Questions tagged [set-theory]

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 relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.

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Long chains of amorphous cardinalities

An amorphous set is an infinite set that cannot be partitioned into 2 infinite subsets. An amorphous cardinality is the cardinality of an amorphous set. Working in $\sf ZF$, it is consistent that ...
Ynir Paz's user avatar
  • 153
5 votes
0 answers
93 views

Do precipitous ideals "always" come from collapsing?

It's well-known that if $\kappa$ is a measurable cardinal, then there is a poset $\mathbb{P}$ that forces $\kappa$ to carry a precipitous ideal. Suppose that $\omega_1$ carries a preciptous ideal $I$. ...
Toby Meadows's user avatar
  • 1,101
1 vote
0 answers
47 views

Uniformization and functions on Turing degrees

Assuming Martin's Conjecture on functions between Turing degrees, is AD + DC consistent with existence of an $f:\mathcal{D}_t → \mathcal{D}_t$ of rank $Θ$ ? $\mathcal{D}_t$ is the set of Turing ...
Dmytro Taranovsky's user avatar
1 vote
1 answer
326 views

If parameters in ZFC are restricted to definable sets, can existence of uncountable sets be proven?

If we restrict all parameters in the set building axioms of $\sf ZFC$ to definable sets, would the celebrated Cantor's theorem still apply? Can existence of uncountable sets be proven at all? The set ...
Zuhair Al-Johar's user avatar
10 votes
1 answer
266 views

Building the real from Dedekind finite sets

It is well known that the real numbers can be countable union of countable sets by starting with GCH and taking a finite support permutations while collapsing all of $\aleph_n$ for natural $n$. The ...
Holo's user avatar
  • 1,633
9 votes
1 answer
435 views

Does proper forcing preserve properness under PFA?

I'm interested in forcing classes $\Gamma$ which preserve membership in themselves, i.e. for all posets $\mathbb{P}, \mathbb{Q}\in \Gamma$, we have $\Vdash_{\mathbb{P}}\check{\mathbb{Q}}\in\Gamma$. ...
Ben Goodman's user avatar
5 votes
1 answer
342 views

Second-order ordinal definability

As is familiar, a set $S$ is ordinal definable ($S \in \mathsf{OD}$) just in case there exists a formula $\varphi(x, \vec{z})$ of the first-order language of set theory with parameters $\vec{z} = z_1, ...
Beau Madison Mount's user avatar
-4 votes
0 answers
81 views

What is the Measure of the Permutations of the Real Numbers? [closed]

Since the permutations of the real numbers would form a set of cardinality $\aleph_2$, do we just say it has infinite measure? It would seem to be the case for a Lebesgue measure. Is there a standard ...
Carl Gueck's user avatar
8 votes
1 answer
303 views

Consistency strength of strongly compact cardinal

Where can I find a proof that strongly compact cardinal has higher consistency strength than Woodin cardinal, or even just strong? Recall that a strongly compact cardinal itself may not be strong, ...
Lxm's user avatar
  • 313
0 votes
0 answers
115 views
+50

What is the strength of adding this de-schematizing inference rule to Ackermann's set theory?

Language: first order logic with equality, membership, and a constant symbol $W$. Axioms: Extensionality: $\forall z \, (z \in x \leftrightarrow z \in y) \to x=y$ Comprehension: $\exists x \forall y \,...
Zuhair Al-Johar's user avatar
-5 votes
0 answers
76 views

Can ur-elements be used as/to construct infinitesimals?

Background material: Truss[95], "The structure of amorphous sets." Harrison-Trainor and Kulshreshtha[22], "The Logic of Cardinality Comparison Without the Axiom of Choice." ...
Kristian Berry's user avatar
0 votes
0 answers
104 views

Is this modified H. Friedman theory bi-interpretable with ZFC + ORD is Mahlo?

