All Questions
1,142 questions
3
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161
views
Making the powerset into a topological monoid
Every monoid $X$ induces a monoid structure $\circledast$ on $\mathcal{P}(X)$ via
$$U\circledast V := \{uv\ |\ u\in U,v\in V\}.$$
Moreover, a morphism of monoids $f\colon X\to Y$ induces a morphism of ...
- 11.8k
3
votes
0
answers
231
views
Is $\sf ZFC + Classes$ finitely axiomatizable?
$\sf ZFC + Classes$ is a bi-sorted theory with lower cases standing for sets and upper cases for Classes; axioms include all $\sf ZFC - Extensionality$ axioms written in lower case, and the following ...
- 11.3k
3
votes
2
answers
568
views
Can we write Tangled Type Theory without reference to type sequences?
I just want to know if Tangled Type Theory $\mathsf{TTT}$ of Randall Holmes ([see: Holmes - NF is consistent, p:11, Holmes - The equivalence of NF-style set theories with “tangled” type theories; the ...
- 11.3k
3
votes
0
answers
303
views
Pseudomodules, "general coherence theorem"
A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...
- 2,312
3
votes
1
answer
248
views
A question on recursion in Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection
$\Sigma_{3}KP\omega$ be Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection. What strengthening of Barwise's Definition by $\Sigma$ Recursion (Theorem 6.4 on ...
- 3,574
3
votes
2
answers
990
views
Theory interpreted in non-set domain of discourse may be consistent?
Following the blow. I will try to ask question in order to check if I well understand what was pointed. I decide to ask another question, because mathoverflow is not projected to be good environment ...
- 1,626
3
votes
1
answer
946
views
On statements provably independent of ZF + V=L
Are there any known statements that are provably independent of $ZF + V=L$? A similar question was asked here but focusing on "interesting" statements and all examples of statements given in that ...
- 33
3
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0
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249
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Independence through forcing vs generic collapses
Are there known statements in $V_{ω+ω}$ independent through forcing after $\mathrm{Col}(ω,<κ_1)*\mathrm{Col}(κ_1,<κ_2)*\mathrm{Col}(κ_2,<κ_3)*...$ where $κ_1<κ_2<κ_3<...$ are ...
- 7,545
3
votes
0
answers
262
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"Matryoshka" sets and the Axiom of Choice
Consider the following two very similar statements in ${\sf ZF}$:
(Mat_1) There is a set $A$ a map $\alpha: \omega \to {\cal P}(A)$ such that for all $n\in \omega$ we have $\alpha(n+1) \subseteq \...
- 48.9k
3
votes
1
answer
144
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Levels of L resembling each other, take 2
(Everything below is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\}...
- 20.5k
3
votes
1
answer
961
views
Extensions of fast-growing hierarchy
In recent weeks, I have been fascinated by the possible extensions of the fast-growing hierarchy. But is there a way to define it for all recursive ordinals? I saw a statement of this sort on ...
- 3,738
3
votes
1
answer
1k
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About Grothendieck Universe and Tarski's A and A' Axioms
A-The addition of the Grothendieck Universe Axiom (for every set x, there exists a set y that is a universe and contains x as member element) to ZFC (ZFC+GU) is considered as giving an almost good ...
- 2,655
3
votes
2
answers
480
views
A Baire subset of reals that is not Suslin measurable
EDIT: The definition of a Suslin measurable set I wrote here is incorrect. It should be that $\mathcal{S}$ contains the field (or algebra) of open subsets of ${}^\omega\omega$ (or, in other words, it ...
- 1,442
3
votes
1
answer
275
views
Is ZF + Def a conservative extension of ZFC+HOD? If not, what are counter-examples?
$\sf ZF + Def$ is the theory that extends $\mathcal L(=,\in)_{\omega_1,\omega}$ with axioms of $\sf ZF$ (written in $\mathcal L(=,\in)_{\omega, \omega}$) and the axiom of definability:-
$\textbf{...
- 11.3k
3
votes
1
answer
492
views
Does the partition principle imply (DC)?
For sets $x, y$ we write $x\leq y$, if there is an injection $\iota: x \to y$, and we write $x \leq^* y$ if either $x = \emptyset$ or there is a surjection $s: y \to x$. In ${\sf (ZF)}$ we have that $...
- 48.9k
3
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1
answer
404
views
How many monoids with $n$ arrows exist?
How many monoids with strictly $n$ arrows exist? Is this known? I ask this only out of curiosity. Looking at $n=1,2,3,4$, this number doesn't appear to be very large relative to $n$.
