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.
1,056
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Does pointwise convergence imply uniform convergence on a large subset?
Suppose $f_n$ is a sequence of real valued functions on $[0,1]$ which converges pointwise to zero.
Is there an uncountable subset $A$ of $[0,1]$ so that $f_n$ converges uniformly on $A$?
Is there a ...
71
votes
13
answers
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Logic in mathematics and philosophy
What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second ...
64
votes
3
answers
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Forcing as a new chapter of Galois Theory?
There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was ...
56
votes
6
answers
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Can the symmetric groups on sets of different cardinalities be isomorphic?
For any set X, let SX be the symmetric group on
X, the group of permutations of X.
My question is: Can there be two nonempty sets X and Y with
different cardinalities, but for which SX is
isomorphic ...
53
votes
2
answers
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How to add essentially new knots to the universe?
A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, tame and ...
52
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2
answers
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Silver's approach to the inconsistency of $\mathrm{ZFC}$
As all probably know, Jack Silver passed away about one month ago. The announcement released, with delay, by European Set Theory Society includes a quote by Solovay about his belief on inconsistency ...
48
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0
answers
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How many algebraic closures can a field have?
Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...
47
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10
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What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
There are many equivalent formulations of the Continuum Hypothesis, but I think the most standard one is that
there is no infinite cardinality lying strictly between the cardinality of the natural ...
45
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4
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The origin of sets?
The history of set theory from Cantor to modern times is well documented. However, the origin of the idea of sets is not so clear. A few years ago, I taught a set theory course and I did some digging ...
43
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4
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Lists as a foundation of mathematics
I am wondering if there is a foundation of mathematics where not sets or "set-like objects" (such as objects of a suitable topos as in ETCS) are the primitive notion, but rather lists. These ...
41
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2
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On the difference between two concepts of even cardinalities: Is there a model of ZF set theory in which every infinite set can be split into pairs, but not every infinite set can be cut in half?
An interesting question has arisen over at this
math.stackexchange
question
about two concepts of even in the context of infinite
cardinalities, which are equivalent under the axiom of
choice, but ...
39
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7
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What is the general opinion on the Generalized Continuum Hypothesis?
I'm community wikiing this, since although I don't want it to be a discussion thread, I don't think that there is really a right answer to this.
From what I've seen, model theorists and logicians ...
39
votes
5
answers
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Is there a constructive proof of Cantor–Bernstein–Schroeder theorem ?
An important feature of the Cantor-Schroeder-Bernstein theorem is that it does not rely on the axiom of choice. However, its various proofs are non-constructive, as they depend on the law of excluded ...
38
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4
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Understanding the countable ordinals up to $\epsilon_{0}$
in a recent MO question, link, discussing the current foundations of mathematics, the author linked a video lecture by Prof. Voevodsky, which argues against the principle of $\epsilon_{0}$-induction ...
36
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6
answers
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Distinct well-orderings of the same set
An easy consequence of the Erdős-Dushnik-Miller theorem $\kappa\to(\kappa,\omega)^2$ is the following, that will denote $(*)$ (it appears as an exercise in Kunen's book, it was probably mentioned ...
36
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14
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What are interesting families of subsets of a given set?
Motivation
The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$.
Indeed, one defines a topology on $S$ to be a family of subsets ...
35
votes
7
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How would set theory research be affected by using ETCS instead of ZFC?
In "Rethinking Set Theory", Tom Leinster argues in favor of teaching axiomatic set theory via Lawvere's Elementary Theory of the Category of Sets with 10 axioms (but phrased in a way that requires no ...
35
votes
3
answers
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Counterintuitive consequences of the Axiom of Determinacy?
I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty ...
34
votes
5
answers
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Forcing as a replacement of induction and diagonal arguments
Let me give some examples motivating the question.
The use of forcing instead of induction: For this consider Cantor's theorem:
Theorem 1. Any two countable dense linear orders $I, J$ without end ...
34
votes
3
answers
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Latest stand of core model theory?
What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004:
http://www.math.cmu....
33
votes
15
answers
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What's a magical theorem in logic?
Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less straightaway&...
33
votes
3
answers
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Wiki for consequences of axiom of choice?
I raised the following question as part of another MO question, but I am following the suggestion of Nate Eldredge to make it a question in its own right.
For many years, there has a been a valuable ...
32
votes
4
answers
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Do there exist non-PIDs in which every countably generated ideal is principal?
The title pretty much says it all: suppose $R$ is a commutative integral domain such that every countably generated ideal is principal. Must $R$ be a principal ideal domain?
More generally: for ...
32
votes
2
answers
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Translates of null sets
Does there exist a null set of reals $N$ such that every null set is covered by countably many translations of $N$?
