All Questions
1,460 questions with no upvoted or accepted answers
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Collapsing the Linear Time Hierarchy and finite axiomatizability of bounded arithmetic
It is well known that if ${\bf T_2}$ (or $I\Delta_0+\Omega_1$) is finitely axiomatizable, then the Polynomial Hierarchy collapses.
Q. Is there any similar relation between $I\Delta_0$ and Linear ...
- 1,689
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478
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Cohen's model yet again
It has been discussed already whether a countable OD set necessarily contains an OD element. See e.g.
A question about ordinal definable real numbers
.
A negative answer was obtained in Archive for ...
- 2,281
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363
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Inner models and strongly compact cardinals
The following question is motivated by a result of Magidor that it is consistent that the least strongly compact cardinal is the least measurable cardinal.
Question. Assume $\kappa$ is a strongly ...
- 32.1k
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276
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A variant of strong ideals, is it consistent?
Is it consistent relative to large cardinals that there is a precipitous ideal on $\omega_1$ forcing a generic elementary embedding $j : V \to M \subseteq V[G]$, such that $j(\omega_1) = \omega_n^V$ ...
- 18.6k
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508
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Riemann hypothesis in Zilber's field
Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?
- 32.1k
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216
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Can we find minimal-diameter metrics without computability?
A beautiful argument by Nabutovsky and Weinberger (see http://math.uchicago.edu/~shmuel/fractal.ps) shows that, if $M$ is any smooth compact manifold of dimension $\ge 5$, then the diameter functional ...
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292
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When does $HOD^{V[G]} \subseteq V$?
Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$
satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$.
...
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331
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preserving saturated ideals
A reliable source made the following claim:
Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$.
Question 1: How do you ...
- 18.6k
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306
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The Chang model after collapsing an inaccessible limit of Woodins
If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...
- 5,471
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465
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Does Sageev's result need an inaccessible?
In 1981, building on work by Ellentuck in 1974, Sageev showed ("A model of ZF + there exists an inaccessible, in which the Dedekind cardinals constitute a natural non-standard model of arithmetic," ...
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270
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Maximality of linear orders in the Keisler order on theories
Recently Malliaris and Shelah (see their preprint http://math.uchicago.edu/~mem/Malliaris-Shelah-CST.pdf) have shown that theories with $SOP_2$ are maximal in the Keisler order. A preceding result of ...
- 489
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514
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Existence of a regular subposet which collapses everything except the top cardinal
Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < \delta$)...
- 2,231
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314
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How much do idempotent ultrafilters generate in terms of semigroups?
It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
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317
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full support iteration of semiproper forcings
Suppose $\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\omega_1\rangle$ is a full support iteration of (semi)proper forcings. Is the full limit $P_{\omega_1}$ (semi)proper or at least stationary set ...
- 489
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Consequences of recent claims of Ordinal Analysis of $Z_2$
Recently Toshiyasu Arai submitted "An ordinal analysis of $\Pi_{N}$-Collection" and Henry Towsner submitted "Proofs that Modify Proofs", both of which claim ordinal analysis of ...
- 161
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274
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Has a computer search for inconsistency of large cardinals been carried out before?
In discussion about the consistency of large cardinals, a justification which occasionally appears is that despite years of work, no inconsistency has currently been discovered. For example, here are ...
- 2,031
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168
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Algebraic structures on spaces of ultrafilters
The space of ultrafilters on $\omega$ has a natural semigroup structure, and ultrafilters that are idempotent in that algebra have seen applications in combinatorics on the natural numbers, for ...
- 18.6k
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138
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Locally presentable and accessible categories without the axiom of choice?
Is there a good reference for the study of locally presentable and accessible categories without the axiom of choice? For instance, it seems one will need to understand:
What is a good notion of $\...
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287
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The approximate mean value theorem / Rolle's theorem in pure constructive mathematics
In the replies of this very similar question, there is a fascinating answer that is beautiful in its simplicity. In particular, it seems to use perhaps the most minimal assumptions one can possibly ...
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210
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Is there an Arithmetized Completeness theorem for intuitionistic theories?
For classical theories, Henkin's completeness proof can be arithmetized. This leads to the result that for classical theories $T$ and $S$ if $\sigma$ is a formula enumerating $S$ in $T$ then $S \leq T ...
