All Questions
1,458 questions with no upvoted or accepted answers
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Chain condition in iterated forcing
(I apologize for the long question, which has no mathematical content. Just looking for the right reference.)
In their celebrated paper [ST1971] introducing iterated forcing, Solovay and Tennenbaum ...
- 14k
5
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137
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Almost-disjoint sequence of sets at singular cardinals and stationary reflection
Let $\mu$ be a singular cardinal of countable cofinality. Let $ADS_\mu$ be the assertion that there exists $\langle A_\alpha\subset \mu: \alpha<\mu^+\rangle$ such that for all $\beta<\mu^+$, ...
- 1,006
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106
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Collapsing the Exponential time Hierarchy with a complete language as oracle
It is known that $\mathsf{P^A=NP^A}$ is true for every $\mathsf{EXP}$ complete language $\mathsf{A}$. The question is the whether the similar things hold for Exponential time Hierarchy.
Is there ...
- 1,689
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64
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Characters on monotone functions
Characters on the semigroup $(C_{+}^{b}(\mathbb{R}^{d}),+)$, i.e. on bounded positive continuous functions with the ususal pointwise addition, are known to be of the form $C_{+}^{b}(\mathbb{R}^{d})\ni ...
- 289
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395
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Derived tensor products and Tor of commutative monoids
Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
- 889
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196
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How to count Isomorphism Types of arbitrary structures?
For all relational signatures $\sigma$ and nonnegative integers $n$, I want to count the number of isomorphism types of structures of order $n$ of the signature $\sigma$.
What I mean by structure is ...
- 391
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241
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A possible characterization of weakly compact cardinals
Aside from the well-known characterization of weakly compact cardinals in terms of the usual partition calculus, I've been wondering if there are other characterizations that are variants of the ...
- 3,038
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143
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Normal ideals on $[\lambda]^{<\kappa}$ concentrating on maximal cardinality
Is the following consistent? $2^\omega > \omega_2$, and there is a normal precipitous ideal $I$ on $[\omega_2]^\omega$ such that every $X \subset [\omega_2]^\omega$ of size $< 2^\omega$ is in $...
- 18.6k
5
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165
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Almost elementary equivalence
Let $ M_1 $ and $ M_2 $ be stuctures over a language $ L $.
Let $ forms_n(L) $ be the set of all first-order formulas over $ L $ with length at most $ n $ (let assume we use a reasonable definition ...
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126
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Collapsing the Intuitionistic Bounded Arithmetics Hierarchy
Let $iT$ be the intuitionistic first order theory with non-logical axioms of classical first order theory $T$.
Theorem1. If $\mathsf{T^i_2}\vdash \mathsf{T_2}$, then $\mathsf{T^i_2}$ proves that the ...
- 1,689
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99
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Zappa-Szép products of the monoid of integers with itself
Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \...
- 5,482
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386
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Did Kleene constructively prove Brouwer's axioms?
Harvey Friedman's request on the FoM-forum for an overview of current intuitionistic foundations revived the following question, which I have been meaning to ask for five years. (I'm no expert on ...
- 1,616
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192
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The club filter in definable preorders
So this is an embarrassing question. Call a preorder $\mathbb{P}$ good if it has the following properties:
Every countable chain in $\mathbb{P}$ has a least upper bound.
$\mathbb{P}$ is directed (any ...
- 20.5k
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428
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Inductive generation of non-spatial locales
Is there an example of a locale/formal space which is not spatial (i.e., one can prove it is not spatial, rather than something like $\mathbb{R}$, where spatiality is independent in a constructive ...
- 303
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249
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Natural combinatorial properties of $\omega_1$ and weakly compact cardinals
One of the magnificent theorems of $\sf ZFC$ is that there exists an Aronszajn tree on $\omega_1$. Namely, a tree of height $\omega_1$ in which every level is countable, but no branch is cofinal.
On ...
- 39.8k
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222
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Semi-algebraicness of cells involved in integrals of semi-algebraic functions
Background: In "Stability under integration of sums of products of real globally subanalytic functions and their logarithms", by R. Cluckers and D.J. Miller, it is shown that the integral of a ...
- 701
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228
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Proper full submodels of full models of type theory
Let $N$ be the standard full model of the simply typed lambda calculus with infinite base type $o$ and let $X$ be an infinite and coinfinite subset of $N(o)$. I want to know if there's a full ...
