All Questions
1,459 questions with no upvoted or accepted answers
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Category whose morphisms are commutative monoids but not enriched
In a recent investigation, I constructed a category $\mathcal{C}$ with the following property. For objects $X,Y \in \mathcal{C}$, the morphism set $\text{Mor}(X,Y)$ is a commutative monoid with ...
- 161
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76
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Directly proving the extensionality principle for product types without quasi-inverses
In section 2.6 of the Univalent Foundations Project's Homotopy Type Theory book, the extensionality principle of product types is proven by showing that for all elements $a:A$, $a':A$, $b:A$, $b':A$, ...
- 2,624
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96
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Is Jaskowski's paraconsistent system moderate if sparked?
Stanislaw Jaskowski published a non-adjunctive paraconsistent logic, which does not have the inference rule $\vdash A \ \& \ \vdash B\Rightarrow \ \vdash A\wedge B$. The paper first appeared in ...
- 3,574
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70
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Another matrices for a semigroup with intermediate growth
Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where
$
A=\begin{bmatrix}
1&1\\
0&1\\
\end{bmatrix}
,
B=\begin{bmatrix}
1&0\\...
- 883
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92
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Doing reverse mathematics by regarding modal logic as weak first-order logic
Reverse mathematics seeks to find subsystems of second-order logic that are equivalent to certain mathematics theorems, say over $\mathsf{RCA}_0$.
Modal logic can be regarded as a weak version of ...
- 331
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52
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Can we define the notion of ordinal\cardinal definable set in Z + Ranks?
Working in Zermelo + Ranks, can we define the notions "ordinal definable" set, "cardinal definable" set? Or does it beg Replacement\Reflection to be defined?
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154
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Can we have a spectrum of intermediate choice properties between set choice and global choice?
Global choice is equivalent to saying $|V|=ON$, while ordinary choice is equivalent to saying that $V= H_{< ON}$. So the relative cardinalities of $V$ and $ON$ seems to affect the degree of choice ...
- 11.3k
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188
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Can we have a proper class of infinitely descending infinite ordinals?
Working in $\sf ZF-Reg.$, can we have a transitive model $M$ of $\sf ZF$ such that there exists a proper class (i.e. a subset of $M$ that is not an element of $M$) of infinitely descending infinite ...
- 11.3k
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52
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Term for degrees realizing least possible first n jumps
Is there a term for (Turing) degrees which realize the least possible jump (in the following sense) for the first n jumps.
That is degrees which satisfy for all $0 < m \leq n$:
$$X^m \equiv_T X \...
- 3,029
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274
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Functional equation $f(x*y) = f(f(x)*f(y))$
Find all endo-functions $f$ on a commutative semigroup $(\mathbb{S},*)$ such that
$f(x*y) = f(f(x)*f(y))$.
Typical case of interest are $(\mathbb{N},+)$ or $(\mathbb{Z}/k\mathbb{Z},+)$ or $(\mathbb{Z}/...
- 1,306
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The "hyperbolicity preserving" probabilities
A classical fact (due to Polya ?) is that if $P\in{\mathbb R}[X]$ has only real roots (one says that $P$ is hyperbolic), and $a$ is a real number, then the roots of
$$L_aP(X):=\frac12(P(X+ia) +P(X-ia))...
- 52.3k
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195
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A variant of Buchholz's ordinal notation
Buchholz here introduced an ordinal notation, consisting of a set $\mathcal{T}$, a linear order $\prec$ on $\mathcal{T}$ and some $\mathcal{OT} \subset \mathcal{T}$ such that $(\mathcal{OT}, \prec)$ ...
- 704
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165
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Question regarding ultrafilter extension of $\tau$-model
Since I'm not native speaker, my writing is probably difficult to read. Hence please point out any mistakes.
I'm reading page 96 and 97 of Modal Logic written by Patrick Blackburn.
$\textbf{...
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What is the consistency strength of this addition on simple type-set theory?
Language: multi-sorted first order logic with equality and membership, where for each natural $n$ there is a set $x^n$ of sort $n$. Equality "$=$" only occurs between variables of the same ...
- 11.3k
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Is $\sf \Gamma_0$ the proof theoretic ordinal of this kind of predicative class theory?
Adopting the approach of Mono-sorted $\sf NBG$, define sets as elements of classes, then axiomatize:
Extensionality, Predicative Class comprehension, emptyset, in the usual manner along mono-sorted $\...
