All Questions
6,027 questions
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Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
7
votes
1
answer
561
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How are real numbers defined in elementary recursive arithmetic?
I am currently reading about elementary function arithmetic and Harvey Friedman's grand conjecture.
In Number theory and elementary arithmetic, Jeremy Avigad expressed Fermat's last theorem, ...
user507115
24
votes
4
answers
3k
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A Löwenheim–Skolem–Tarski-like property
I am interested in the following Löwenheim–Skolem–Tarski-like property.
Given a cardinal $\kappa$, what (if any) is a property $\phi(x)$ such that if $\phi(\kappa)$ holds, then we can prove the ...
- 343
6
votes
2
answers
406
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Axiomatic strength of the cumulative hierarchy
In the 2021 paper Level Theory Part I: Axiomatizing the Bare Idea of a Cumulative Hierarchy of Sets by Tim Button, a first order theory of the cumulative hierarchy is explored. Initially no axioms ...
- 10.1k
21
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1
answer
1k
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Are the real numbers isomorphic to a nontrivial ultraproduct of fields?
Let $K_1, K_2, \dots$ be a countable sequence of fields, and let $\prod_{\mathcal F} K_i$ be the ultraproduct with respect to some nonprincipal ultrafilter $\mathcal F$.
Question: Can there be a field ...
- 64k
3
votes
1
answer
151
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Large almost disjoint family on $\mathbb{N}$ with property $\mathbf{B}$
Let
$\newcommand{\oo}{[\omega]^\omega}\oo$ denote the collection of all infinite subsets of the set of nonnegative integers $\omega$. We say that $\newcommand{\ss}{{\cal S}}\S\subseteq \oo$ ...
- 48.9k
5
votes
1
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170
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Can we see quantifier elimination by comparing semirings?
This question came up while reading the paper Hales, What is motivic measure?. Broadly speaking, I'm interested in which ideas from motivic measure make sense in arbitrary first-order theories (or ...
- 20.5k
2
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1
answer
169
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Smallest ${\mathbf B}$-function $f:\omega\to( \omega\setminus\{0\})$
Motivation. Every hypergraph $(\omega, E)$ where $E$ is countable and consists of infinite sets has property $\newcommand{\B}{\mathbf{B}}\B$. On the other hand, if the members of $E$ are allowed to be ...
- 48.9k
5
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1
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311
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Is there a statement in Presburger arithmetic about primes this simple heuristic fails for?
I came up with the following conjecture while thinking about ways to formulate some heuristics about primes:
Conjecture: Given a statement $s$ in Presburger arithmetic, using an additional unary ...
- 3,319
10
votes
2
answers
240
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Additive, multiplicative, and Dedekind infiniteness in ${\sf (ZF)}$
We call a set $X$
Dedekind-infinite if there is an injective map $f:X\to X$ that is not surjective,
addititvely infinite if $X \neq\emptyset$ and there is an injective map $f:\big((X\times\{1\})\cup(...
- 48.9k
4
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0
answers
143
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Part II to Ketonen's "Set Theory for a Small Universe I. The Paris-Harrington Axiom"
There is an unpublished manuscript "Set Theory for a Small Universe I. The Paris-Harrington Axiom" by Ketonen which appeared early in the study of the Paris-Harrington theorem, around 1979. ...
- 2,031
3
votes
1
answer
256
views
Can these short set-building expressions of the finite set world extend to the infinite set world?
A formula of the form $\forall \vec{p}\, \exists x \, \forall y\, (y \in x \leftrightarrow \phi(y,\vec{p}))$
is to be named a "set-building" formula.
Now, when $\vec{p}$ includes a predicate ...
- 11.3k
4
votes
0
answers
149
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Isomorphism between the reduced C*-algebra of a groupoid and the crossed product of inverse semigroups
In Paterson's book "Groupoids, Inverse Semigroups and their Operator Algebras" he proves that for any r-discrete groupoid $G$ with unit space $G^0$, its full $C^* $-algebra $C^* (G)$ is ...
- 303
1
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0
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244
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Christoph Benzmüller and Gödel's ontological proof?
Are there any notable mathematical or logical issues within Christoph Benzmüller and Bruno Woltzenlogel-Paleo formalized Gödel's ontological proof (pdf) that has been identified by the community?
1
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1
answer
180
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Natural functions outside $\sf PA$?
Can theories stronger than $\sf PA$ manage to define functions from the naturals to the naturals, that $\sf PA$ cannot? If so, what are examples of those functions?
- 11.3k
8
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157
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How to define Dedekind reals and Eudoxus reals such that they are equivalent to unmodulated Cauchy reals
In constructive mathematics without choice, we have three different versions of the real numbers (each embedding into the next).
Regular Cauchy reals (functions $f : \mathbb N \to \mathbb Q$ such ...