- 357
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151
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Automorphism group of a structure without the SAP
A few years ago, a number of examples were given of Fraisse structures without the SAP in answer to the question raised in A Fraïssé class without the strong amalgamation property.
It is ...
- 2,943
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320
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Unbounded towers and combinatorial cardinal characteristics of the continuum
Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations.
This question assumes familiarity with combinatorial cardinal characteristics of ...
- 3,104
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287
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Is the two variable fragment of arithmetic, i.e., theory of ($\mathbb{N}, + ,\times$), decidable?
Any references would be appreciated. Most places only address different vocabularies (e.g. a survey of arithmetical definability by Bes).
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637
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Unique product groups (and semigroups)
A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a ...
- 18.7k
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307
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Proving that a subgroup is normal
This question is partly inspired by the recent question on measurability and the axiom of choice.
Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary ...
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431
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Cardinal characteristics without choice
(I'm taking my definition of a cardinal characteristic from Blass' excellent article http://www.math.lsa.umich.edu/~ablass/need.pdf, which cites Vojtas/Fremlin/Miller; theirs is more general, but I'm ...
- 20.5k
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137
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Pseudovarieties of monoids
All (pseudo)varieties considered here are (pseudo)varieties of monoids.
It is known that any (finite or infinite) monoid that satisfies the identities
\begin{equation}
xhxyty = xhyxty, \quad xhytxy=...
- 563
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227
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A variant of Chang's model with choice
Let $M_n$, $n < \omega$, be a models of $ZFC$ with the same ordinals, closed under countable sequences. Let $\alpha_n$ be an ordinal which is a regular cardinal in $M_n$.
Question 1: Is it ...
- 5,112
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185
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The (global) theory of Borel equivalence relations
What do we know about the complexity of the theory of Borel equivalence relations, with the Borel reducibility order $\leq_B$?
That is, let $\mathcal{B}$ be the set of all Borel equivalence ...
- 3,115
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295
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Orbit-Stabilizer theorem for continuous groups
The orbit-stabilizer relationship (also known as the orbit-stabilizer theorem) is very clear for finite groups. Is there an equivalent relation for continuous groups?
Also, is there a similar notion ...
- 615
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449
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How many Dedekind-finite sets can $\mathbb{R}$ be partitioned into?
Building off Asaf Karagila's answer to my previous question (Can $\mathbb{R}$ be partitioned into dedekind-finite sets?) on partitioning $\mathbb{R}$ into strictly Dedekind-finite sets:
(1) What ...
- 20.5k
5
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149
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If M is an inner model containing all the reals, might every game in M be "strongly" determined in V?
QUESTION
Let M be an inner model (of height Ord) containing all the reals. For each $X \in M$, define $S_X = \{x \in X^\omega : x_I \in M \land x_{II} \not \in M\}$. ($x_I$ is the set of plays in $x$ ...
- 2,146
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599
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Existence of Non-Borel sets in models of "All sets measurable"
We know that the consistency of ZFC+"Exists an inaccessible cardinal" implies the consistency of ZF+DC+"All sets are Lebesgue measurable"; and DC proves the existence of non-Borel sets.
J. Truss ...
- 39.8k
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404
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Proof of "AD + every set of reals is Suslin" implies AD$_\mathbb{R}$
Could someone point me toward a proof that "ZF + AD + every set of reals is Suslin" (+ $\mathsf{DC}\_\mathbb{R}$?) implies $\mathsf{AD}\_\mathbb{R}$, either with a reference or a hint?
I am ...
- 5,471
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154
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Original source for the undecidabity of the first order theory of finite dimensional representations of the free algebra on 2 generators
Many books and papers on the representation theory of finite dimensional algebras state that the first order theory for finite dimensional modules for the free algebra on two generators is undecidable ...
- 38.6k
5
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784
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ZFC+``every analytical set is measurable"
I know that "ZFC+the existence of an inaccessible cardinal" is equconsistent to
"ZFC + every $\mathbf{\Sigma}^1_3$ set is measurable".
Then how about the light face case?
Without large cardinal ...
- 4,201
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162
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What propositions about sets can be transfered to propositions about a presheaf category?
When I work with various presheaf categories, and I need some lemma, I often am able to prove the lemma by proving the analogous lemma for sets. As a simple example, let $f_i :X_i\hookrightarrow Y$ be ...
