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.
5,256
questions
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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 ...
14
votes
0
answers
559
views
Explicit and complete list of Lean's Axioms
I'm a big fan of the idea of fully formalizing mathematics. So the Lean proof checker appeals to me.
Relating to this, one of the biggest appeals of mathematics to me is that there is a (largely) ...
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 $\mathsf{SCH}$ ...
12
votes
3
answers
4k
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Existence of prime ideals and Axiom of Choice.
One of the must obvious equivalences of Axiom of Choice is the converse of Krull Theorem.
Bernhard Banaschewski in the Article titled by A New Proof that “Krull implies Zorn” showed a very simple ...
6
votes
0
answers
146
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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 ...
4
votes
0
answers
142
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Slicing large countable ordinal properties, from $\Pi_3$-reflection to $\Sigma_2$-admissibility
Edit 2024: This post was based on an incorrect premise, as can be seen by my conversation with Farmer S in the comments. However the mistake I made and the conversation in comments may be instructive (...
0
votes
0
answers
100
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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 ...
6
votes
1
answer
209
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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-...
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 ...
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 $\sf FOL(=,\in)$ ...
6
votes
2
answers
419
views
Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal
Is anything known about the consistency strength of the following statement?
$\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ such that $a \cap \...
1
vote
3
answers
945
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Explicitly constructing an infinite set with particular size
I would like to preface by saying that I have no significant experience working with set theory, so I'm probably making an intuitive mistake. I have figured out where the mistake probably is, but I ...
5
votes
1
answer
198
views
On the number of complete Boolean algebras
In their 1972 paper On the number of complete Boolean algebras Monk and Solovay showed that if $\lambda$ is an infinite cardinal, then there are $2^{2^\lambda}$ many isomorphism types of
complete ...
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 ...
5
votes
1
answer
145
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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 ...
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
$\langle x \rangle = x$;
$\...
1
vote
0
answers
112
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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 ...
6
votes
1
answer
145
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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 ...
10
votes
1
answer
348
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1970 question of Reinhardt - how large is this ordinal?
On page 241 of William Reinhardt's paper "Ackermann's set theory equals ZF" (Annals of Math. Logic vol. 2, 1970), question 4.15 is the following:
How large is the first ordinal $\gamma$ ...
3
votes
1
answer
88
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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 ...
4
votes
0
answers
103
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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(\...
2
votes
1
answer
221
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A possible ${\sf (ZF)}$-theorem in the spirit of the $3$-set-lemma
The number $3$ plays an interesting role in the following statement:
$\newcommand{\S}{\sf(S_3)}\S$ Let $X$ be a non-empty set and let $f:X \to X$ be fixpoint-free (that is $f(x) \neq x$ for all $x\in ...
4
votes
1
answer
193
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Weak Power Hypothesis and Dependent Choice
Consider in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$ the following statement:
Weak Power Hypothesis (WPH): if $X,Y$ are sets and there is a bijection between $\newcommand{\P}{{\cal P}}\P(X)$ and $\P(Y)$, ...
-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 ...
4
votes
2
answers
141
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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 ...
4
votes
1
answer
169
views
Generic absoluteness
In Theorem 14 of "On The Question Of Absolute Undecidability" Peter Koellner describes a generic absoluteness result which could be summed up as $\Sigma^2_1(\Gamma^\infty)$-generic ...
8
votes
2
answers
1k
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Is there one binary operation foundational for set theory?
The membership relationship "$\in$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\in$". Naturally, the ...
5
votes
1
answer
163
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Can a generic ultrafilter over $\mathrm{NS}^+_{\omega_1}$ witness $\omega_1$ is Ramsey-like?
Suppose that $\kappa$ is an appropriate large cardinal (preferably a Woodin cardinal, but possibly something stronger) and let $G$ be a $\operatorname{Col}(\omega_1,<\kappa)$-generic filter over $V$...
109
votes
10
answers
23k
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Set theories without "junk" theorems?
Clearly I first need to formally define what I mean by "junk" theorem. In the usual construction of natural numbers in set theory, a side-effect of that construction is that we get such theorems as $...
12
votes
0
answers
191
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Are there times when replacement is "more natural" than collection?
There are a couple examples I'm aware of where choosing to axiomatize $\mathsf{ZF(C)}$ using collection instead of replacement results in a much nicer (or at least less surprising) picture:
Let $\...
14
votes
1
answer
570
views
Changing the cofinality of a regular cardinal without collapsing any cardinals?
I have a short but hopefully interesting question on cardinal arithmetic and collapsing cardinals:
Is it possible to change the cofinality of a regular cardinal without collapsing any cardinals?
Is ...
10
votes
2
answers
673
views
Is a Borel image of a Polish space analytic?
A topological space $X$ is called analytic if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$.
We say that a topological ...
5
votes
0
answers
200
views
Are there Dedekind-infinite amorphous sets?
An amorphous set is an infinite set (i.e. cannot be put into bijection with any finite set $\{ 1, \dots, n \}$ for any $n$) that cannot be partitioned into two mutually disjoint infinite subsets. ...
