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.
1,112 questions
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Can local $0^\#$ exists in L?
Assume $0^\#$ exists and there is an inaccessible cardinal.
Are there two transitive sets $M,N$ s.t. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?
- 1,490
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Incompleteness and nonstandard models of arithmetic
The following are a collection of doubts, some of which shall have concrete answers while others may have not. Any kind of help will be welcome.
Reading Peter Smith's "Gödel Without (Too Many) Tears",...
- 1,465
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Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?
It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can ...
- 3,982
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Pontryagin dual of the surreal numbers?
Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown.
Alternatively, has this ...
- 4,907
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The global dimension of fields
In the absence of the Axiom of Choice, it is not necessarily true that all vector spaces over a field have bases.
What are the possible global dimensions of fields in a model of ZF in which AoC ...
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Is it possible to prove that any two points of a convex complete metric space are connected by some metric segment without the axiom of choice?
We say that a point $m$ is between points $p$ and $q$ of a metric space $(M, d)$ if $d(p, q) = d(p, m) + d(m, q)$ and $p ≠ m ≠ q$.
A metric space $M$ is said to be metrically convex if given any two ...
- 307
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862
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Harrington's unpublished note "The constructible reals can be anything"
Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...
- 32.2k
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453
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Does $Add(\kappa,1)^L$ ever collapse cardinals?
In general, we know that adding a subset to a regular cardinal $\kappa$ can collapse cardinals. If, for example, there is $\gamma < \kappa$ with $2^\gamma >\kappa$, then $Add(\kappa,1)$ will ...
- 1,146
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Is it a theorem of ZF that a non-empty countable Cartesian product of finite non-singleton sets has the cardinality of the continuum?
I think that, without countable choice, I can prove quite easily that for a sequence $(S_n)_{n \in \mathbb{N}}$ of sets $S_n$ with $|S_n|=2$, the Cartesian product $\,\prod_{n=1}^\infty S_n\,$ is ...
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Model of ZF + $\neg$C in which Solovay's Theorem on stationary sets fails?
It is a theorem of Solovay that any stationary subset of a regular cardinal, $\kappa$ can be decomposed into a disjoint union of $\kappa$ many disjoint stationary sets. As far as I know, the proof ...
- 1,174
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489
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Intuition behind Pincus' "injectively bounded statements"
In
David Pincus, Zermelo-Fraenkel Consistency Results by Fraenkel-Mostowski Methods,
The Journal of Symbolic Logic, Vol. 37, No. 4 (Dec., 1972), pp. 721-743
Pincus introduces the notion of ...
- 35.5k
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Is there any forcing free proof for hard independence results?
We are forced to use forcing for almost all "hard" independence results such as: $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $. The question simply is:
Primary Question: Is there any "forcing free" ...
user36136
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Existence of infinite groups that are too reluctant to be topological
With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...
- 1,145
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Road to Solovay's Land.
In the first semester of 2012 I took a course in General Topology and Set Theory, at undergraduate level. For topology, I was instructed to use Engelking's General Topology; albeit I had a great ...
- 405
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540
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Vaught's conjecture for partial orders
In
``Steel, John R. On Vaught's conjecture. Cabal Seminar 76–77, pp. 193–208''
the following is proved:
Theorem. Let $\phi\in L_{\omega_1,\omega}.$ If every model of $\phi$ is a tree, then $\phi$ has ...
- 32.2k
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466
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Properness of quotient forcing
It is well known that if $P$ is a proper notion of forcing and $\Vdash_P \dot{Q} \text{ is proper}$ then the iteration $P \ast \dot{Q}$ is proper. Is the converse also true, i.e
Suppose that we have ...
- 2,180
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Does fast function forcing really have $\kappa$-Knaster property?
I ran into a claim concerning Woodin's fast function forcing in the following paper of Apter and Cummings which sounds no right to me:
A. Apter, J. Cummings, Blowing up the power set of the least ...
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507
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Making all cardinals countable and its HOD
Suppose that $G$ is $Col(\omega, <Ord)$-generic over $V$ and let $W=HOD^{V[G]}$. Is $CH$ true in $W$? In general, what can we say about the behaviour of the power function in $W$?
