All Questions
1,460 questions with no upvoted or accepted answers
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Which functions have all the common $\forall\exists$-properties of continuous functions?
This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well.
For a ...
- 20.5k
14
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500
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The Ax-Kochen isomorphism theorem and the continuum hypothesis
Let's recall that:
(1): The Ax-Kochen principle says that if $\mathcal{U}$ is a non-principal ultrafilter over prime numbers, then $\prod_{\mathbb{U}} \mathbb{F}_p((t)) \equiv \prod_{\mathbb{U}} \...
- 32.2k
14
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297
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Ordinal-valued sheaves as internal ordinals
Let $X$ be a topological space (feel free to add some separation axioms like “completely regular” if they help in answering the questions). Let $\alpha$ be an ordinal, identified as usual with $\{\...
- 32.5k
14
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654
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Reverse Mathematics of Euclid's theorem
Euclid's theorem that there are infinitely many prime numbers has multiple proofs, ranging from Euclid's original theorem that constructs a new prime from a finite list of such, to Euler's proof that ...
- 35.5k
14
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404
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O-minimality and forcing
It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it.
In an ongoing project with Will ...
- 32.2k
14
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810
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Cardinality vs. isomorphism type of vector spaces without choice
One of the classical uses of the existence of bases of vector spaces (which is equivalent to the axiom of choice) is the following theorem:
If $V$ is an infinite vector space over a field $F$, and $...
- 39.8k
14
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353
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The failure of GCH al $\aleph_\omega$ by nice forcing
There are several ways to force $GCH$ below $\aleph_\omega$ and $2^{\aleph_\omega}> \aleph_{\omega+1},$ say:
1) The Gitik-Magidor's extender based forcing, see Prikry type Forcings.
2) Woodin's ...
- 32.2k
14
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850
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Algebraic proofs of algebraic theorems about algebraically closed fields
It is well-known that the first order theory of algebraically closed fields admits quantifier elimination, whence the theory $ACF_p$ of algebraically closed fields of given characteristic $p$ is ...
- 2,550
14
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504
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How to measure the strength of Zermelo over bounded Zermelo?
Bounded Zermelo is Zermelo set theory with only bounded separation. It has the same strength as simple type theory or MacLane set theory or ETCS. It is a finitely axiomatized fragment of Zermelo, so ...
- 11.1k
14
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815
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Does Fermat hold in non-standard models?
Let $n \geq 3$ be a natural number and $PA$ denote Peano arithmetic. Do we have
$PA \models \forall x,y,z \geq 1 : x^n + y^n \neq z^n$?
In other words, does Fermat's Last Theorem hold also in non-...
- 63.1k
14
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1
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2k
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Finite dimensional real division algebras
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
- 2,550
13
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0
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260
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Big list of Hochster dual concepts
Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and ...
13
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252
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Intuitionistic proofs of propositional formulae versus natural transformations between finite sets
The setup: Given a formula $\varphi$ of intuitionistic propositional logic (i.e., made from the connectors $\Rightarrow$, $\land$, $\lor$, $\top$ and $\bot$ from propositional variables $A,B,C,\ldots$)...
- 32.5k
13
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325
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When does HSP reduce to SPH?
This is actually a poorly camouflaged attempt to use the answers to When is the opposite of the category of algebras of a Lawvere theory extensive? (all very interesting) for the purposes of my ...
- 18.4k
13
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571
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Why is it so hard to give examples of differentially closed fields?
The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...
- 32.5k
13
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545
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Is there a logical relationship between constructions of the reals and proof methods in real analysis?
In my elementary real analysis course three years ago, I remember noting that there seemed to be 3 main ways of proving the main theorems about continuity. There was Bolzano-Weierstrass, continuous ...
- 295
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800
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Reference request for a complete and formal Duality Principle in category theory
Most textbooks on category theory only sketch the meaning of the Duality Principle. But even when they do it more formally, I have only seen a version so far which concerns the language of a (single) ...
- 63.1k
13
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362
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Context of set theory in which one doesn't have to worry about size issues
In this beautiful talk by Colin McLarty, McLarty quotes Grothendieck:
It would be nice to have a context where one doesn't add any real axioms to set theory, and yet one can work with categories ...
- 395
13
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447
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The original Erdős-Volkmann ring problem
The Erdős-Volkmann ring problem and its solution are famous, but the original problem is actually still open. I'll describe this and a related problem from geometric measure theory, I think both of ...
- 1,689
13
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248
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Birkhoff's HSP theorem in categories other than $\mathbf{Set}$
Fix a category $C$ with finite products and a set $L$ of function symbols (each equipped with an arity in $\mathbb N$).
An $L$-algebra in $C$, $\mathbf A=(A,(f^\mathbf{A})_{f\in L})$, is given by some ...
- 131
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707
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Applications of Set theory vs. model theory in mathematics
I have a question that has occupied my mind for some time.
Let's first consider applications of set theory and model theory in mathematics.
Major applications of set theory are in topology, Banach ...
- 32.2k
13
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229
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Is there a finite equational basis for the join of the commutative and associative equations?
I asked this on math stack exchange, but I was told to post it on mathoverflow. Consider the lattice of equational theories of a single binary operation $*$. The meet of the theory axiomatized by the ...
- 2,023
13
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429
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Is it an open problem whether fast-growing hierarchies can be defined without fundamental sequences?
Googology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to ...
- 4,597
13
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229
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Primitive recursive and feasible presentations for nonstandard models of arithmetic
Let us define a countable model $\cal{M}$ = $(M,+_M ,\cdot_M, <_M)$ of $Q$ (Robinson arithmetic) to have a (primitive) recursive presentation if $\cal{M}$ is isomorphic to $(\omega, \oplus, \...
- 17.7k
13
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466
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Does Foundation increase the strength of second-order logic?