- 1,325
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Is the radical of a homogeneous ideal homogeneous?
Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: is it true that $\sqrt{I}$ is homogeneous? In this generality,...
- 1,811
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301
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When does $\operatorname{Aut}(M)$ preserve a linear order?
I have a general-type question:
Let $M$ be a countable structure that is ultrahomogeneous, i.e. every (partial) isomorphism between finitely generated substructures of $M$ extends to an automorphism ...
- 2,882
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647
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Adjunction between classic and intuitionistic logic
Let $\Sigma$ be a (classic, single-sorted) signature. Denote by $\mathit{Mod}\_H(\Sigma)$ the category of $H$-valued models over $\Sigma$, where $H$ is a complete Heyting algebra. Then for any first-...
- 4,096
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336
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Defining a topology by means of closed subsets in a topos
In the following we fix a topos. I'll speak of sets instead of objects and of subsets instead of subobjects.
Let $X$ be a set and assume $F$ is a set of subsets of $X$ that contains $\emptyset, X$, ...
- 63.1k
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350
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Chain/Hierarchy of Monoids
Let's assume that we have the following collection of structures:
Some space $P$.
Monoids $(M_{i+1},\circ_{i+1})$, and
Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
And $\bullet_{0}:...
- 1,882
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390
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What structure do all kinds of theories, models, interpretations, proofs and all that form?
This is a question about a structure that can be used to investigate all kind of structures that can be investigated. Many years ago with Joseph Gubeladze we discussed something similar but I only ...
- 18.4k
4
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152
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Are the natural powers of two conservatively embedded in $\mathbb{C}$?
This is a followup to this question.
Consider $\mathbb{C}$ as a structure - in the sense of first-order logic - with the graphs of addition and multiplication. Let $\mathcal{X}$ be the substructure ...
- 20.5k
4
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107
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Partial uniformization under AD
Under ZF + AD, and especially $\text{AD}^+$, even if uniformization fails for reals, in some ways it must almost hold.
For a notion of small, we say that uniformization holds on a co-small set of ...
- 7,545
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100
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Explicit superexponential growth for Presburger Arithmetic
Fischer and Rabin proved a superexponential bound $2^{2^{cn}}$ for the worst-case length of a proof of a proposition of length $n$ in Presburger arithmetic. The result is in
Michael J. Fischer and ...
- 16.6k
4
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214
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Algebraic logic in the style of algebraic geometry
I am writing a thesis on algebraic logic, I wonder if there is any recent research on an idea mentioned in Yuri Manin's book on algebraic geometry and in another Russian textbook on differential ...
- 558
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206
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Fine structure without choice
In set theory, are there approaches to fine structure that give fine-structural models that do not satisfy the axiom of choice?
We can build fine-structural models above a given set (such as $\mathbb ...
- 7,545
4
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103
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Unstable structures with unstable $\aleph_0$-categorical reducts
Suppose $M$ is a first-order structure which is unstable. If necessary, assume it is $\aleph_0$-saturated (or more, but I don't think it matters beyond that).
Are there any interesting criteria for ...
- 1,338
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261
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Läuchli's "intermediate thing"
On page 230 of An abstract notion of realizability ..., Läuchli writes the following:
If we drop the restrictions put on $\Theta$, then we get classical logic in one case and an intermediate thing in ...
- 20.5k
4
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143
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Part II to Ketonen's "Set Theory for a Small Universe I. The Paris-Harrington Axiom"
There is an unpublished manuscript "Set Theory for a Small Universe I. The Paris-Harrington Axiom" by Ketonen which appeared early in the study of the Paris-Harrington theorem, around 1979. ...
- 2,031
4
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148
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Isomorphism between the reduced C*-algebra of a groupoid and the crossed product of inverse semigroups
In Paterson's book "Groupoids, Inverse Semigroups and their Operator Algebras" he proves that for any r-discrete groupoid $G$ with unit space $G^0$, its full $C^* $-algebra $C^* (G)$ is ...
- 301
4
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146
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Can one formalize the prevalence of the Big Five systems of reverse math?
Simpson's systems of second order arithmetic turn out to be five in
number; to simplify notation let's denote them A, B, C, D, E. What
seems to be an empirical observation is that most theorems in
...
- 16.6k