Update. Are the ...
- 32.2k
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On the number $n_0$ in Shelah's construction of a Jonsson group
In the paper "On a Problem of Kurosh, Jonsson Groups, and Applications", Shelah proved the following
Theorem 2.1. There exists a number $n_0$ such that for every infinite cardinal $\lambda$ with $\...
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360
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A combinatorial property of uncountable groups, II
Problem 1. Is it true that each uncountable group $G$ contains two subsets $A,B\subset G$ such that
1) for any $x,y\in G$ the intersection $xA\cap yB$ is finite and
2) for any function $\...
- 41.9k
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Covering compact Hausdorff spaces with closed $G_\delta$ sets
I'm thinking about results of the form: Under assumption $A$, if $X$ is a compact Hausdorff space and $C$ is a cover of $X$ by closed $G_\delta$ sets, then there is a subcover of cardinality $\leq\...
- 12.5k
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If there is a non-constructible real, is there an $L$-generic real?
If we assume that $\Bbb R^V\neq\Bbb R^L$, can we deduce that there is some $x\in\Bbb R^V$ which is $L$-generic?
Of course if $V$ is a generic extension of $L$ this is true, but if $V=L[0^\#]$ this is ...
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A question about the Axiom of Choice
Let AC denote the Axiom of Choice. Let PP denote the so-called "Partition Principle" which states that "If S is a non-empty set and T is a non-empty set of pairwise disjoint subsets
of S, then S can ...
- 6,215
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417
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On a weak tree property for inaccessible cardinals
Suppose that $\kappa$ is inaccessible and consider a tree of height $\kappa$ whose levels have size strictly below some cardinal $\gamma < \kappa$. Does this type of tree always have a $\kappa$-...
- 5,902
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449
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Does Vizing's conjecture hold for the infinite graphs?
In finite graph theory, there are many (in)equalities which relate the integer value of a certain graph invariant (e.g. domination or chromatic number) for the product of two finite graphs (e.g. ...
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693
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Generalizing Feferman - Levy
The Feferman - Levy model makes $\aleph_1$ singular by a cardinal collapse $\aleph_1 = \aleph_{\omega}^L$. Unless I've got something wrong, the same thing would work to make any well-orderable ...
- 11.1k
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374
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Order types of models of theories of ordinals
For $C$ a set of ordinals, let $\mathcal{L}(C)$ be the language with identity, a relation symbol for less than, function symbols for successor, addition, multiplication, and exponentiation, and a ...
- 1,155
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360
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Models of ZFA corresponding exactly with a particular class of groups
I recently read [1], in which Blass exhibits a correspondence between:
Permutation models of ZFA in which the axiom of choice (AC) fails but the Boolean prime ideal theorem (BPIT) holds; and
...
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Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?
A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...
- 5,471
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643
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Explicit counter example to Vopěnka's principle in the constructible universe?
Vopěnka's principle is a large cardinal axiom which has many equivalent formulations. One of them, which I find especially appealing, is the following: if the universe is satisfies Vopěnka's principle ...
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514
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How big is the least non-$\Sigma^1_1$-pointwise-definable ordinal?
There's a large countable ordinal which has cropped up (as a lower bound!) in a computable structure theory problem I'm playing with. At present I don't really understand how big it is, and I'm ...
- 20.5k
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Does the class category of ZF-algebras satisfy the Multiverse axioms?
I read with interest both Hamkins Multiverse Axioms and Joyal and Moerdijk's algebraic set theory. Both of these perspectives takes a set-theory universe as an object, and consider collections of set-...
user2529
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393
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Where can I find the following S. Shelah's paper?
I've been trying to find the following article: "S. Shelah, Remarks on cardinal invariants in topology, General topology Appl. 7(3) (1977), 251-259". I tried to go directly to the journal ...
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Ill-founded models of set theory with well-founded ordinals
Let $(\mathcal{M},E)$ be an internally non-well-founded model of set theory i.e of $ZFC^{\neg f}=ZFC\setminus \mathrm{foundation}+\neg \mathrm{foundation}$, then $\mathcal{M}$ includes an infinite ...
