All Questions
1,460 questions with no upvoted or accepted answers
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Entailment and implication
Over on the nlab, I was looking at the page for fully formal ETCS and one clause stood out, namely the one for "well-pointedness", namely:
$$(s(f) = s(g) \wedge t(f) = t(g)) \vdash \forall_h (s(h) = ...
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Defining filters in closure algebras: reference request
A closure algebra C is a boolean algebra B together with a unary closure operator, and additional axioms, the Kuratowski axioms, that the closure operator must satisfy. (The Wikipedia article prefers ...
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203
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Maximizing the number of 'correct' literals in planar monotone 3SAT
I'm trying to find the complexity of this optimization problem:
Given an instance of planar monotone 3SAT, with positive clauses $C_i = v_{i1} V v_{i2} V v_{i3}$ and negative clauses $D_i = not(w_{i1}...
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Single logic foundation vs. multi-logic foundation
Dear all,
I have always wondered why I have never read anything about this topic. My question is, are there are any books or articles covering this subject?
With this topic I mean the philosophical ...
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A question about set theory and Frege logic
Does there exist a very weak axiomatic theory of arithmetic-weaker than (but possibly a sub-theory of)
Robinson's theory Q-which can be interpreted in the first order fragment of Frege logic? If so, ...
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Concept of synchronizability
This thread is about the concept of synchronizability. It's a concept I tried to formalize in its most general sense but without success. The goal of this thread is therefore to try to formalize it in ...
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Is there any current development of a first order formalization of metamathematics?
I hope that this post isn't off topic, but I already asked math.stackexchange about first order formalizations of first order logic. There are provability logics and extensions in modal logic's that ...
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How near are a groupoid and its 'preorderification'?
As remarks, a groupoid is a category with only (categorical) isomorphisms as its morphisms and a preorder is a category only having one morphism between each object. If we choose one isomorphism by ...
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Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
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the (indirect) deduction theorem
$\DeclareMathOperator\Cn{Cn}\DeclareMathOperator\Sb{Sb}$I would like to ask about the Deduction Theorem for an inconsistent system. This is a very well-known fact that for the classical propositional ...
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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 ...
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143
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Introductory resources on rewriting logic
Hi I would like to grasp the theory behind Maude [1], [2]
Are there any recommended video lecture notes, talks or introductory notes?
I have been exposed to Functional Analysis, Topology and some Term ...
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245
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Is this mereotopology theory consistent?
$ \newcommand{\Pt}{ \ \mathbb P \ }
\newcommand {\cz}{\ C_z \ }
\newcommand {\eps}{\ \varepsilon \ }$Logic: first order logic with equality
Extra-logical primitives:
"$\varepsilon$" ...
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Is Acyclic ZF consistent with downshifting automorphisms?
Recall the criterion of acyclic comprehension. This is shown to be equivalent to stratified comprehension for language $\sf FOL(=, \in)$, given minimal assumptions. [See here, and here].
Let Acyclic ...
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Which algebraic structure characterizes the set of non-trivial qudratic residues in a finite field?
I understand this question may be too naive to ask, but I am unable to figure it out.
Suppose, $\mathbb{QR^*}$ denotes the set of all quadratic residues in a finite field except the identity element $...
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Is stratified Z - Infinity + there is a set as big as its powerset, consistent if NF is consistent?
The question of consistency of $\sf NF$ can be seen to be equivalent to the question of whether the theory "Stratified $\sf Z$ - Regularity - Infinity + There exists a set as big as its powerset&...
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135
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Provability predicates
We know that there are provability predicates, that is, predicates derived from the recursive relation "x is a demonstration of y", with which Godel's second incompleteness theorem would not ...
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What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known?
What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known?
Given that there are $3{,}684{,}030{,}417$ semigroups of order $8$, I guess $n\...
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63
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A construction that sort of merges two semigroups to build a new one
Suppose $H$ and $K$ are semigroups and assume without loss of generality that (the underlying sets of) $H$ and $K$ are disjoint. We can then extend the operations of both $H$ and $K$ to a binary ...
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Is there a name for this condition on a monoid?
Suppose we have a commutative monoid ${\mathcal M}=\langle M,\otimes\rangle$ such that the usual divisibility relation $\leq_\otimes$ given by $a\leq_\otimes b\Leftrightarrow \exists c(a\otimes c=b)$ ...
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Is ZFC interpretable in bisimilar ZFC?
If we add a primitive binary relation $\sim$ to denote "bisimilarity" relation. Remove the axiom of Extensionality from axioms of $\sf ZFC$, and add:
Bisimilarity: $\forall x \, (x \sim x) \\...
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Is there a clear inconsistency with this general assertion about n-internalizations of external bijections?
Define: $j^1[x]= j(x) \\ j^{n+1}[x] = \{j^n[y]: y \in x\} \\ j^{-n}[x] = \{y : j^n[y] \in x\}$
Define:
$n=1,2,3,...\\ _n\mathsf{Forth}_j(S)=\{j^n[x] : x \in S\} \\ _n\mathsf{Back}_j(S)=\{j^{-n}[x] : ...
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Polyextremal groups
A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form
$f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
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To find a DFT for complex functions on a semigroup
For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
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Countably infinite monoids with minimal right ideals
Is there any classification of countably infinite monoids with minimal right ideal? or at least in some classes of monoids?
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211
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What are the hidden assumptions behind Harvey Friedman's claim, CSR?
I'm doing some archeology and trying to understand a claim. As summed up by David Roberts, on the FOM list in 2011:
Let the statement "every infinite sequence of rationals in [0,1] has an ...
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137
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Is definable bounded separation equivalent to bounded separation?