The following theory is a modification of Harvey Friedman $\sf K(W)$ theory. Language: first order logic with equality, membership, and a constant symbol $W$. Axioms: Extensionality: $\forall z \, (z ...
Zuhair Al-Johar's user avatar
6 votes
0 answers
107 views

From HODs to corresponding models of AD

If $M$ is HOD of a model $N$ of $\text{AD}^+ + V=L(P(ℝ))$, what kind of forcing construction in $M$ gives back such an $N$? HODs for $\text{AD}^+ + V=L(P(ℝ))$ are conjectured (and under anti-large ...
Dmytro Taranovsky's user avatar
-4 votes
0 answers
155 views
+50

CH vs Not CH, What is the Consequence?

EDIT: Altered (3) to say that $\beth_1$ can't be $\aleph_{\omega}$ or $\aleph_0$ but removed statement that it must be less than $\aleph_{\omega}$. Let us assume ZFC. We now consider 2 transfinite ...
E8 Heterotic's user avatar
6 votes
1 answer
154 views

An iteration of proper forcing without proper iterands

Famously, a countable support iteration with proper iterands is again proper. This is mostly stated as follows: Suppose $(\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha})_{\alpha<\lambda}$ is a ...
Hannes Jakob's user avatar
  • 1,612
5 votes
0 answers
181 views

Friedman's proof of covering lemma for $L$

There is a two-page proof of the covering lemma for $L$ using $\Sigma_n$ substructures (Theorem 3.10) in Sy Friedman's Fine Structure and Class Forcing, compared to the proof that spans about twenty ...
Lxm's user avatar
  • 313
6 votes
1 answer
133 views

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
Alexander Osipov's user avatar
2 votes
2 answers
191 views

Name for a certain type of cardinal

I'm not a set-theorist, but I hope this question is appropriate. This is just a question about names: Fix a cardinal $\lambda$. I'd like to know if there is a name for regular cardinals $\kappa$ such ...
Maxime Ramzi's user avatar
  • 13.3k
12 votes
1 answer
673 views

Can proper classes have different sizes?

I'm presently working in a non-ZF set theory, where there are proper classes. (Think MK or VNBG.) And I'm interested in how to think about the possibility (or impossibility) of proper classes with ...
Anonymous grad student's user avatar
1 vote
0 answers
158 views

How does the cardinality of a set and its powerset compare in the hereditarily rank-concordant constructible world?

Working in the constructible universe "$L$", if we define two kinds of ranks for any constructible set $x$, one being the ordinal index of the first $L_\alpha$ where $x$ appears as a subset ...
Zuhair Al-Johar's user avatar
5 votes
1 answer
205 views

Is the partition tiling relation transitive?

The following is motivated by an (as of yet) unanswered question on optimal colorings of graphs. I am convinced that the question below has a positive answer in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$, but ...
Dominic van der Zypen's user avatar
4 votes
1 answer
192 views

Simplified method of building an Aronszajn tree

There is a very interesting method to build an Aronszajn tree in Judith Roitman's "Introduction to Modern Set Theory", on pages 100-102. In short, we build a tree $T$ in which the nodes are ...
Mike Battaglia's user avatar
8 votes
1 answer
186 views

A reference for forcing projections

The idea of a projection $\pi\colon\mathbb{Q}\to\mathbb{P}$ of forcing notions is something like a combinatorial stand-in for the fact that forcing with $\mathbb{Q}$ produces a generic for $\mathbb{P}$...
Miha Habič's user avatar
  • 2,289
3 votes
1 answer
170 views

Is there a metric separable space with the following properties...?

Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$. Is there a metric separable space $X$ with the following properties: $|X|\geq\...
Alexander Osipov's user avatar
6 votes
0 answers
217 views

Do maximal compact logics exist?

By "logic" I mean regular logic in the sense of abstract model theory (see e.g. the last section of Ebbinghaus/Flum/Thomas' book). My question is simple: Is there a logic $\mathcal{L}$ ...
Noah Schweber's user avatar
-3 votes
0 answers
216 views

Are there known examples like this almost official exposition of ZFC that is very weak?

Pseudo-ZFC is a theory written in the usual language of set theory, i.e. mono-sorted first order logic with equality and membership. The extra-logical axioms are: Extensionality: $\forall x \forall y:...
Zuhair Al-Johar's user avatar
5 votes
1 answer
240 views

How far does the intuition "non-stationary = null sets (of certain measure)" go?; what position do non-stationary ideals take in measure theory?