Thinking about the recent threads on structural consequences of the Axiom of Foundation (AF) over ZF-AF, I've been trying to find some conservativity result which explains why AF doesn't seem to have ...
- 4,316
13
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251
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Is every simply connected finite complex the classifying space of a finite monoid
On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any finite simply connected complex should [have the same weak homotopy ...
- 665
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345
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Can you define a probability measure on the set of countable transitive models of ZFC?
It is well known that the set of hereditarily countable sets $H(\omega_1)$ —or, if you prefer, $H_{\omega_1}$— has cardinality $2^{\aleph_0}$, and I understand that every countable ...
- 1,124
13
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530
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Large cardinals arising from alternate set theories
My question is whether there are model-theoretic large cardinal axioms associated with $NFU$ - or more generally, with set theories other than $ZFC$.
Large cardinal properties generally come in one ...
- 20.5k
13
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387
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On sentences true in all finite groups (revisited)
Let $w$ be a group word with variables $\bar x, \bar y$, where
$\bar x=(x_1,\dots ,x_m)$ and
$\bar y=(y_1,\dots ,y_n).$
I am interested in the following questions.
(1) Is the sentence $(\forall\bar ...
- 893
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1k
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Has anyone read/debunked Yessenin-Volpin–Hennix “Beware of the Gödel-Wette paradox”?
A student recently asked me about the status of a 2001 arXiv post, Beware of the Gödel-Wette paradox!, by Alexander Yessenin-Volpin (aka Esenin-Volpin and several other transliterations) and Catherine ...
- 21.3k
12
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210
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Are there times when replacement is "more natural" than collection?
There are a couple examples I'm aware of where choosing to axiomatize $\mathsf{ZF(C)}$ using collection instead of replacement results in a much nicer (or at least less surprising) picture:
Let $\...
- 20.5k
12
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542
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Does Wedderburn's Little Theorem hold constructively?
Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative.
The proofs that I am aware ...
- 63.1k
12
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444
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On the division paradox
This question is partly motivated by Timothy Chow's recent question on the division paradox.
Say that a set $X$ admits a paradoxical partition if and only if there is an equivalence relation $\sim$ on ...
- 32.5k
12
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253
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Pointwise convergence of trigonometric series
$f$ is said to have trigonometric expansion if some series $\sum_{n\in\mathbb{Z}}c_ne^{inx}$ converges pointwise to $f(x)$. On the second page of the article Trigonometric series and set theory, ...
- 221
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324
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Sentences preserved under inverse limits
One of the classical theorems of model theory is the Chang-Łos-Suszko preservation theorem that states that the theories formulated in FOL (first order logic) that are preserved under direct limits (...
- 17.7k
12
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574
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Harvey Friedman's minimalist axioms for set theory
[This is a question on the FOM mailing list.]
In 1997, Harvey Friedman introduced the following theory: Let $\in$ be a binary predicate and $U$ be a constant. Add the following axioms:
Subworld ...
- 2,203
12
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406
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An internal notion of freeness for complete Boolean algebras
Background and Definition
Gaifman and Hales showed that there are no infinite free complete Boolean algebras.
But let a complete Boolean algebra $B$ be internally free if there is a set $X\subseteq B$ ...
- 121
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552
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When does Matiyasevich's theorem "kick in"?
Hilbert's 10th problem was famously resolved by the Matiyasevich–Robinson–Davis–Putnam theorem: the theorem implies that there is no algorithm which decides whether a given polynomial equation with ...
- 26.9k
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506
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Can a generic $\mathbb{R}$ have a new cardinality?
This question was asked and bountied at MSE, without success.
My main question is whether, starting with a model of determinacy, a "generic $\mathbb{R}$" could be different in cardinality ...
- 20.5k
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321
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Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field
Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
- 38.6k
12
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241
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Is there a characterization of the class of first-order formulas that are closed in every compact Hausdorff structure?
Fix a relational language $\mathcal{L}$. (I don't think relational really matters that much but I don't want to worry about it.) A topological $\mathcal{L}$-structure is an $\mathcal{L}$-structure $M$ ...
- 12.5k
12
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329
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Topology is to semi-decidability, coarse structures are to what?
There is a folklore correspondence between topology as semi-decidability amongst computer scientists, which is explained in places like:
The monograph Synthetic Topology: of Data Types and Classical ...
- 1,221
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538
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What happens when you iterate Cohen reals?
There are a few classical theorems in set theory:
The finite support iteration of ccc forcing is ccc.
The countable support iteration of proper forcing is proper.
The finite support iteration of ...
- 39.8k
12
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0
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357
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Undetermined copy/diagonalize games without CH
This question was the motivation behind an earlier question of mine; having thought about it some more, that question seems nontrivial and the connection is actually pretty tenuous anyways, so I've ...
- 20.5k
12
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468
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A question concerning model theory of groups
Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not ...
- 4,201
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263
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Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings
In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...
- 121
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458
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Adding a saturated ideal
Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?
Some motivation:
If $\delta$ is a Woodin cardinal, then it remains ...
- 18.6k
12
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0
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377
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Singular Jonsson cardinals
Is the following consistent?
$(*)$: There exists a singular cardinal $\kappa$ such that :
(1) $\kappa$ is a Jonsson cardinal,
(2) $\kappa$ is not a fixed point of the $\aleph-$function, i.e., $\kappa ...
- 32.2k
12
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0
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1k
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Hrushovski's Construction
Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s).
Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:
(a) Trivial (...
- 1,872
11
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0
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442
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c.c.c forcing notions and adding minimal generic reals
Is the following statement consistent:
``There is no non-trivial c.c.c forcing notion adding a minimal generic real''?
The question is related to Prikry's question: Is it consistent that any non-...
- 32.2k