All Questions
1,460 questions with no upvoted or accepted answers
9
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239
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Which semirings have enough injectives in their category of modules?
Let $R$ be a semiring and $Mod_R$ its category of modules. That is, $R$ is a monoid in the monoidal category of commutative monoids and $Mod_R$ is its category of modules in the usual sense.
Question ...
- 63.9k
9
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440
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A new maximality principle and its consequences
Let us consider the following maximality principle:
$(MP_*):$ For all uncountable regular cardinals $\kappa, 2^{<\kappa}=\kappa^{+}$
and all trees of height and size $\kappa$ are specialized.
It ...
- 32.1k
9
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0
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471
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(A little bit) Beyond the E-recursive
The E-recursive functions are a particular generalization of classical recursion theory to the entire set-theoretic universe, $V$. They are defined via a schemes: see Sacks' $E$-recursive intuitions. ...
- 20.5k
9
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240
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Homogeneity of a variant of Prikry forcing
Prikry forcing is easily seen to be cone homogeneous (for any $p, q \in \mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: \mathbb{P}/p' \simeq \mathbb{P}/q'$); in particular for ...
- 32.1k
9
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373
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Embedding $\beta\mathbb{N}$ into a product of Cantor sets
Let us consider $\beta\mathbb{N}$, the Stone-Čech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ...
- 297
9
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242
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Two cardinal obstructions
Given a theory $T$ and a formula $\phi(x)$ we say that they admit a $(\kappa, \lambda)$ model if there is a model $M$ such that $|M| = \kappa$ and $|\phi(M)| = \lambda$.
In all examples that I know ...
- 866
9
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299
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On an unpublished result of Magidor
In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$
is strong limit and $2^{\aleph_\omega}=\aleph_{\...
- 32.1k
9
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0
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271
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Which forcing types preserve the axiom of determinacy?
Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy?
To be more specific, in Which forcings ...
- 39.8k
9
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358
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Is there a "hereditary" construction for $L$?
Recall that $L$, Godel's constructible universe is constructed by defining the following hierarchy:
$L_0=\varnothing$, for a limit ordinal $\delta$, $L_\delta=\bigcup_{\alpha<\delta}L_\alpha$, and ...
- 39.8k
9
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0
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304
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Co-Heyting Valued Models of Paraconsistent Set Theory
I've been trying to do some forcing arguments in intuitionistic ZF using Heyting valued models where the Heyting algebra I'm using is actually a bi-Heyting algebra (both a Heyting algebra and a co-...
- 631
9
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0
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319
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From Frege to Gödel - German equivalent?
I know this question does not quite fit here, but I felt it could best be answered here. I recently stumbled upon the book From Frege to Gödel, which is a sourcebook containing some of the most ...
- 213
9
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0
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526
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"Hard" separation results in reverse mathematics (or similar)
This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...
- 20.5k
9
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965
views
Has anyone pursued Frege's idea of numbers as second-order concepts?
Gottlob Frege was a pivotal figure in the history of mathematical logic. He gave an analysis of numbers that proceeded along roughly the following lines, in his books "The Foundations of Arithmetic" (...
- 4,597
9
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369
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Is there Ultracoproduct-like construction for topological spaces in general?
In
http://arxiv.org/pdf/math/9704205.pdf
they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...
- 241
9
votes
1
answer
861
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$\Pi_0^1$-weakly indescribable cardinals are exactly the regulars
Hi,
I'm not sure if I should ask here or over at math.stackexchange.com, but I think here it's a bit more fitting. This question stems from a homework problem:
Definition:
Given some class of formulas ...
- 331
8
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0
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199
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Can $\mathbb{C}$ have a "doppelganger" in $L(\mathbb{R})$ with countable automorphism group?
Working in $\mathsf{ZFC}$ + large cardinals (a proper class of Woodins, to be precise), is there a field $F\in L(\mathbb{R})$ such that $V\models F\cong\mathbb{C}$ and $L(\mathbb{R})\models\vert\...
- 20.5k
8
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0
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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
8
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0
answers
164
views
Is there a substructure-preservation result for FOL in finite model theory?