When I was taught undergraduate set theory I was told that the idea for club sets is that they are "of full measure" and the non-stationary sets are "of null measure". (There were ...
sobach'e_pole's user avatar
2 votes
0 answers
70 views

(MK) universal class = von Neumann universe?

I am studying MK set theory. By the axiom schema of class comprehension, which roughly states that Given a monadic predicate $\phi$ of MK, then there is a class $C = \{x: \phi(x)\}$, the universal ...
Wenchuan Zhao's user avatar
11 votes
1 answer
202 views

Is ground model $\Sigma_n$ truth $\Sigma_n$-definable in every forcing extension?

It is well-known (and quite useful) that for any formula $\phi$, forcing poset $\mathbb{P}$, $p\in\mathbb{P}$, and $\mathbb{P}$-name $\dot{a}$, $p\Vdash_{\mathbb{P}}\phi(\dot{a})$ is definable as a ...
Ben Goodman's user avatar
7 votes
2 answers
608 views

Ideals generated by Turing independent sets

Recall that $X \subseteq 2^{\omega}$ is Turing independent if no $y \in X$ is computable from the Turing join of any finite subset of $X \setminus \{y\}$. Question 1. Can we construct a Turing ...
Fiona's user avatar
  • 71
5 votes
1 answer
188 views

Chromatic number of the infinite Erdős–Hajnal shift-graph

For any set $X$, let $[X]^2= \big\{\{x,y\}: x\neq y \in X\big\}$. Let $\kappa$ be an infinite cardinal. Let $G_\kappa = ([\kappa]^2, E_\kappa)$ where $E_\kappa = \big\{\{a,b\}\in \big[[\kappa]^2\big]^...
Dominic van der Zypen's user avatar
9 votes
0 answers
215 views

Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$

I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ ...
new account's user avatar
6 votes
0 answers
145 views

Complexity of transfinite 5-in-a-row and other games

Suppose that 5-in-a-row is played on an infinite board, and after an infinite number of moves, if no one won yet and there is an empty square, the game just continues. At limit steps, it is the first ...
Dmytro Taranovsky's user avatar
12 votes
1 answer
368 views

Partition into antichains

I've read that the following statement is a result of Balcar, but I am unable to find a reference or a proof: Theorem: If $\kappa\ge \lambda$ are infinite cardinals, then $[\kappa]^{<\lambda}$ can ...
Lajos Soukup's user avatar
  • 1,415
0 votes
0 answers
100 views

What are the primitive notions and axioms in model theory? [migrated]

I know every theory has its primitive notions and axioms. Now I am reading Basic Model Theory, and there is no term or sentence referred as to a primitive notion or an axiom. But, I think I know that ...
Wenchuan Zhao's user avatar
2 votes
1 answer
112 views

Closed unbounded sets and partitions

Let $\kappa$ be a regular, uncountable cardinal. Let $S\subseteq \kappa$ be a closed and unbounded set. Suppose that we partition $S$ into $<\kappa$ pieces. Does one of those pieces contain a ...
Pace Nielsen's user avatar
6 votes
1 answer
209 views

Subset of the reals with zero inner measure and "full" outer measure in $\mathsf{ZF}+\mathsf{DC}$

Working in $\mathsf{ZF}+\mathsf{DC}$ (that is, we are allowed to use Dependent Choice but not full choice), suppose that there exists a non-measurable subset of the unit interval $[0,1]$ (just non-...
David Fernandez-Breton's user avatar
18 votes
1 answer
1k views

Let $R, S$ be infinite sets. Alice and Bob will play a game over $R\times S$

Let $R, S$ be infinite sets. Alice and Bob will play a game over $R\times S$. Alice's color is red and Bob's color is blue. In each step, for each $s\in S$, a player will choose finitely many ...
Alma Arjuna's user avatar
2 votes
0 answers
127 views

Namba forcing, one Cardinal up

The classical Namba forcing collapses $\omega_2$ to have cofinality $\omega$ while preserving the cardinal $\omega_1$. Higher analogs were constructed to (under some additional hypotheses) give a ...
Hannes Jakob's user avatar
  • 1,612
5 votes
1 answer
143 views

Infinite sequences/ordered tuples of proper classes in NBG

The question is originally from math stack exchange here. Basically, what I am asking is if we can define ordered tuples of proper classes in NBG. My idea, for finite tuples of proper classes, was to ...
Shthephathord23's user avatar
2 votes
1 answer
177 views

Does inductive definitions must be supported by the set theoretical definition of natural numbers?