- 6,353
-4
votes
2
answers
399
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Two equivalent statements about formulas projected onto an Ultrafilter
Question 1:
In the same language, let $ X $ be a nonempty set, and let $ \{ (\forall x_{x(i)} f(i)) \ | \ i \in X \} $ be a set of formulas. We use $ x(i) $ to denote the index of the variable on ...
- 127
53
votes
7
answers
8k
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Zorn's lemma: old friend or historical relic?
It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this ...
- 18.7k
13
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1
answer
2k
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Are some interesting mathematical statements minimal?
Gödel's set $\mathrm{L}$, of constructible sets, decides many interesting mathematical statements, as the Continuum hypothesis and the Axiom of Choice.
Are some interesting mathematical questions, ...
- 3,574
2
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1
answer
119
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Does $\mathrm{L}_{s_{n+1}}$ contain a surjection from $\omega$ to $\mathrm{L}_{s_n}$?
Let $s_n$ be the least $\Sigma_n-$admissible ordinal, so that $\mathrm{L}_{s_n}$ is a model of Kripke-Platek set theory with $\Sigma_n-$collection and $\Sigma_n-$separation.
Does $\mathrm{L}_{s_{n+1}}$...
- 3,574
1
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1
answer
138
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Is there inconsistency with having countable models of Z with these internalizing properties?
Is there a clear inconsistency with the following?
There exists a countable transitive model of Zermelo set theory, such that for all external bijections between sets the images and preimages of sets ...
- 11.3k
3
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0
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161
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On generators of the multiplicative semigroup $\{r\in\mathbb Q:\ r>1\}$
The set $M=\{r\in\mathbb Q:\ r>1\}$ is a commutative semigroup with respect to the multiplication. For any integers $a>b\ge1$, we clearly have
$$\frac ab=\prod_{n=b}^{a-1}\frac{n+1}n.$$
So the ...
- 15.6k
1
vote
1
answer
146
views
Can PA define functions related to higher theories?
Working in $\sf ZFC$ we can define a function $F$ from naturals to naturals such that $F(0) = \, ^\circ\ulcorner r({\sf Z}) \urcorner$, where $r({\sf Z})$ is the Rosser sentence of Zermelo set theory $...
- 11.3k
7
votes
1
answer
290
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Equivalence of omniscience principles for natural numbers and analytic omniscience principles for Cauchy real numbers
In constructive mathematics, a proposition $P$ is decidable if $P \vee \neg P$, and a proposition is stable if $\neg \neg P \implies P$. We have the following principles of omniscience for the natural ...
- 2,624
7
votes
1
answer
334
views
Why include $0$ and $1$ in the signature of Presburger arithmetic?
I recently became interested in some problems concerning decidability of extensions of Presburger arithmetic. However, being a number-theorist rather than a logician, I am confused by some notational ...
- 1,914
20
votes
2
answers
2k
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Tennenbaum's Theorem and polynomials
Tennenbaum's Theorem theory says that in a countable non-standard model of arithmetic with an underlying set consisting of standard numbers, neither the polynomial $A(x,y):=x+y$ nor the polynomial $M(...
- 17.6k
12
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1
answer
684
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Graphs $G$ with $G \cong \text{Aut}(G)$
Let $G=(V,E)$ be a simple, undirected graph. By $\newcommand{\Aut}{\text{Aut}}\Aut(G)$ we denote the collection of graph isomorphisms $\varphi:G\to G$. We let $$E(\Aut(G)) =\big\{\{\varphi, \psi\}:\...
- 48.9k
10
votes
1
answer
502
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Must strange sequences wear Russellian socks?
This is an attempt to make more precise a vague guess at the end of this answer of mine. We work in $\mathsf{ZF}$ throughout.
Say that a sequence $\mathcal{A}=(A_i)_{i\in\omega}$ of disjoint sets is ...
- 20.5k
2
votes
1
answer
154
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The Dirichlet principle and arithmetical induction
Let us consider the Dirichlet principle as follows: for all natural numbers $n > k > 0$, there is no injection from $\{0, \dots, n-1\}$ into $\{0, \dots, k-1\}$.
Is it true that in some non-...
135
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43
answers
38k
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What are the most attractive Turing undecidable problems in mathematics?
What are the most attractive Turing undecidable problems in mathematics?
There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on ...
1
vote
1
answer
276
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About having one axiom schema for ZFC motivated after the iterative conception of sets?
This posting is related to this posting, and builds its motivation from this answer to it.
Define: $\operatorname {History}(x) \iff \\\forall y \in x: y=\{c \mid \exists z : z \in y \cap x \land (c \...
- 11.3k
8
votes
2
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774
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Does PA prove (Artemov-style) the consistency of a stronger system?
There was a recent question on Artemov's paper here on MO Situation with Artemov's paper?