2
votes
0
answers
113
views
Adding partitions of one but not the other kind
Say that two partitions $(P_i)_{i\in I}, (Q_j)_{j\in J}$ are isomorphic iff there is a bijection $f: I\rightarrow J$ such that $\vert P_i\vert=\vert Q_{f(i)}\vert$ for all $i\in I$. (Note that in the ...
9
votes
2
answers
1k
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Truth in a different universe of sets?
I understand that provability and truth as different concepts.
Provability is syntactic, it only concerns whether the given
sentence can be derived by reiterating the inference rules over a
collection ...
47
votes
5
answers
7k
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What axioms are used to prove Gödel's Incompleteness Theorems?
I understand Gödel's Incompleteness Theorems to be statements about effectively generated formal systems, which basically makes them theorems about algorithms. This is cool, because despite being ...
1
vote
0
answers
62
views
Can the proper/whole domain relationship in bi-interpretations be reversed for non-synonymous theories?
Suppose we have theories $T$ and $H$ that are bi-interpretable, now suppose that the relevant interpretations achieving that bi-interpretability are: $\tau: T \to H$, and $\pi: H \to T$. Now suppose ...
3
votes
0
answers
130
views
On a connectivity property of set systems
Let $X\neq \emptyset$ be a set and let ${\cal A}\subseteq {\cal P}(X)$ with $\bigcup {\cal A}=X$. We say that $S\subseteq X$ is indivisible if for all $T\subseteq S$ with $\emptyset \neq T \neq S$ ...
19
votes
2
answers
2k
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Did Hilbert discuss his 23 problems with Felix Klein?
Hilbert's lecture at the ICM in Paris in 1900 presented 10 of the famous 23 open problems. It is well known that the idea of the lecture came from Hermann Minkowski. Hilbert was at Göttingen at the ...
3
votes
1
answer
270
views
When does $\Pi_2$-reflection on $X$ fail to imply iterated $\Pi_1$-reflection on $X$?
Let lowercase Greek letters denote ordinals. Recall from Richter and Aczel's "Inductive definitions and reflecting properties of admissible ordinals", for a set of formulae $\Gamma$ and a ...
11
votes
2
answers
780
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Undefinable inner model
What are some examples of a pair $M\subseteq N$ of transitive set models of $\mathsf{ZFC}$ with the same ordinals, such that $M$ is not a definable class (with parameters) in $N$? Is it possible that $...
4
votes
0
answers
133
views
Proof of: No rapid filter is Lebesgue measurable
I'm studying the following theorem in (Schindler, 2014: Set Theory
Exploring Independence and Truth), p. 178-180:
Theorem 9.16 (Mokobodzki) No rapid filter F $\subset$ ${}^\omega 2$ is Lebesgue ...
4
votes
1
answer
456
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Truth Values of Statements in non-standard models
Excuse me, if the question sounds too naive.
Non-standard models of PA will have statements of non-standard lengths, basically infinite. And it is also true that every statement of a theory will have ...
0
votes
1
answer
212
views
Is it consistent that $2^{(\cdot)}$ is "surjective" on the class of uncountable ordinals?
$\newcommand{\Z}{{\sf (ZFC)}}$
It is consistent in $\Z$ that there is an uncountable cardinal $\kappa$ such that for no cardinal $\lambda$ we have $2^\lambda = \kappa$: Take any model in which $2^{\...
8
votes
1
answer
1k
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Worst of both worlds?
It's well known that $\mathsf{AC}$ implies the existence of non-measurable sets. And it's also true that, if all sets are measurable, then $|\mathbb{R}/\mathbb{Q}| > |\mathbb{R}|$. But is there a ...
2
votes
0
answers
142
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Is it consistent to have these kinds of acyclic hereditarily size sets?
Working in $\sf ZFC-Reg. {}+ Acyclicity$. Where :
Acyclicity: $\neg \exists x_1, \cdots, \exists x_n: x_1 \in x_2 \in\cdots\in x_n \in x_1$
We add the following kind of weird non-well founded sets.
$\...
3
votes
0
answers
135
views
Lindström's theorem part 2 for non-relativizing logics
By "logic" I mean the definition gotten by removing the relativization property from "regular logic" — see e.g. Ebbinghaus/Flum/Thomas — and adding the condition that for every ...
4
votes
1
answer
157
views
Which of the known variants of Replacement can survive DeExtensionality?
Starting with $\sf ZF$. If we replace the power set axiom by the axiom stating that for any set $A$ there exists a set $x$ such that for every $y \subseteq A$ we have a set $y' \in x$ such that $\...
3
votes
1
answer
171
views
Is the universe of ZFA rigid? The pairing axiom implies that even atoms have a unitary set which discern them from all other atoms. So, is it rigid?
Although any permutation of atoms induces an automorphism of the whole universe, atoms seem to be indiscernible only within the permutation models. Can a permutation model be extended to a rigid ...
10
votes
1
answer
343
views
Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?
Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures.
Consider the endomorphism $\hat{\Phi}$ ...