It's well-known$^*$ that the Los-Tarski theorem ("Every substructure-preserved sentence is equivalent to a $\forall^*$-sentence") fails for $\mathsf{FOL}$ in the finite setting: we can find ...
- 20.5k
8
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152
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Which sentences are "strategically preserved"?
Below, everything is first-order.
Say that a sentence $\varphi$ is strategically preserved iff player 2 has a winning strategy in the following game:
Players 1 and 2 alternately build a sequence of ...
- 20.5k
8
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0
answers
386
views
Can the p-adic be countable?
Recently arxiv submitted a new paper (Andrej Bauer, James E. Hanson, The Countable Reals) claiming an incredible theorem that Dedekind reals are not sequence-avoiding, and furthermore obtaining a ...
- 695
8
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368
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An obscure case of Curry-Howard
It is a theorem of the Intuitionistic Propositional Calculus that
$$
(p\to q)\to p = (q\to p) \land ((p\to q)\to q).
$$
The Curry-Howard correspondence realizes this as a pair of operators (for any ...
- 18.4k
8
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246
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Large cardinals beyond choice and HOD(Ord^ω)
Are Reinhardt and Berkeley cardinals (and even a stationary class of club Berkeley cardinals) consistent with $V=\mathrm{HOD}(\mathrm{Ord}^ω)$ ?
It seems natural to expect no, but I do not see a proof....
- 7,545
8
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148
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What is this quotient of the free product?
Previously asked at MSE. The construction here can generalize to arbitrary algebras (in the sense of universal algebra) in the same signature with the only needed tweak being the replacement of "...
- 20.5k
8
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0
answers
196
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Reference request: choiceless cardinality quantifiers
There is a substantial literature on the logic of cardinality quantifiers. (E.g., the quantifier $Q_\alpha$ where $M \vDash Q_\alpha x \, \varphi (x)$ iff $\vert \{a \in M : M \vDash \varphi[a] \} \...
- 1,155
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157
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How strong is exponentiation with only open induction? (Or: "how low can we go?")
Do the strongest theories currently known to be unconstrained by Tennenbaum's theorem ($IOpen$ and some modest extensions) remain so when augmented with a definition of exponentiation and axiom $\...
- 3,612
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523
views
What is the relationship (if any) between constructivism, finitism and predicativism?
The terms “constructivism”, “finitism” and “predicativism” refer to ideas / currents in the philosophy of mathematics (or loosely defined conditions on a system of logic) that I think I understand ...
- 32.5k
8
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97
views
Is the hypotenuse operation associative in every Tarski plane?
By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...
- 41.9k
8
votes
0
answers
169
views
Upper-bounding determinacy
While the converse of Borel determinacy ("If a set of reals is determined, then it is Borel") is boringly disprovable, I'm curious if there is a sense in which something like it is ...
- 20.5k
8
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244
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First order formula describing connected components
I ask this question here after no answer came up in the original MathSE question.
Let $\mathcal{L}$ be the language $\{+,-,\cdot,0,1,P\}$ where $P$ is some $n$-ary relation symbol. Is there a formula $...
- 478
8
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0
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411
views
Semigroups of matrices closed under conjugate transposition
An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
- 563
8
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187
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Intuition for branch uniqueness in inner model theory
In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage?
At the level ...
- 7,545
8
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0
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285
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Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 4,943
8
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348
views
Curry-Howard isomorphism: What is the logical counterpart of closure conversion?
Continuation-Passing Style (CPS) translation in programming languages corresponds to double-negation translation in logic (and the Yoneda lemma in category theory). Then what in logic corresponds to ...
- 193
8
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0
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280
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Inner models from highly saturated ideals
Solovay proved that a measurable cardinal is equiconsistent with the existence of an extension of Lebesgue measure to a full measure on $P(\mathbb R)$. The inner model direction is relatively simple. ...
- 18.6k
8
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0
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416
views
Choice function for elementary embedding $j: V_\lambda\prec V_\lambda$
Work in ZF. Let $\lambda$ be strongly inaccessible. Let $\kappa$ be such that for every $\alpha\lt\lambda$, there is some $j: V_\lambda\prec V_\lambda$, with critical point $\kappa$, such that $j(\...