- 481
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Are there good criteria for the topological models where BD-N and BD hold?
A (non-empty/inhabited) subset $S$ of $\mathbb{N}$ is said to be pseudo-bounded if for every sequence $x_n$ in $S$ we have
$\lim_{n\to \infty} \frac{x_n}{n} = 0$
Clearly all bounded subsets are pseudo-...
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Can we have a 'universal class' for elementary embeddings $j\colon V\to V$
Work over $\mathsf{GB}$, Gödel-Bernays set theory (without choice). My question is the following:
Question. Is the following statement consistent with $\mathsf{GB}$? There is a universal class for ...
- 3,042
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164
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Parallelizability of Lie monoids
A Lie monoid is a monoid, together with a structure of a smooth manifold (possibly with a boundary), such that the monoid multiplication is smooth.
If all left (or right) translations in a Lie monoid $...
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237
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Continuum hypothesis analogue for substructures
This question was previously asked and bountied at MSE. Throughout, "theory" means "possibly-incomplete first-order theory in a countable language."
Say that a theory $T$ has CHS (...
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252
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Another determinacy-related cardinal characteristic
This question is a kind of "dual" to an earlier one of mine.
Although I don't know a reference for this, it's easy to show the following result:
Suppose $G$ is a game in which neither ...
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178
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Does determinacy imply unravellability for the Borel sets (over a weak base theory)?
As far as I know, the only way we currently know how to prove Borel determinacy in $\mathsf{ZFC}$ is to go through unravelability (a rather technical property whose definition can be found in Martin's ...
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593
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How much condensed mathematics can be founded on finite order arithmetic (or ETCS) instead of ZFC?
I recently learnt from David Roberts' answer that, there is a way, due to Colin McLarty, to set up the foundations on finite order arithmetic for EGA & SGA. In particular, all usages of ...
- 2,806
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243
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Is this cardinal characteristic trivial? (Number of strategies needed to guarantee at least one win)
(Previously asked at MSE.)
Let the determinacy number, $\mathfrak{g}$ (for "game"), be the smallest cardinal such that for every (two-player, perfect-information, length-$\omega$) game on $\...
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275
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Is “simplicity is elementary” still hard? (Felgner’s 1990 theorem on simple groups, and subsequent work)
I came across a reference in this MathOverflow answer to an intriguing result of Ulrich Felgner [1]: among finite non-Abelian groups, the property of being simple is first-order definable. According ...
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Can "$\exists\mathcal{X}(R\cong C(\mathcal{X}))$" be expressed in "large" infinitary second-order logic?
Originally asked and bountied at MSE without success:
Say that a ring $R$ is spatial iff there is some topological space $\mathcal{X}$ such that $R\cong C(\mathcal{X})$, where $C(\mathcal{X})$ is the ...
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161
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Number of tautologies of a given size?
Fix some complete set of $L$ logical connectives such as $\{ \wedge, \neg \}, \{\Rightarrow, \neg \}, \{\ \vee, \wedge, \neg \}, \{ \uparrow\}, \{\wedge, \vee, \neg, \Rightarrow \}$ - I'll assume all ...
- 1,075
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314
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Is there a relationship between the $\Omega$-conjecture and choiceless large cardinals?
Woodin’s $\Omega$-conjecture is absolute under set-forcing. One proposal to decide it is that some large cardinal hypothesis might refute it. On the other hand, Woodin is seeking extensions of the ...
- 18.6k
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Non-closed Neeman forcing
This question is something of a follow-up to this one:
Iterating Neeman's forcing
It regards the work of Itay Neeman, MR3201836.
Neeman formulates his two-type models forcing seemingly in greater ...
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347
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What is the precise connection between logarithmic algebraic geometry and the field with one element?
Monoid schemes (a.k.a. $\frak M$-schemes) have been introduced by Deitmar as a possible approach to geometry over the field with one element. These build upon monoids as the basic building blocks for ...
- 11.8k
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306
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Coding third-order objects via second-order ones
As is well-known, the language of second-order arithmetic only has variables for natural numbers and sets of natural numbers. Higher-order objects, like functions on $\mathbb{R}$, have to be ...