In one of the answers there it was asserted (apparently incorrectly - see Noah Schweber's comments and ...
- 1,974
8
votes
1
answer
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
2
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0
answers
142
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Can a path in Kleene's $\mathcal{O}$ enumerate all of the computable reals via uniform diagonalization?
It's a well-known fact that there are computable diagonalization functions on Baire space $\mathbb{B} = \mathbb{N}^\mathbb{N}$ (i.e., functions which take a sequence $(r_i)_{i\in \mathbb{N}}$ of ...
- 13.1k
18
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3
answers
3k
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What's the earliest result (outside of logic) that cannot be proven constructively?
Although mathematicians usually do not work in constructive mathematics per se, their results often are constructively valid (even if the original proof isn't).
An obvious counter-example is the law ...
- 6,353
1
vote
1
answer
216
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Reference about cancellation property for semigroups
Have the semigroups with the following cancellation property been studied?
Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
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107
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36
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21k
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Interesting examples of vacuous / void entities
I included this footnote in a paper in which I mentioned that the number of partitions of the empty set is 1 (every member of any partition is a non-empty set, and of course every member of the empty ...
7
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1
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795
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Must there be a proper class of Reinhardt cardinals if there is a Reinhardt cardinal?
A cardinal is Reinhardt if $\kappa$ is the critical point of a nontrivial elementary embedding of $V$ to itself, where $V$ is the class of all sets. As Reinhardt cardinals are inconsistent with $\...
- 2,031
11
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3
answers
582
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What would you do with a new model of linear logic?
I have been working for some time with collaborators developing some models of linear logic which we are confident are new. However, none of us is deep enough in the field to answer the sceptic's ...
- 519
6
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2
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470
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Does Well-Ordered Interval Power Set "WOIPS" principle , prove $\sf AC$ in $\sf ZFA$?
Does $\sf ZFA + WOIPS$ prove $\sf AC$?
Where $\sf WOIPS$ is phrased as: for every infinite set $X$ the set of intervening cardinals between $|X|$ and $|\mathcal P(X)|$ is well-ordered.
In $\sf ZF$, I ...
- 11.3k
4
votes
1
answer
150
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Comparing semiring of formulas and Lindenbaum algebra
This is motivationally related to an earlier question of mine.
Given a first-order theory $T$, let $\widehat{D}(T)$ be the semiring defined as follows:
Elements of $\widehat{D}(T)$ are equivalence ...
- 20.5k
5
votes
0
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192
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Do most semigroups have a zero?
It is widely believed in finite semigroup theory that asymptotically almost all finite semigroups $S$, up to isomorphism, are 3-nilpotent, i.e., they satisfy $\#\{abc\,:\,a,b,c\in S\} = 1$. My ...
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3
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92
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Reference for the monoidal category structure $X \otimes Y = X + Y + X \times Y$ on a distributive category
Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X ...
- 10.7k
2
votes
0
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100
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Realizing arithmetic hierarchy in algebraic number theory
Is it possible to realize arithmetic hierarchy in algebraic number theory?
For example, consider a $\Pi^0_4$ statement of the form $\forall x \exists y \forall z \exists w \phi(x,y,z,w)=0$ where $\phi$...
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14
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1
answer
1k
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Hilbert's sixth problem and QFT description
The Wikipedia entry on Hilbert's sixth problem about QFT description is “Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, modern quantum field theory can also be considered ...
- 3,104
6
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0
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173
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Measurable functions from logical formulas
Let $A(X, Y)$ be an arithmetical formula with (only) second-order variables $X, Y\subset \mathbb{N}$.
Assuming $(\forall X\subset \mathbb{N})(\exists Y\subset\mathbb{N})A(X, Y)$, there is a choice ...
- 4,359
1
vote
1
answer
143
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What determines non-finite axiomatizability of a class extension of a set theory?
Suppose $T$ is a set theory, i.e. doesn't have proper classes. And $T$ can interpret $\sf PA$, and $T$ is an effectively generated consistent first order set theory. Now, let $T^+$ be a class theory ...
- 11.3k
4
votes
1
answer
534
views
How to settle the Generalized Continuum Hypothesis when there are urelements?
Work in $\sf ZFCA$ and permutation models has preceded forcing by several decades. Was it used to settle the question of the Generalized Continuum Hypothesis $\sf GCH$ when urelements are admitted? I ...
- 11.3k
11
votes
1
answer
722
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Existence of finite powerset
Consider ZFC with both powerset and infinity removed and collection and $\in$-induction included. The well-ordering principle is not assumed.
Does this theory prove that, for every set $X$, there is ...
- 1,382
4
votes
1
answer
432
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How to settle continuum hypothesis like questions for impure sets?
How questions similar to the continuum hypothesis can be solved if we work in a set theory that admits urelements and that permit impure sets that are not injective to any pure set, for example $\sf ...
- 11.3k