- 11.3k
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127
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What is the proof theoretic ordinal of this kind of predicative type-set theory?
The following is a kind of Predicative Type Set Theory.
The question is about what is exactly the proof theoretic ordinal of this theory? Is it lower than the one expected for predicative theories, i....
- 11.3k
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117
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Can this type theory interpret second order arithmetic?
Language: multi-sorted first order logic with equality and membership, where for each natural $t$ there is a set $x^t$ of sort $t$. Equality "$=$" only occurs between variables of the same ...
- 11.3k
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94
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Is definability in $V$ in $\sf Ack+MK$ expressible in its language?
Recall Ackermann set theory. If we extend Ackermann's set theory by adding all axioms of $\sf MK$ to it. We shall denote the universe of all elements by $W$, while $V$ is the primitive constant symbol ...
- 11.3k
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112
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What's the consistency strength of adding this inference rule to Ackermann's set theory?
Working in the language of Ackermann set theory:
Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\...
- 11.3k
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Can we add the singleton map to $\sf NFP$?
In $\sf NF(U)$ it is known that the singleton map $(x \mapsto \{x\})$ is not a set, which is a source of a lot of extremely counter-intuitive results in $\sf NF(U)$. However, a weakening of the ...
- 11.3k
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Does ${\sf ZC + Universes + ZFC}^V$ meet Muller's criteria for a founding theory of Mathematics?
I was re-thinking Muller's criteria in Sets, Classes and Categories: page 14 for a theory that founds Mathematics. To him, it should be able to be a foundation of Category Theory. He lays down six ...
- 11.3k
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Can all stages of the cumulative hierarchy beyond $V_\omega$ violate the weak partition principle?
This question is a follow up of this.
Is it consistent for ALL infinite stages $V_{\alpha > \omega}$ of the cumulative hierarchy of $\sf ZF$, to violate the weak partition principle? That is, each ...
- 11.3k
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119
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Can all relations and functions be implemented as sets in some fragments of set theory?
Define wholly stratified $\sf NF$ to be $\sf NF$ with its language restricted to stratified expressions.
In this theory we can arrive at a general implementation of tuples, that is:
$\langle x_1,..,...
- 11.3k
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131
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What is the consistency strength of iterated sharps?
I've been interested in sharps such as $0^\sharp$, $0^{\sharp \sharp}$ and $\mathbb{R}^\sharp$, so I wondered about iterating sharps. Let $n \in \omega$ and $x \in V$. Define $x^{\sharp n}$ like so:
...
- 704
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103
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What is the set of axioms for this theory extracted from what is provable in the minimal model of ZFC?
This post is a follow up of this one posted to MathStackExchange.
Working in $\sf ZFC + \exists M \, (M \overset{trs} \models ZFC)$, lets define a theory $T_0$ as:
$$(\varphi \ \epsilon \ T_0) \iff \...
- 11.3k
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146
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What's the consistency strength of this kind of cardinal?
Bumped, since I was recently thinking about these again.
My friend introduced the following notion:
Let $\xi > 0$, $\eta$ be ordinals, $n$ be a natural number and $\mathcal{A}, X$ be classes. A ...
- 704
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76
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Proof of the Local Deduction Theorem, for one of many logics
I'm looking for a proof of the Local Deduction Theorem for any one of many logics. That is, a proof of the statement:
$\Sigma \cup \{\phi\} \models \psi$ iff for some positive $n,$ $\Sigma \models \...
- 421
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182
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Thinning chains of elementary extensions
I'm bumping this question, since I'm still curious regarding the answer but this question seems to have gone unnoticed. Bumped again.
This is a follow-up to this question, regarding a stronger variant ...
- 704
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100
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Are those two theories equivalent?
Lets denote any set that is "the set of all strictly smaller subsets of it" as size- unreachable. Formally: $$\operatorname {size-unreachable}(X) \iff \\\forall Y (Y \in X \iff Y \subset X \...
- 11.3k
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139
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Can Reinhardt cardinals be compatible with Choice in absence of Extensionality?
Is the proof of existence of Reinhardt (and higher) cardinals violating Choice dependent on Extensionality in an essential manner?
What I mean is if we work in $\sf ZFA$ would it be possible to have a ...
- 11.3k
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170
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Can Jensen's covering lemma be proven easier in generic extensions of L?
Jensen's covering lemma, stating that if there is no $0^\#$ in V, then some covering property holds true, has a very complex proof.
