forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence ...

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2
votes
1answer
41 views

Suslin lines hereditarily Lindelof

I need to prove that every suslin line is hereditarily Lindelof. Any idea will be helpfull.
3
votes
1answer
179 views

What year was Hechler forcing created?

Hechler forcing is described on page 278, Jech. Does anyone know when Hechler forcing was first used in a publication?
4
votes
0answers
95 views

$\infty$-Borel Determinacy?

An $\infty$-Borel set is a set $X\subseteq\mathbb{R}$ which has an $\infty$-Borel code - a set $r$ coding the construction of $X$ via open sets, complementation, and well-ordered unions (see ...
-2
votes
0answers
76 views

Formal definition of function: equality [on hold]

Consider this formal definition of function, which does not require $A$ to be domain. \vspace{0.5cm} \textit{Let $A$, $B$ be sets, then $F\subseteq A \times B$ is said to be a function iff ...
13
votes
0answers
158 views

A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$? This question is motivated ...
4
votes
2answers
145 views

Generic Ultrapower as a Class

If $X$ is a set and $I$ is an ideal on $X$. Let $\mathbb{P}$ be the forcing poset consisting of $I^+$ subsets of $X$ with the subset partial ordering. Let $G$ be $\mathbb{P}$-generic filter over $M$, ...
8
votes
0answers
188 views

cardinals below the critical point of a generic embedding

This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension? To focus on the ...
5
votes
0answers
338 views

Different approaches to the multiverse of sets

There are some different approaches to the multiverse of sets, in particular: 1) The approach by Woodin, 2) The approach by Sy Friedman, ..., 3) The approach by Hamkins. I wonder to know if ...
9
votes
1answer
289 views

Applications of SCH outside of set theory

Recall that the Singular Cardinals Hypothesis (SCH) says that if $\kappa$ is a singular cardinal and $2^{cf(\kappa)}<\kappa,$ then $\kappa^{cf(\kappa)}=\kappa^+.$ Clearly it has many applications ...
7
votes
2answers
197 views

Forcing with Nontransitive Models

A common approach to forcing is to use countable transitive model $M \in V$ with $\mathbb{P} \in M$ and take a $G \in M$ (which always exists) to form a countable transitive model $M[G]$. Another ...
7
votes
2answers
441 views

$\aleph$ looks like $\mathbb N$?

We all know the notation $\aleph_\lambda$ for the $\lambda$th (or, I guess, $\lambda+1$st) infinite cardinal number; in particular $\aleph_0$ is the cardinality of the the set of natural numbers ...
-5
votes
0answers
64 views

How to replace set of large number of elements with set of smaller number of elements? [closed]

Suppose s={12,45,150,320,400,550,600} applying any mathematics formula or theorem i)find p={a,b,c,d}(number of elements in p is lesser than s) such that applying any mathematical formula or theorem ...
11
votes
0answers
290 views

Does every Aronszajn tree has a Suslin or a Special subtree?

Question: Does every $\omega_1$-Aronszajn tree contains a Suslin sub-tree or a special Aronszajn sub-tree? Recall that Suslin trees are $\omega_1$-trees (trees of height $\omega_1$, and countable ...
14
votes
2answers
386 views

Pathological behavior of Borel sets?

Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the ...
5
votes
1answer
167 views

$\omega$ universally Baire sets, tree representations

I've recently encountered the notion of a universally Baire set, and I've tried to look at the paper by Feng, Magidor and Woodin where this notion is studied. There are several points that confuse me. ...
0
votes
0answers
12 views

the lexicographix order [migrated]

If it is given ordinals $\alpha$ and $\beta$, the lexicographix order on $\alpha \times \beta$,$\leq_{lex}$ is given by: $(\gamma_0,\delta_0)<_lex(\gamma_1,\delta_1)$ if and only if either ...
6
votes
1answer
174 views

regularity of ultrafilters

An ultrafilter $U$ is $(\mu,\kappa)$-regular if there is a sequence $\langle X_\alpha : \alpha < \kappa \rangle \subseteq U$ such that for all $y \in [\kappa]^\mu$, $\bigcap_{\alpha \in y} X_\alpha ...
8
votes
2answers
326 views

Covering the space by disjoint unit circles

Sierpinski has proved the following two interesting theorems. Theorem 1. The Euclidean plane $\mathbb{R}^2$ is not a union of nondegenerate disjoint circles. Theorem 2. The Euclidean space ...
7
votes
2answers
269 views

Which linearly ordered sets have the property that their completion is equipotent with their powerset?

As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the ...
9
votes
0answers
207 views

Souslin trees and weakly compact cardinals

In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$. In this question I would like ...
9
votes
1answer
284 views

Does there exist a supercompactness theorem?

Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of ...
8
votes
2answers
244 views

The (non-)absoluteness of second-order elementary equivalence

Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, ...
6
votes
0answers
131 views

Canonical model for $\neg\mathsf{CH}$ and $\Omega$-logic

Recently I found this book by Woodin. In the introduction of it the author writes the following: The main result of this book is the identification of a canonical model in which the Continuum ...
5
votes
0answers
104 views

A result of Steel on characterizing lightface pointclasses

In the article Projectively wellordered inner models, Steel proves the following theorem (4.12): Theorem: Let $n < \omega$ and suppose $\mathcal{M}_n^{\sharp}$ exists. Let ...
7
votes
1answer
118 views

On the definition of $\alpha$-proper poset

I am reading Uri Abraham's chapter on Proper Forcing in the Handbook of Set Theory and I have a quite trivial question on the definition of $\alpha$-proper forcing. Since there are many equivalent ...
5
votes
2answers
151 views

Can we have a $\kappa$-Suslin tree where $\kappa$ is above a measurable cardinal?