In page 4 of Gödel's book The Consistency Of The Axiom Of Choice and Of The Generalized Continuum Hypothesis With The Axioms Of Set Theory, Gödel defined the $n$-tuple as $\langle x \rangle = x$; $\...
Wenchuan Zhao's user avatar
1 vote
0 answers
112 views

Is the existence of elements (sets) postulated in MK?

The Background of My Question I am learning MK (Morse-Kelley set theory). In MK, the primitive notions are: class and $\in$. No doubt, the existence of classes is implicitly postulated in MK as class ...
Wenchuan Zhao's user avatar
-4 votes
1 answer
175 views

Is Bounding Reflection consistent?

Working in the first order language of set theory. Let $\varphi^{*B}$ be the formula obtained from $\varphi$ by merely bounding all open quantifiers in $\varphi$ by the symbol "$B$". Here a ...
Zuhair Al-Johar's user avatar
6 votes
1 answer
145 views

Is the definition of the arbitrary union of proper classes valid in Morse-Kelley set theory?

I am recently studying Morse-Kelley set theory (MK). There is an axiom called the axiom of class comprehension, which states that, given a predicate $\phi(x)$ written in the language of first-order ...
Wenchuan Zhao's user avatar
5 votes
1 answer
334 views

What is the proof of consistency of anterior reflection?

Let Anterior Reflection be the following principle: $$\forall \vec{v}~ \exists X: \operatorname {transitive} (X) \land \, (\varphi \to \varphi^{X"}) $$ where $\varphi$ is a formula in $\sf FOL(=,\in)$ ...
Zuhair Al-Johar's user avatar
-3 votes
1 answer
283 views

Can this form of reflection be consistent?

Is this form of reflection consistent? First I'll begin by clarifying the notation I'm using here: By a quantifier being relativized or bounded it means that the first occurrence of the quantified ...
Zuhair Al-Johar's user avatar
4 votes
0 answers
103 views

Closure of a pointclass under universal real quantification

Let us assume $\mathsf{AD}^+$ and let $\Gamma$ be a pointclass such that $P(\mathbb{R})\cap L(\Gamma)=\Gamma$ and $L(\Gamma)\models\mathsf{AD}_\mathbb{R}+\mathsf{DC}$. Since the cofinality of $o(\...
Obrad Kasum's user avatar
3 votes
1 answer
88 views

Is this form of replacement suitable for ZF - Powerset + well-ordering principle?

The following scheme can be understood as a form of replacement. Axiomatizing $\sf ZF$ with it instead of the usual replacement schema renders it immune to removal of extensionality; see here. In an ...
Zuhair Al-Johar's user avatar
-2 votes
0 answers
98 views

Extending a map between $A$ and $B$ to a map between $L(A)$ and $L(B)$

Are any known results about extending a map $\phi:A\to B$ to a map $\overline{\phi}:L(A)\to L(B)$ or $\phi':HOD(A)\to HOD(B)$? This seems like something that would have been investigated already, and ...
blark's user avatar
  • 97
4 votes
2 answers
140 views

Are there outer models $V \subset W$ of $L$ such that $V$ is "far" from $L$ but $W$ is "not too far" from $V$?

In the following, whenever I say "$V_1$ is an outer model of $V_2$", I mean $V_1, V_2$ are transitive models of $\mathsf{ZFC}$, $V_2 \subset V_1$,and $ORD^{V_1} = ORD^{V_2}$. I am curious ...
Zoorado's user avatar
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