

### Absoluteness of completeness

Suppose $V_0, V_1$ are (not necessarily well-founded) models of ZFC and suppose $\varphi$ is a first order sentence in a finite language $L$ (in our background model of set theory). Because every true ...

**10**

votes

**1**answer

152 views

### n odd: $\bf\Delta^1_n$ wadge degrees are $< \bf\delta^1_{n+1}$

My adviser is out of town and there is a comment in the Van Wesep paper "wadge degrees and descriptive set theory" that I can't figure out.
Work in ZF+AD throughout.
As stated in the title, the ...

**2**

votes

**0**answers

238 views

### Sorting of countabe set [closed]

Let $X$ be a countable ordered set. My question is very simple - Can we sort $X$ in countable number of steps?
When $X$ is finite, the answer is obviously yes. But what is the answer when $X$ is ...

**17**

votes

**2**answers

483 views

### Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr ...

**8**

votes

**1**answer

179 views

### Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...

**5**

votes

**1**answer

170 views

### Can we always permute Cohen reals?

Consider the Cohen forcing, and suppose that $\dot x,\dot y$ are names for reals, which are not in the ground model (i.e. $1$ forces that neither is in the ground model).
Can we always find an ...

**4**

votes

**1**answer

188 views

### What is the probability of an arbitrary nonlinear dynamical system to be chaotic?

Particularly, how to characterize a set of chaotic nonlinear dynamical systems as a subset of nonlinear dynamical systems with respect to the set cardinality?
To explain the question more, a simple ...