Question: Can we have a set theory in which there exists a $\kappa$-Suslin tree with $\kappa$ larger than the least measurable cardinal? A $\kappa$-Suslin tree is a tree with levels indexed by ...
-3
votes
1answer
117 views

is the existence of an inaccessible cardinal stronger than just CON(ZFC)? [closed]

is it even stronger than that ZFC has a transtitive model?
1
vote
3answers
262 views

Why doesn't choice imply global choice (in NBG)?

I thought ZFC proved the existence of an inductive well-ordering that is itself a set for any stage of V. NBG with only the regular AC should then prove/assert the existence of a class R of ordered ...
5
votes
1answer
160 views

questions about worldly cardinals

A cardinal $\kappa$ is worldly if $V_\kappa$ is a model of ZFC. How many $\beth$-fixed points are there smaller than the smallest worldly cardinal? How many worldly cardinals are there smaller than ...
15
votes
1answer
232 views

Linear maps between arbitrarily chosen vectors of vector spaces $V$ and $W$

I recently came across this question: Is the axiom of choice needed to prove the following statement: Let $V, W$ be vector spaces, and suppose $V \neq \{0\}$. Let $v \in V$, $v \neq 0$, $w \in W$. ...
5
votes
1answer
227 views

Woodin Cardinals and Inner Models

I have a few questions I have been thinking about that I could definitely use some insights on: Question 1. Since a Woodin cardinal is a "local" notion, defined with respect to some rank-initial ...
8
votes
2answers
1k views

Is there a Hotel California of set-theoretic geology?

Is there a universe which can always be forced to, which never can be forced from?
5
votes
1answer
307 views

Order homomorphism functions on $\omega_1$

Let $\omega_1$ be the first uncountable ordinal, same as the set of all countable ordinals. Let $F$ be the set of all regressive functions $f$ from $\omega_1$ minus singleton $0$ into $\omega_1$, ...
1
vote
0answers
148 views

Countable axiom of choice: why you can't prove it from just ZF [migrated]

This is a follow-up question to the discussion about the finite axiom of choice here. Suppose we have a countable collection of non-empty sets {A_1, A_2, A_3, ....} Reasoning as indicated in that ...
9
votes
1answer
307 views

Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups?

Let $X_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Also ...
1
vote
1answer
150 views

Would a non-constructible set become constructible if we had oracles of arbitrarily high cardinality for the halting problems of ordinal computers?

I still have trouble to grasp the concept of a non-constructible set, my intuition is that we could "avoid" the non-constructibility of many of them if we assume we have "ordinal computers" extended ...
-3
votes
0answers
22 views

What if union of disjoint sets results in universal set? [migrated]

I have a question related to set theory. If A1,A2,A3....An belongs to universal set U And If all of the sets are disjoint i.e. Ai intersect Aj = {} for all i and j. And If their union equals to ...
3
votes
1answer
158 views

Adding Generic Reals to Forcing Extensions

I'm following the Jech's Multiple Forcing for a seminar group and I intend to show how to add one or some reals to extensions. I studied Solovay's model and I can see why learning how to add random ...
16
votes
2answers
431 views

Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point. Is the notion of FPP for topological spaces an absolute notion? More ...
1
vote
1answer
269 views

Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...
3
votes
1answer
144 views

Is there a well-defined notion of dimension for $\mathcal{L}$-structures? [closed]

Model theorists usually refer to a $\mathcal{L}$-structure $M$ as a "world". Note that in physics and many parts of mathematics when we refer to an object as a "world" or "space" we usually can ...
10
votes
1answer
368 views

Ground Axiom and behaviors of continuum function

The Ground Axiom ($GA$) is the assertion that the universe of sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing $P\in W$. Is $GA$ consistent with any possible ...
4
votes
2answers
434 views

What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?

Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...
1
vote
1answer
74 views

Is there such a sufficient condition for “$X$ is a stationary subset of uncountable regular $\kappa$” involving limit points?

If $X$ is a set of ordinals, define $f(X, \alpha)$ such that $f(X, 0) = X$, $f(X, \beta + 1) = f(X, \beta) \cap Lim(f(X, \beta))$, where $Lim(Y) = \{$limit points of $Y\}$, and $f(X, \gamma) = \bigcap ...
4
votes
1answer
199 views

Perfect set property implies $\omega_1$ is a limit cardinal in $L$

Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal. The original proof is in German, and I've been ...
2
votes
0answers
97 views

Does the Lévy collapse obey this nice characterization? [duplicate]

This question is related to an issue in my answer to Monroe Eskew's question on the failure of Cantor-Bernstein for the Lévy collapse. Question. Is the Lévy collapse $\text{Coll}(\omega,\lt\kappa)$ ...
6
votes
1answer
161 views

Failure of Cantor-Bernstein for the Levy Collapse

Related to this question, is it possible to give an example of the failure of Cantor-Bernstein for complete embeddings of forcing notions involving the Levy Collapse $Col(\omega,<\kappa)$? Suppose ...
8
votes
1answer
140 views

Does being special on a club imply being special?

Let $T$ be an Aronszajn-tree, $C\subset \omega_1$ a club set and $f:\bigcup\limits_{\alpha\in C}T_\alpha\longrightarrow \mathbb Q$ a strictly increasing function (where $T_\alpha$ is the ...
5
votes
1answer
74 views

Property of $L$ Relating to Reflection

The idea of the question is whether it is ever possible that $L$ is so nice in the sense that $\{L_\alpha\}$ does not incorrectly "guess" a bigger inaccessible than $L$ really has, as long as ...
5
votes
1answer
203 views

A model of Krivine

In a paper by J.-L. Krivine, Modèles de ZF+AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue [C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A549–A552, ...