If we have all axioms of Mac Lane set theory except Separation and add to them the schema of definable bounded separation, then would the resulting theory be equivalent to Mac Lane set theory?
...
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152
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What is the strength of allowing multiple predecessor numbers?
If we have a theory of numbers, pairs of numbers, and sets of those, and axiomatize that the relation $<$ on numbers is both extensional and well founded, then this theory would prove all PA axioms ...
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How to define BHO alternatives below admissible ordinals?
Bachmann-Howard ordinal is a recursive ordinal. It's not that large compared to those proof-theoretic ordinals of stronger theories, but the definition of BHO is sufficient to illustrate how ...
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Is ZF equivalent to Specification + Sentence Reflection?
Examine the first order theory with the following extra-logical axioms:
Specification:: $$[\forall A \, \exists! X \, \forall Y \, (Y \in X \iff Y \in A \land \phi)]$$; whenever $\phi$ doesn't use the ...
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Does existence of those cyclic cardinals imply a kind of Choice?
Working in $\sf ZF - Fnd$, add the following axiom:
AntiFoundation: $\forall x: x \neq \emptyset \to \exists! y: y \in y \land y \sim x$
where "$\sim$" stands for existence of a bijection.
...
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An arithmetic hierarchy without bounded quantifiers
I posted this question on math.stackexchange.com with no answers (https://math.stackexchange.com/questions/4337852/an-arithmetic-hierarchy-without-bounded-quantifiers).
Let $\exists_n$ formulas in the ...
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ZF plus class-choice?
$\DeclareMathOperator\Cls{Cls}\newcommand{\ZFC}{\mathrm{ZFC}}$Let ZF be extended with class and set predicates, and extend the theory with a class abstraction schema so that for all formulas $\phi$:
$$...
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168
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Can we have a bijection between a set and its powerset with the following properties?
This question is related to a question Is this internalization of a bijection between a set and its powerset possible? lately posted to $\cal MO$. Similarly, we add one primitive unary partial ...
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119
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Scott-replacement and transitive closure
In D. Scott: More on the axiom of extensionality, in Essays on the foundations of mathematics, dedicated to A. A. Fraenkel on his seventieth anniversary, edited by Y. Bar-Hillel, E. I. J. Poznanski, M....
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Axiom of Choice and bases of $k$-vector spaces, $k$ fixed
I know that from ZF + the Axiom of Choice (AC) follows that every vector space has a basis.
And, conversely, Blass proved that in ZF set theory, the assumption that every vector space has a basis ...
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Inconsistency in a modal logic
I need a first order modal logic, where inconsistency between formulas in not binary: a pair of formulas may be more or less inconsistent. The modal operators express uncertainty. So the formulas ...
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Erotetic inference and extrinsic justification?
Gödel introduced his notion of what has come to be called extrinsic justification in the following terms:
Furthermore, however, even disregarding the intrinsic necessity of some new axiom, and even ...
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113
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Empty context-sensitive language independent of ZFC?
Is there a simple context-sensitive grammar $G$ such that $L(G)=\emptyset$ is independent of ZFC?
$L(G)$ is the formal language generated by $G$.
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Where does intuitionistic predicate logic live in the arithmetical hierarchy?
I started reading Plisko's papers on arithmetic complexity on the arithmetic complexity of constructive logic (see for example here or here). In this context, I started wondering about the following ...
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Can this theory that internalize sets from external functions be consistent?
I want to know if the following theory stand a chance of being consistent?
$Language:$ Mono-sorted first order predicate logic + primitives of equality $``="$ and class membership $``\in"$:
Define: $...
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Quotient of monoids and monoid algebras
Let $ X $ be a monoid and $ R $ be a (two-sided) congruence relation on $ X $ which is generated by some relations $ u_i \equiv_R v_i $ for any $ i $ in some index set $ J $. Let $ K $ be a field, $ K[...
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Reflective embeddings outside separation?
One of the known ways to axiomatize ZF is via these two axiom schemata:
Separation: $\forall \vec{w} \forall A \exists! x: x=\{y \in A: \phi\}$
Reflection: $\forall \vec{w} \exists \alpha: \phi \to \...
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184
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Proper class of nested rank into rank embeddings
I propose the following large cardinal axiom:
There exists a proper class of cardinals $\lambda$, such that for each $\lambda$, there exists a rank-into-rank embedding $j: V_\lambda \rightarrow V_\...
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Flag variety as monoid and Schubert calculus
The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation.
When looking ...
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What are the handy, go-to methods of proving consistency of a proof system?
Suppose you face a proof system portraying some notion or knowledge that you haven't encountered, or others haven't studied before. What would be your first attempts to examine the consistency of the ...
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276
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Axiom of determinacy as setting for studying rigs with $\operatorname{Aut}(\mathcal{M})\cong\operatorname {Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$?
I read years ago in the great French book "l'aventure des nombres" (Gilles Godefroy, Odile Jacob, 1997) that the axiom of determinacy was incompatible with the axiom of choice. The latter ...
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Can this Ackermann like set theory formulated without adding a primitive of set-hood reach the consistency of ORD is Mahlo?
The following is a theory that uses a reflection principle similar to Ackermann but on a size notion that is definable in the language of set theory, my question is about its consistency strength ...
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250
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Has this theorem on cancellative monoid actions been discovered and published?
Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?
Theorem 1. Let $W$ be a non-trivial cancellative invertible-free [1] ...
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Proof that $\omega_1^{CK}$ is admissible
An ordinal $\alpha$ is admissible if $L_\alpha\vDash KP$ (Kripke–Platek set theory). $\omega_1^{CK}$ is the least non-recursive ordinal; the set of all recursive ordinals. It is known that $\omega_1^{...
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