Deciding homomorphic images of De Bruijn graphs
The De Bruijn graph$B_n$ ofdimension $n$ (on the two-letter alphabet) is defined as the directed graph on$2^n$ vertices and $2^{n+1}$ edges, where for every $w = w_0 \dots w_n \in2^{n+1}$ we put an...
View ArticleDoes there exist a geometric morphism between the effective and topological...
I'm presenting in final projects for my computability and computational topology courses on the connections between computability, continuity, and logic. As a mathematician/unmentored baby logician...
View ArticleIs Bauer–Hanson’s result “there is a topos where the Dedekind reals are...
Last year, Andrej Bauer gave a talk showing that there is a topos in which the set of Dedekind reals is (sub)countable, and thus, you cannot prove that $\mathbb{R}$ is uncountable without LEM. He...
View ArticleSubcountability
In these slides of a talk Giovanni Curi shows that the generalized uniformity principle follows from Troesltra’s uniformity principle and from the subcountability of all sets, which are both claimed to...
View ArticleExamples of concrete games to apply Borel determinacy to
I'm teaching a course on various mathematical aspects of games, and I'd like to find some examples to illustrate Borel determinacy. Open or closed determinacy is easy to motivate because it proves the...
View ArticleSituation with Artemov's paper?
Artemov's paper on Goedel's theorem has been on the arxiv since 2019. There was a (less than fully friendly) discussion of this on FoM. At stackexchange, I found only a brief mention at this MSE post....
View ArticleDoes PA prove (Artemov-style) the consistency of a stronger system?
There was a recent question on Artemov's paper here on MO Situation with Artemov's paper?Apparently the main mathematical claim of the paper is (where PA is 1st-order Peano Arithmetic):for each $n$, PA...
View ArticleHarvey Friedman: The expanding mind
In reference 1, Friedman writes:I discuss my efforts concerning 3 crucial issues in the foundations of mathematics that are deeply connected with the great work of Kurt Gödel.[...]B. Are there...
View Article$\Pi^0_1$ sentences modulo "schematic entailment"
Let $\mathfrak{P}$ be the preorder of $\Delta^0_0$ (= only bounded quantifiers) formulas with one free variable in the language of arithmetic with a symbol for each primitive recursive function, under...
View ArticleWeak extender models for supercompactness without choice
Assume ZFC and a supercompact cardinal $κ$. Is it consistent that there is a weak extender model $N⊨\text{ZF}$ for supercompactness of $κ$ such that the axiom of choice (and well-ordering of $P_κ(λ)$...
View ArticleThe "first-order theory of the second-order theory of $\mathrm{ZFC}$"
$\newcommand\ZFC{\mathrm{ZFC}}\DeclareMathOperator\Con{Con}$It is often interesting to look at the theory of all first-order statements that are true in some second-order theory, giving us things like...
View ArticleLarge 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....
View ArticleAre some interesting mathematical statements minimal?
Gödel's set $\mathrm{L}$, of constructible sets, decides many interesting mathematical statements, as the Continuum hypothesis and the Axiom of Choice.Are some interesting mathematical questions, which...
View ArticleA question about the "information-content" of a very simple type of Turing...
All the Turing machines we consider have (1) a two-way infinite tape (2) one and only one haltingstate (3) an alphabet of exactly two symbols-"1" and ""(or "blank"). Let n be any positive integer.Let...
View ArticleBig 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...
View ArticleExistence of unknowable algorithms ?
Here by «algorithm» I mean a (halting) Turing machine with finite alphabet and memory.Is it possible to obtain by purely existential (i.e. non-constructive) means the existence of an algorithm...
View ArticleMembership problem in monoids
What is the simplest example of a monoid with undecidable membership problem? In other words, I'm looking for a concrete monoid $S$ such that there is no algorithm which takes elements $s_1,...,s_n$...
View ArticleComputable nonstandard models for weak systems of arithmetic
By Tennenbaum's theorem, PA itself does not have any computable nonstandard models. The integer polynomials which are 0 or have a positive leading coefficient form a computable nonstandard model of...
View ArticleNatural Numbers
Let $L$ be a countable language and $M$ be a model of $L^N$ (the realization of $L$ in the natural numbers $N$) in which every recursive unary relation is expressible. Show that $M$ is not recursive.
View ArticlePeriodicity in the cumulative hierarchy
Under Reinhardt cardinals in ZF, the cumulative hierarchy exhibits a periodicity in that for large enough $λ$, certain properties of $V_λ$ depend on whether $λ$ is even vs odd. See Periodicity in the...
View ArticleSemilattices in atomless boolean algebras
Let S be a bounded semilattice without maximal elements. Can we always construct an atomless boolean algebra B, containing S as a subsemilattice, such that S is cofinal in B-{1}? That is, for every x≠1...
View ArticleIn finite model theory, is "invariant FOL with $\epsilon$-operator"...
Throughout, all structures are finite.Say that a class of finite structures $\mathbb{K}$ is $\mathsf{FOL}_{\epsilon}^{inv}$-elementary iff it is the class of finite models of a sentence in the...
View ArticleMust there be a proper class of Reinhardt cardinals if there is a Reinhardt...
A cardinal is Reinhardt if $\kappa$ is the critical point of a nontrivial elementary embedding of $V$ to itself, where $V$ is the class of all sets. As Reinhardt cardinals are inconsistent with...
View ArticleIs factorial definable using a $\Delta_0$ formula?
The factorial function is primitive recursive, and therefore definable by a $\Sigma_1$ formula. Is it also definable by a $\Delta_0$ formula (i.e. bounded quantifiers)?If not, why?
View ArticleAn 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 objects...
View Article(Weakly) minimal subcovers of linear covers
Motivation. The starting point of this question is the trivial observation that if we cover $\mathbb{N}$ with $$\big\{\{0,\ldots n\}: n\in \mathbb{N}\big\},$$ then this cover doesn't have a minimal...
View ArticleWhat's the smallest $\lambda$-calculus term not known to have a normal form?
For Turing Machines, the question of halting behavior of small TMs has been well studied in the context of the Busy Beaver function, which maps n to the longest output or running time of any halting n...
View ArticleIs there a stepwise manner of adding big sets over the cumulative hierarchy?
Take a countable transitive model $M$ of $\sf ZFC+ inacc$, let $M^-$ be the segment of $M$ below $V_\alpha \in M$ where $\alpha$ is the first ordinal seen as inaccessible within $M$, i.e. for each...
View ArticleWhat would internalizing stratification be useful for?
What could that method be useful for? Would it useful in automations? Like proof checkers?The relation $\supset^1$ means a proper superset with exactly one extra element. That is$$y \supset^1 x \iff...
View ArticleDoes every linear cover contain a minimal cover?
This is a follow-up question to an older question.Let $X\neq \emptyset$ be a set. We say that ${\cal C}\subseteq {\cal P}(X)$ is a cover if $\bigcup {\cal C} = X$, and we call ${\cal C}$linear if $|A...
View Article