In any generic extension L[G] of L, $0^\#$ don't exist, so the ...
- 1,490
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155
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Countable $L_\alpha$ model for $S$ if $S$ has a countable well founded model?
Let $S$ be a proper fragment of $\mathit{ZFC}$, and let $S$ properly extend $\mathit{ZFC}^-$, i.e. $\mathit{ZFC}$ minus the power set axiom. Is there a countable ordinal $\alpha$ so that $L_\alpha$ is ...
- 3,574
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183
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Can GCH fail everywhere in every finite way?
Since the $\sf GCH$ cannot fail everywhere everyway (see here), the question here is if it can fail everywhere in every finite manner, that if we have a strictly increasing function $f$ on the ...
- 11.3k
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231
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Can we plausibly interpret the empty set in Mereology as a distant atom?
The Mereological grounding of set theory that follows Lewis's general outlines have various possible approaches to the definition of the empty set. But the choice of these approaches are (admited by ...
- 11.3k
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43
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Interleaving in Viennot's Heaps models?
I am looking for past results on interleaving of heaps (in the sense of Viennot). For a very simplified example, suppose I have two pieces, (b1 a1 b1), and (b2 c2 b2), where the letter represents a ...
- 11
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66
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The history of suprema involving parameters
I had the following historical question (with implications for logic/reverse math):
Who first used the notion of supremum explicitly involving parameters?
Let me provide a positive example of the ...
- 4,359
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145
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Can we entangle two external injections this way?
This question is related to a question Is this internalization of a bijection between a set and its powerset possible? lately posted to $\cal MO$, as well as to Can we have a bijection between a set ...
- 11.3k
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192
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Does foundationless Ackermann set theory prove replacement?
From Ackermann's set theory equals ZF (1970) by William N. Reinhardt:
Let A be the theory determined by the following axioms:
Extensionality: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$
...
- 2,203
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140
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Kleene-fixpoint, transfinite recursion or sui generis?
In the context of type free set theories with set abstracts, as e.g. investigated by Andrea Cantini, Logical Frameworks for Truth and Abstraction, Elsevier 1996, one has the following construction (...
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To be is to be an element?
The distinction between sets and proper classes has interesting ontological consequences.
One way to define a proper class, in the context of set theory, is to state that it is not an element, in a ...
- 3,574
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70
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Can all sets in stratified L above some stage be proximate?
Define stratified $L$, denoted by $^S L$, as:
Let $S$ be the set of all stratified formulas in first order language of set theory.
Define:
${ }^S Def (X) = \{\{y \in X \mid (X, \in) \models \phi(y,z_1,...
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223
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Shortest proof of inconsistency of first order theory
Suppose we are studying recursively axiomatizable first order theories in some metatheory (e.g. PA or ZF).
If we have a proof that a given recursively axiomatizable first order theory is complete and ...
- 11
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56
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Effect on finite transformation semigroup under a particular modification of the generators
The following question arises in connection with problems in automata theory related to the road problem. Let $f_1, f_2: [N] \to [N]$ be maps such that the transformation semigroup $S = \langle f_1, ...
- 695
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First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?
In this old paper D. B. McAlister has introduced another class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the ...
- 1,093
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170
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What is this algebraic object (special case of a semigroup)?
Let $(M,*)$ be a finite semigroup. Further we demand the following:
Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$.
Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$.
...
- 696
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268
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The most simple proof of projective determinacy
I want to read the proof of projective determinacy.
But every proof I could find (martin-steel original, koepke's, the proof in schindler's book, martin's new book) is too long.
Are there a simple ...
- 1,490
1
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answers
172
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May we axiomatize by means of Gödel codes?
Suppose we have a classical arithmetical theory $\mathbf{R}$, at least as strong as Robinson arithmetic, with $\tau$ an extra dummy monadic predicate in its language $\mathcal{L}$. Suppose $\mathbf{R}$...
- 3,574
1
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139
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Terminology for an kind-of principal fibration
My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets.
Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
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261
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Is this theory using defined notions of classes, sets, and membership interpretable in ZFC?
The main difference with this formal theory is that it depends in an essential manner on a defined notion of class, set, and set membership $\in$, rather than the usual appraoch of leaving them ...
- 11.3k
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121
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Why $\beta S$ is not a semigroup when $S$ is a (directed) partial semigroup?
Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as
$$
\mathcal{U} * \mathcal{V} = \left\{ A \...
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