- 41
3
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1
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285
views
Cancellative semigroup on a distributive lattice
Let $(S,\le)$ be a distributive lattice. Is there a semigroup structure on $S$ such that $S$ is cancellative and always $(x\wedge y)(x\vee y)=xy$?
- 1,145
3
votes
1
answer
188
views
A sequence of cardinal characteristics constructed with hypergraph coloring
Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$.
A hypergraph $H=(V,E)$ consists of a set $V$ and a collection of subsets $E \subseteq {\cal P}(V)$. A coloring is a map $c: ...
- 48.9k
3
votes
1
answer
382
views
How to change the successor of a singular with a Woodin?
I'm looking for references on how to change the successor of a singular cardinal from "more or less" minimal assumptions. If possible, then without adding bounded subsets to the singular either.
In ...
- 39.8k
3
votes
1
answer
168
views
Is the consequence relation of a finite set of boolean connectives finitely generated?
I asked this question on math stack exchange, but it didn't receive any answers. Consider a countably infinite set of variables called $PROP$. We augment $PROP$ with a finite set of boolean ...
- 2,023
3
votes
1
answer
625
views
Are the paradoxes of material or strict implication used anywhere to prove theorems in mathematics
In the Stanford Encyclopedia of Philosophy entry "Relevance Logic", the following inference is listed as classically valid:
The moon is made of green cheese. Therefore, it is raining in Ecuador ...
- 6,132
3
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0
answers
99
views
Counting Eilenberg/Schutzenberger-type definitions of pseudovarieties
See Eilenberg/Schutzenberger, On pseudovarieties for background on pseudovarieties. I've phrased things in terms of pairs-of-sets to avoid some annoying language about multisets. Also, I'm aware that ...
- 20.5k
3
votes
1
answer
374
views
Is a finitely generated residually free group "almost LERF"?
Let $G$ be a finitely generated residually free group.
(i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.)
Let $...
- 11.3k
3
votes
1
answer
509
views
Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality
The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...
- 867
3
votes
1
answer
444
views
Number of non-isomorphic models
I had this question up on Math stackexchange: https://math.stackexchange.com/questions/1349247/number-of-non-isomorphic-models/1350763#1350763 . While it was answered partially there, I'm posting here ...
user75685
3
votes
1
answer
294
views
Tuple machinery in I-Sigma_0
After thinking on Joel's answer at Computable nonstandard models for weak systems of arithemtic for a few days, I do not see how to develop enough tuple machinery in I-Sigma_0 (PA with induction ...
user5810
3
votes
2
answers
331
views
Time functions of non-deterministic Turing machines (a better question)
This is a more precise version of that question.
Let $M$ be a non-deterministic Turing machine which recognizes a language $L$, that is, for every input word $u$ there is an accepting computation ...
user6976
3
votes
2
answers
233
views
${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$
Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define
$f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$;
$f\leq^* g$ if there is $N\in\omega$ ...
- 48.9k
3
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0
answers
314
views
Certain conditions on cancellative semigroups
This is extracted from this question following Benjamin Steinberg's suggestion.
For a semigroup $S,$ let $P(S)$ denote the power semigroup of $S,$ which is made up of all non-empty subsets of $S$ ...
- 1,445
3
votes
1
answer
802
views
Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2
While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an ...
- 95
3
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1
answer
313
views
When can you canonically extend an ultrafilter after forcing?
Suppose that $V$ is a model of $\sf ZFC$, and fix some regular $\kappa$, say $\omega_1$ for practical purposes.
Let $\cal U$ be an ultrafilter on $\omega_1$ in $V$ which is non-principal and even ...
- 39.8k
3
votes
0
answers
115
views
Are "equi-expressivity" relations always congruences on Post's lattice?
Given a clone $\mathcal{C}$ over the set $\{\top,\perp\}$, let $\mathsf{FOL}^\mathcal{C}$ be the version of first-order logic with (symbols corresponding to) elements of $\mathcal{C}$ replacing the ...
- 20.5k
3
votes
2
answers
720
views
Shortest axiom of infinity for foundationless set theory
Let $T$ be the theory with a binary symbol $\in$, an unary symbol $S$, and the following axioms:
Axiom of extension:
\begin{equation}
\forall x \forall y (\forall z (z \in x \leftrightarrow z \in ...
- 2,203
3
votes
1
answer
140
views
Does "productive = dimension $\omega$" for computable structures?
