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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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 ...
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Latest status 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:
https://...
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Does finite mathematics need the axiom of infinity?
A statement referring to an infinite set can sometimes be logically rephrased using only finite sets/objects. For example, "The set of primes is infinite" <-> "There is no largest prime". ...
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How far is Lindelöf from compactness?
A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the ...
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The set-theoretic multiverse as a (bi)category
Joel Hamkin's The set-theoretic multiverse has featured in MO questions before, e.g., here and here. But I was wondering about the best category theoretic angle to take on it.
In the paper Joel ...
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Why not adopt the constructibility axiom $V=L$?
Gödelian incompleteness seems to ruin the idea of mathematics offering absolute certainty and objectivity. But Gödel‘s proof gives examples of independent statements that are often remarked as having ...
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Are there examples of statements that have been proven whose consistency proofs came before their proofs?
I'm wondering if there are examples of statements that have been proven whose consistency proofs came before the proofs of the statements themselves.
More informally, I'm wondering how promising in ...
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Are all sets totally ordered ?
The question is the title.
Working in ZF, is it true that: for every nonempty set X, there exists a total order on X ?
If it is false, do we have an example of a nonempty set that has no total ...
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Paradoxical Mathematical Objects Pending for Construction [duplicate]
The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description ...
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Interpretation of the Second Incompleteness Theorem
For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:
ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not ...
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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 ...
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Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
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Chromatic number of a topological space
Here is a question I asked myself years ago. Since it is not really in my field, I hope to find some (partial) answers here... Since it was unclear, I precise that I am looking for an answer in ZFC, ...
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Is Lagrange's Theorem equivalent to AC?
Lagrange's Theorem is most often stated for finite groups, but it has a natural formation for infinite groups too: if $G$ is a group and $H$ a subgroup of $G$, then $|G| = |G:H| \times |H|$.
If we ...
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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 ...
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What would remain of current mathematics without axiom of power set? [closed]
The power set of every infinite set is uncountable. An infinite set (as an element of the power set) cannot be defined by writing the infinite sequence of its elements but only by a finite formula. By ...
user34913
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How should I think about presentable $\infty$-categories?
Let me start out with a confession. I have never cared much for set-theoretic size issues, for they seem not to cause much trouble in my day-to-day mathematical life. Despite that, I have always been ...
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How much choice is needed to show that formally real fields can be ordered?
Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you would guess if you ...
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Is it possible to define higher cardinal arithmetics
In number theory there are several operators like addition, multiplication and exponentiation defined from $\omega\times\omega$ to $\omega$. Each of them is defined as an ...
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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 ...
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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. ...
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Ur-elemental surprises
For most of my (mathematical) life, I believed that there was really no essential difference between set theory without urelements and set theory with urelements. However, while that may be true in ...
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Is the theory Flow actually consistent?
Recently the paper
Adonai S. Sant'Anna, Otavio Bueno, Marcio P. P. de França, Renato Brodzinski, Flow: the Axiom of Choice is independent from the Partition Principle, arXiv:2010.03664
appeared on ...
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Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?
This question was motivated by an answer to this question of Dominic van der Zypen.
It relates to the following classical theorem of Sierpiński.
Theorem (Sierpiński, 1921). For any countable partition ...
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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&...
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Reasons to believe Vopenka's principle/huge cardinals are consistent
There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...
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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 ...
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Are there any interesting surreal constants?
In $\mathbf R$, we have all sorts of fascinating constant, like $e$, $\pi$, $\gamma$, ... For ordinal numbers, we have $\omega$, $\epsilon_0$, $\omega_1^{CK}$, $\omega_1$, ... Have we discovered any ...
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Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?
On page 205 of his Topology textbook, James Munkres made an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...
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Are all free ultrafilters 'the same' in some sense?
Consider the set of ultrafilters $\beta(\mathbb N)$ on $\mathbb N$.
Any function $f\colon\mathbb N\to\mathbb N$ extends to a function $\beta f\colon \beta \mathbb N \to \beta\mathbb N$. We say that ...
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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 ...
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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 ...
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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:
$$0=\{\ \}$$
$$1=\{0\}$$
...
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How many of the true sentences are provable?
Is there a natural measure on the set of statements which are true in the usual model (i.e. $\mathbb{N}$) of Peano arithmetic which enables one to enquire if 'most' true sentences are provable or ...
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Similarities between Post's Problem and Cohen's Forcing
Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated 4/...
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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 $N$?
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Should axiomatic set theory be translated into graph theory?
Recently I saw the abstract of a paper by Nash-Williams: ``Should axiomatic set theory be translated into graph theory?''. The abstract, taken from Mathscinet says the following:
The author ...
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Bidi: A new cardinal characteristic of the continuum?
This question assumes familiarity with combinatorial cardinal
characteristics of the continuum.
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural ...
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Can ZFC → NBG be iterated?
von Neumann-Bernays-Gödel set theory (NBG) is a conservative extension of ZFC which contains "classes" (such as the class of all sets) as basic objects. "Conservative" means that anything provable in ...
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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 ...
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On independence and large cardinal strength of physical statements
The present post is intended to tackle the possible interactions of two bizarre realms of extremely large and extremely small creatures, namely large cardinals and quantum physics.
Maybe after all ...
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Is "all categorical reasoning formally contradictory"?
In the December 2009 issue of the newsletter of the European Mathematical Society there is a very interesting interview with Pierre Cartier. In page 33, to the question
What was the ontological ...
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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 ...
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How can category theory help my research in set theory?
How can category theory help my research in set theory?
I rarely use category theory as such in my current work, and one almost never sees any category theory in set-theoretic research papers or at ...
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Why should we care about "higher infinities" outside of set theory?
Let's say you are a prospective mathematician with some addled ideas about cardinality.
If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :)
If you ...
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Is the fixed point property for posets preserved by products?
Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.
Theorem. Suppose $P$ and $Q$ are posets ...
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Unique existence and the axiom of choice
The axiom of choice states that arbitrary products of nonempty sets are nonempty.
Clearly, we only need the axiom of choice to show the non-emptiness of the product if
there are infinitely many ...
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How (non-)computable is set theory?
Here is a naive outsiders perspective on set theory: A typical set-theoretical result involves constructing new models of set theory from given ones (typically with different theories for the original ...
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Does Fermat's last theorem hold in the ordinals?
My question is whether there are no nontrivial solutions in the ordinals of the equations arising in Fermat's last theorem $$x^n+y^n=z^n$$
where $n\gt 2$, and where we use the natural ordinal ...
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Hahn's Embedding Theorem and the oldest open question in set theory
Hans Hahn is often credited with creating the modern theory of ordered algebraic systems with the publication of his paper Über die nichtarchimedischen Grössensysteme (Sitzungsberichte der ...
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