32
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3
answers
5k
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Should there be a true model of set theory?
As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
32
votes
1
answer
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Does "every" first-order theory have a finitely axiomatizable conservative extension?
I originally asked this question on math.stackexchange.com here.
There's a famous theorem (due to Montague) that states that if $\sf ZFC$ is consistent then it cannot be finitely axiomatized. ...
31
votes
4
answers
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Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
31
votes
1
answer
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Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?
This is a question in two parts.
Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...
31
votes
3
answers
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Is "compact implies sequentially compact" consistent with ZF?
Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...
31
votes
2
answers
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Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?
There are many interpretations of arithmetic in set theory. The
Zermelo interpretation, for example, begins with the empty set and applies the singleton operator as successor:
$$0=\{\ \}$$
$$1=\{0\}$$
...
29
votes
2
answers
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What is the dimension of the mathematical universe?
Forcing construction in set theory leads to a new understanding of the mathematical (multi)universe by providing a machinery through which one can construct new models of the universe from the ...
28
votes
0
answers
810
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Can one divide by the cardinal of an amorphous set?
This question arose in a discussion with Peter Doyle.
It is provable in ZF that one can divide by any positive finite cardinal $k$: if $X \times \{1,\ldots,k\} \simeq Y \times \{1,\ldots,k\}$ then $X \...
27
votes
5
answers
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What is induction up to $\varepsilon_0$?
This is a question asked out of curiosity, and because I can't understand the Wikipedia page.
I have often been told that PA cannot prove the validity of induction up to $\varepsilon_0$, which has ...
27
votes
4
answers
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Finite axiom of choice: how do you prove it from just ZF?
The axiom of choice asserts the existence of a choice function for any family of sets F. Suppose, however, that F is finite, or even that F just has one set. Then how do we prove the existence of a ...
27
votes
3
answers
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Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...)
Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...
27
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6
answers
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Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?
We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced ...
27
votes
4
answers
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Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?
This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal.
Question 1. Does every set of reals contain a measure-zero subset
of the same ...
26
votes
2
answers
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When does the choice of the generic matter?
It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases ...
26
votes
2
answers
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Axiom of choice: ultrafilter vs. Vitali set
It is well known that from a free (non-principal) ultrafilter on $\omega$ one can define a non-measurable set of reals. The older example of a non-measurable set is the Vitali set,
a set of ...
26
votes
3
answers
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Using consistency to create new axioms in set theory
As everybody knows, the ZFC axioms may serve as a foundation for (almost)
all of contemporary mathematics, and it is also well-known that several results
are "indecidable" in ZFC, which means that ...
26
votes
9
answers
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Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]
As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...
26
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4
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What "metatheory" did early set theory/logic researchers use to prove semantic results?
Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic.
The modern approach seems to be, usually, to interpret a "model" ...
26
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4
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What would be some major consequences of the inconsistency of ZFC?
Update (21st April, 2019). Removed the reference / initial trigger behind my question (please see comment thread below for the reasons). Am retaining, of course, the actual question, noted both in the ...
26
votes
4
answers
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Nilradicals without Zorn's lemma
It's well known that the nilradical of a commutative ring with identity $A$ is the intersection of all the prime ideals of $A$.
Every proof I found (e.g. in the classical "Commutative Algebra" by ...
25
votes
1
answer
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How far wrong could the Continuum Hypothesis be?
I hear it's consistent with ZFC to have
$$ 2^{\aleph_0} = \aleph_n $$
for any $n = 1, 2, 3, \dots $. How much worse can it get?
More precisely: are there models of ZFC with $2^{\aleph_0} \gt \aleph_n$...
25
votes
2
answers
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Recent claim that inaccessibles are inconsistent with ZF
Here it is mentioned that someone claims to have proven that there are no weakly inaccessibles in ZF.
Question 1: What reasons are there to believe that weakly inaccessibles exist?
Question(s) 2: ...
25
votes
2
answers
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Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$
For a structure $\mathcal{X}=(X;...)$, say that a cardinal $\kappa$ is $\mathcal{X}$-detectable iff there is some sentence $\varphi$ in the language of $\mathcal{X}$ together with a fresh unary ...
25
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2
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Writing a function on $\mathbb{R}$ as a sum of two injections
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
25
votes
2
answers
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Integration in the surreal numbers
In the appendix to ONAG (2nd edition), Conway points that the definition of integration (using Riemann sums as left and right options) gives the "wrong" answer : $\int_0^\omega \exp(t)\thinspace dt=\...
25
votes
4
answers
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Is Monsky's theorem dependent on the axiom of choice?
The extension of the 2-adic valuation to the reals used in the usual proof clearly uses AC. But is this really necessary? After all, given an equidissection in $n$ triangles, it is finite, so it ...