In analogy with the terminology for sets, say that a (countable, computable language) structure $\mathfrak{A}$ is productive if there is a computable way to properly expand any computable list of ...
- 20.5k
3
votes
1
answer
316
views
Unorthodox constructive reasoning: The Kleene Getaway
KG (the Kleene Getaway) is the name I improvised (in my answer to a question on MO on constructive Perron-Frobenius) for a constructive principle which enables the direct constructive use of a ...
- 1,616
3
votes
0
answers
853
views
What is the role of the (formalized) omega rule in Ramified Analysis?
In the 1960's, Feferman and Schutte did groundbreaking proof-theoretic work to find out the strength of predicative systems of second-order arithmetic. They used the ramified theory of types, a ...
- 4,597
3
votes
1
answer
258
views
Strength of BTEE
What is the consistency strength of Basic Theory of Elementary Embeddings (BTEE) from The spectrum of elementrary embeddings j : V → V by Paul Corazza?
BTEE uses the language of $(V,∈,j)$ and asserts:...
- 7,545
3
votes
0
answers
689
views
"Nicely" strong measure zero sets
This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: \omega\...
- 20.5k
2
votes
3
answers
1k
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on the set of numbers generated by integer linear combination of two real numbers.
Let $b > a > 0$ be two real numbers. I am interested in the set of numbers
$X(p,q) = p a + q b$ with $p,q$ positive integers. Basically this is the set $a \mathbb{N} + b \mathbb{N}$.
What ...
- 29
2
votes
1
answer
200
views
Some very weak statements on choice
This is a follow-up question to Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
Consider the statements
$(\text{S}1)$ For any infinite set $X$ there is ...
- 48.9k
2
votes
0
answers
195
views
"Very $L$-like" models, part 2: combinatorics
Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite ...
- 20.5k
2
votes
1
answer
142
views
Does $WKL_0$ plus CON(PA+X) give a binary tree model of PA+X?
In the context of reverse mathematics $WKL_0$ is considered equivalent to Gödel's completeness theorem over $RCA_0$. Does this mean that e.g. $WKL_0$ plus the consistency statement CON(PA+X) gives a ...
- 3,574
2
votes
2
answers
292
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Substructure Argument for Chain Conditions
Once, someone showed me a nice argument using elementary substructures for proving chain conditions about forcings. It was a basic example, maybe that Cohen forcing is ccc. I've been trying to look up ...
- 23
2
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1
answer
122
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If $H$ is essentially equimorphic to $K$, then is $H$ atomic only if so is $K$?
I will first state my question, and then give all the relevant definitions.
Q. Let $H$ and $K$ be monoids, and assume $H$ is essentially equimorphic to $K$. Is it true that $H$ is atomic only if so ...
- 10.5k
2
votes
0
answers
163
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Graph theoretical representation of Wang Tile
We note that for one dimensional tiling problem of Wang Tile could be represented by a graph. Each cycle on the graph represents a periodic solution.
However, is there a well established counter-part ...
- 867
2
votes
1
answer
131
views
A question on Carnap's modal semantics on the basis of Cochiarelli's primary semantics
I believe I learned that Carnap's state description semantics for propositional modal logic suffered from validating $\lozenge p$ for all atomic variables p. Re-reading Nino Cochiarelli's primary ...
- 3,574
2
votes
3
answers
662
views
logics restricted in arithmetic hierarchy
Hello, I would like to know if this already has been researched.
There has been lot of research done, where logics are limited. They are often limited in the axioms or inference rules, which makes ...
- 1,659
2
votes
1
answer
223
views
Possible symmetry groups of power terms
Previously asked and bountied at MSE:
Let $\mathfrak{E}=(\mathbb{N};\mathit{exp})$ be the algebra in the sense of universal algebra consisting of the natural numbers with just exponentiation. To each ...
- 20.5k
2
votes
1
answer
173
views
Chromatic number and taking duals of hypergraphs
If $H=(V,E)$ is a hypergraph and $\kappa\neq \emptyset$ is a cardinal, then a map $c:V\to \kappa$ is said to be a colouring if for every edge $e\in E$ with $|e|\geq 2$ the restriction $c\restriction_e:...
- 48.9k
2
votes
1
answer
307
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Sigma-complete Lindenbaum algebras? [closed]
Is there any calculus whose algebraization is a sigma-complete Lindenbaum algebra, i.e., a sigma-complete Boolean algebra after identification of equivalent formulas?
- 21