- 4,359
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108
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Decidable membership for subgroup generated by three elements in $F_2\times F_2$
Let $F_2$ be the non-abelian free group on two generators. Let $G\subset F_2\times F_2$ be a subgroup generated by three elements. Is there an algorithm deciding if a given element of $F_2\times F_2$ ...
user178109
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A model where the uniformity of the meager ideal is strictly below the almost disjointness number
I'm looking for a model satisfying the inequality described in the title. Recall that the uniformity of the meager ideal, denoted $\operatorname{non}(\mathcal M)$ (or $\operatorname{non}(\mathcal B)$) ...
- 1,882
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298
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Big Zariski topos and classifying topos of local rings
In the paper ``Using the internal language of toposes in algebraic geometry'', Blechschmidt mentions several approaches to defining the big Zariski topos over a scheme. I have two questions, which are ...
- 426
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How many steps does it take to "Tarski-Vaughtify" second-order logic?
Given a regular logic $\mathcal{L}$, let $\preccurlyeq_\mathcal{L}$ be the usual elementary submodelhood relation for $\mathcal{L}$. There is also a separate submodelhood relation coming from the ...
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204
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Reverse mathematics of Noetherian rings over $\mathbb{Q}$
Take the Hilbert Basis Theorem over the rational numbers in this form in the language of Second Order Arithmetic: For every $n\in N$ every ideal of the polynomial ring $\mathbb{Q}[x_1,\dots,x_n]$ is ...
- 11.1k
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279
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What logic characterizes relative intrinsic complexity in set recursion?
Short version: Is there an analogue of the Ash-Knight-Manasse-Slaman/Chisholm theorem for $E$-recursion?
Long version: I'm interested in "$E$-recursive structure theory," but it's not ...
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250
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Distributivity of certain infinite products
Suppose we have a sequence of posets $\{\mathbb P_n : n\in\omega\}$ such that for each $n$, $\mathbb P_{n+1}$ is $|\mathbb P_n|^+$-distributive. Is $\prod_{n>0} \mathbb P_n$ necessarily $|\mathbb ...
- 18.6k
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381
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Reflection principle for intuitionistic Zermelo–Fraenkel?
The well-known reflection principle for classical Zermelo–Fraenkel states:
For any formula $\varphi(x_1,\ldots,x_n)$ of the language of ZFC with free variables $x_1,\ldots,x_n$, ZFC proves
$$ \...
- 3,312
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432
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What metatheory proves cut elimination for Simple Type Theory?
Gaisi Takeuti's book Proof Theory proves cut elimination for Simple Type Theory (a combined result of several researchers). Thus it proves consistency of Simple Type Theory with full comprehension in ...
- 11.1k
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270
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Analogue of strong stationary reflection from MM
Foreman-Magidor-Shelah proved that if Martin’s Maximum holds, then for every regular $\kappa>\omega_1$, every stationary $S \subseteq S^\kappa_\omega$ contains a club of ordertype $\omega_1$. This ...
- 18.6k
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346
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Is Videla's solution of Hilbert's tenth problem for rational functions over field of characteristic 2 wrong?
The paper in question.
Quick introduction to the problem: suppose that $F$ is a finite field of characteristic 2
(for purposes of this post $F = \mathbb{F}_2$ will suffice) and let $F[t]$ and $F(t)$ ...
- 543
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ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"
What is known about the theory
($\ast$) ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"?
By "reals" I mean elements of the Baire space $\omega^\omega$. For a cardinal $\kappa$, a set of ...
- 5,471
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539
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The Curry Howard Isomorphism and models for an intuitionistic modal logic and its bimodal translation
My question regards the Curry Howard Isomorphism and how it constrains models in the case of a particular logic.
Consider quantified Lax Logic $QLL$.
https://pdfs.semanticscholar.org/468e/...
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256
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A bi-modal logic related to determinacy
The short version of my question is as follows. There is a natural (I hope!) way to associate a bimodal theory to a game (two-player, perfect-information, length-$\omega$, on $\omega$); are there "...
- 20.5k
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Moschovakis' discovery of E-recursion
E-recursion is a notion of generalized computability theory which seeks to extend computations to allow arbitrary sets as inputs. In contrast with e.g. $\alpha$-recursion, it disallows unbounded ...
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