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A surjection from square onto power: Is limit Hartogs/Lindenbaum number necessary?
I am considering the construction in [Peng—Shen—Wu] in which the authors show the consistency of a set $X$ such that there is a surjection from $X^2$ onto the power set of $X$ (henceforth $\mathscr{P}(...
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535
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Different approaches to forcing
There are many different approaches to the forcing method, and I am looking for all known such approaches. So my question is:
Question 1. Which different approaches to set theoretic forcing are ...
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524
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Analogues of worldly cardinals for (an unusual version of) second-order $\mathsf{ZFC}$
Let $\mathfrak{ZFC}(\mathsf{SOL})$ be the theory in second-order logic (with the standard semantics) gotten from $\mathsf{ZFC}$ by modifying the Separation and Replacement schemes to apply to ...
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Reference request: proof-theoretic strength of $\mathsf{KP}$ with recursively large ordinals and $\mathsf{CZF}$ with large set axioms
Large set axioms are notions corresponding to large cardinals on constructive set theories like $\mathsf{IZF}$ or $\mathsf{CZF}$. The notion of inaccessible sets, Mahlo sets, and 2-strong sets ...
- 3,042
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Cardinality: Why is there no "ℵ½"?
A wikipedia page/paragraph on ℵ₁ states:
"The definition of ℵ₁ implies (in
ZF, Zermelo-Fraenkel set theory
without the axiom of choice) that no
cardinal number is between ℵ₀ and
ℵ₁."
"If the axiom ...
- 207
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338
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Modal logic of "mostly-satisfiability"
For $n\in\omega+1$ let $\mathsf{ZFC}_n$ be $\mathsf{ZC}$ + $\{\Sigma_k$-$\mathsf{Rep}: k<n\}$. Let $\widehat{\mathsf{ZFC}}$ be the strongest consistent theory $\mathsf{ZFC}_n$ (so if $\mathsf{ZFC}$ ...
- 20.5k
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483
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VC dimension of standard topology on the reals
Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is an open set $U$ with $D=S\cap U$?
I'm asking merely out of curiosity, but I'll mention that ...
- 24.8k
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Can iterating countable unions give every set? (ZF)
Does ZF prove that there exists a set S such that S is not in the closure of {{s} : s in S} under at-most-countable unions?
user5810
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Foundations: Existence of uncountable ordinals.
This isn't really a research question, but at least it's research-level mathematics. I'm talking with some other people about the first uncountable ordinal, and I want some facts to inform this ...
- 2,754
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Are there large cardinals for $n$-elementarity?
In July, Asaf Karagila asked three questions about elementary substructures of the universe of sets. The latter two were answered, the upshot being that the hypothesis $V_\kappa \prec V$ doesn't alone ...
- 1,081
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Sunflowers in maximal almost disjoint families
Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. We say ${\cal A}\subseteq [\omega]^\omega$ is almost disjoint if $A \cap B$ is finite whenever $A\neq B \in {\cal A}$. Zorn'...
- 48.9k
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975
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Equivalence of the Banach–Tarski paradox
I am working on the Banach–Tarski paradox and the fact that the Hahn–Banach theorem implies that paradox. The proof involves the equivalence of the Hahn–Banach theorem and the fact that for every ...
- 81
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790
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Are problems in complexity theory dependent on set theory?
I was pondering the fact that maybe the classical hard complexity-theoretic questions are undecidable, not because they are so themselves, but because some set-theoretic foundations makes the ...
- 123
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969
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Probabilities independent of ZFC?
Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
...
- 165
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586
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Shelah's "Can you take Solovay's inaccessible away?"
I was wandering if there was a book, thesis or some notes where Shelah's argument for
$\mathtt{ZF}+\mathtt{DC}+$"All sets of reals are Lebesgue measurable" is equiconsistent with $\mathtt{...
- 2,286
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222
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Mostowski's absoluteness theorem and proving that theories extending $0^\#$ have incomparable minimal transitive models
This question says that the theory ZFC + $0^\#$ has incomparable minimal transitive models. It proves this as follows (my emphasis):
[F]or every c.e. $T⊢\text{ZFC\P}+0^\#$ having a model $M$ with $On^...
- 1,097