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.
5,254
questions
5
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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 ...
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 $I$.
...
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 ...
1
vote
1
answer
326
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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 ...
10
votes
1
answer
266
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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 ...
9
votes
1
answer
435
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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 $\Vdash_{\mathbb{P}}\check{\mathbb{Q}}\in\Gamma$. ...
5
votes
1
answer
342
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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, ...
-4
votes
0
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81
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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 ...
8
votes
1
answer
303
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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, ...
0
votes
0
answers
115
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+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 \,...
-5
votes
0
answers
76
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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."
...
0
votes
0
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104
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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 ...
6
votes
0
answers
107
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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 ...
-4
votes
0
answers
155
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+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 ...
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 ...
5
votes
0
answers
181
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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 ...
6
votes
1
answer
133
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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 ?
2
votes
2
answers
191
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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 ...
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 ...
1
vote
0
answers
158
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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 ...
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 ...
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 ...
8
votes
1
answer
186
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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 $\mathbb{P}$...
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\...
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 $\mathcal{L}$ ...
-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:...
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 ...
2
votes
0
answers
70
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(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 ...
11
votes
1
answer
202
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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 ...
7
votes
2
answers
608
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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 ...
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]^...
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 $\mathsf{SCH}$ ...
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 ...
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 ...
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 ...
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 ...
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-...
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 ...
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
143
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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
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 ...
-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 ...
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 ...
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 $\sf FOL(=,\in)$ ...
-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 ...
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(\...
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 ...
-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
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 ...