- 1,133
8
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0
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452
views
Alternative definition of Kolmogorov complexity
In Kikuchi's paper Kolmogorov complexity and the second incompleteness theorem the Kolmogorov Complexity (KC) of $x$ is defined as
$$ K(x) = \mu e (\varphi_e(0) \simeq x) \, . $$
This seems to give ...
- 189
8
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266
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Are there analogues of real-valued measurability for larger powersets?
Apologies if the question is somewhat naïve - I'm not a set theorist, just an enthusiast.
One possible intuition we could have about the generalized continuum hypothesis is that power sets are so ...
- 8,688
8
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184
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Topological Vaught's conjecture for special theories
As is know, Vaught's conjecture is a special case of topological Vaught's conjecture.
On the other hand, the Vaught's conjecture is true for the following theories:
1- $\omega$-stable theories (...
- 32.1k
8
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451
views
Does any 'logical' theory have a bounded ∞-pretopos as syntactic category?
Stone duality may be understood as providing a duality between syntax and semantics for propositional logic, so that a theory may be recovered from its models. In order to do likewise for first-order ...
- 5,094
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419
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Are most semigroups nilpotent of degree 3?
A semigroup is nilpotent of degree 3 if every product of 3 elements gives the same result. In 2012, Andreas Distler and James D. Mitchell wrote that:
It is part of the folklore of semigroup theory ...
- 22.3k
8
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224
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Large "computably un-simplifiable" computable well-orderings
Question
Suppose $A,X$ are computable well-orderings. Say that $A$ is $X$-unsimplifiable if there is no computable well-ordering $B$ whose ordertype is strictly less than that of $A$ but such that ...
- 20.5k
8
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206
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ladder system uniformization at successors of singulars
Shelah proved (paper 667) that if GCH holds and $\lambda$ is singular, then for every stationary $S \subseteq \{ \alpha < \lambda^+ : \text{cf}(\alpha) = \text{cf}(\lambda) \}$, there is a ladder ...
- 18.6k
8
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0
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581
views
Does second order ZFC conservatively extend first order ZFC?
If I replace the axiom schema of specification in ZFC by a single axiom in second order logic, and similarly do same thing for the axiom schema of replacement, is this version of "second order ZFC" ...
- 81
8
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259
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Monadic second-order theories of the reals
I’m looking for a survey of monadic second-order theories of the reals.
I’m starting from a 1985 survey by Gurevich which says (p 505) that true arithmetic can be reduced to “the monadic theory of ...
user44143
8
votes
0
answers
1k
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Can every set be measurable?
The Solovay model shows that ZF (plus an inaccessible) is consistent with every subset of $\mathbb{R}$ being measurable. How far can we go in that direction? We can always have non-measurable sets ...
- 6,870
8
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0
answers
241
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Topological applications of $\mathfrak{p}=\mathfrak{t}$
I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality.
Searching in ...
- 181
8
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286
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Proper classes in Bounded Zermelo set theory
I want to know if there is a standard terminology for this among set theorists working with element-based set theories like ZFC. I will follow the convention that a class in any given set theory is a ...
- 11.1k
8
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0
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287
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Natural examples of recursive pseudowellorderings
Question: What are some natural examples of recursive pseudowellorderings?
By natural, I mean in the style of reasonable ordinal notation systems as opposed to dependent on a Gödel numbering or an ...
- 7,545
8
votes
0
answers
191
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Specializing fat trees
The discussion is about trees of height $\omega_1$ that are not necessarily thin, namely, no cardinality constraints on the size of each level. A classcial theorem of Baumgartner states that it is ...
- 1,006
8
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0
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155
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Is every total computable function definable by a strongly total lambda term?
Every computable (total) function $f : \mathbb{N} \to \mathbb{N}$ is definable in untyped pure lambda calculus in the sense that there is a term $F$ such that, for every Church's numeral $c_n = \...
- 4,459