lo.logic's questions - English 1answer

3.083 lo.logic questions.

The question is essentially the one in the title. Question. What are some examples of (major) problems in set theory which are solved using techniques outside of mathematical logic?

Questions of predicativity are well-studied in the context of arithmetic. We have a base theory, first-order Peano arithmetic. Some people, like Edward Nelson (in chapter 1 of his book) and Charles ...

For any two structures $\mathcal{M}$ and $\mathcal{N}$ in the same first-order language $\mathcal{L}$ and any ordinal $\theta$, let $G_\theta(\mathcal{M},\mathcal{N})$ be the two-player game of ...

My three computability questions are related to the following group theory question (first asked by Bridson in 1996): For which real $\alpha\ge 2$ the function $n^\alpha$ is equivalent to the Dehn ...

Given a group $G$, one can define the transfinite line of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the ...

Since calculus originated long before our modern function concept, much of our language of calculus still focuses on variables and their interrelationships rather than explicitly on functions. For ...

The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description ...

I'll state my questions upfront and attempt to motivate/explain them afterwards. Q1: Is there a direct way of expressing the relation "$y$ is a function of $x$" inside set theory? More ...

Suppose we have a logic for counterfactuals as with David Lewis. I here use $\Rrightarrow$ for the counterfactual conditional. So suppose we have: Rules: (1) If $A$ and $A\rightarrow B$ are ...

Below I'm working in ZF+DC+AD or similar; I want enough choice that things don't explode, but I also want the Wadge hierarchy to be well-behaved everywhere. Since this question is a bit long, I've put ...

Let $\lambda$ be a singular cardinal of countable cofinality. Is there necessarily a sequence $\{A_\alpha\mid\alpha<\lambda^+\}$ of countable subsets of $\lambda$, such that $\alpha<\beta$ if ...

The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned out to be ...

Let $R$ be a real closed field. Recall that the ladder of $R$ is the divisible, ordered abelian group obtained by quotienting $R$ by a certain equivalence relation. Note that $R$ has trivial ladder ...

A countable structure A is strongly reducible to a structure B if there is a uniform turing functional which, given a copy of the atomic diagram of B, computes a copy of the atomic diagram of A. A is ...

Łoś's theorem: If $F$ is an ultrafilter on $I$, $M_i$ is a model with domain $A_i$, then for any formula $\phi$ of L and any sequence $f/F \in (\prod A_i / F)^\omega$ $$\prod M_i/F \vDash_{f/F} \phi \...

My partner and I are not mathmaticians... We have an interesting question that we think we have come up with an answer to If the earth had a heartbeat... What would it be? We have tried to work it ...

Of all the constructions of the reals, the construction via the surreals seems the most elegant to me. It seems to immediately capture the total ordering and precision of Dedekind cuts at a ...

Suppose that we have an uncountable collection $C_\alpha$ of disjoint Cantor Sets contained in the closed unit interval $I$. Suppose we have ordered the indices $\alpha \in [0,1]$ as well. Then is ...

Let $\mathcal{L}$ be the first order language of ZFC set theory, and let $\mathcal{L}_{\infty,\infty}$ be the usual infinitary extension of the language allowing arbitrary long disjunctions/...

I'm working my way through Rosen's $\textit{Discrete Mathematics and it's Applications}$ (7th edition) after a break of several years. In the discussion of proving set theory identities, an argument ...

Assuming the axiom of choice the following argument is simple, for infinite $A$ it holds: $$2\lt A\leq2^A\implies 2^A\leq A^A\leq 2^{A\times A}=2^A.$$ However without the axiom of choice this doesn't ...

Let us posit that the goal of mathematics is to study mathematical truths, and let us stick to arithmetic for simplicity: so let $\mathscr{T} \subseteq \mathbb{N}$ be the set of (Gödel codes of) true ...

This is a more precise version of that question. Let $M$ be a non-deterministic Turing machine which recognizes a language $L$, that is, for every input word $u$ there is an accepting computation ...

Is it consistent in $\mathsf{ZF}$ that there is a set $X$ with more than $1$ point such that every injective map $f:X\to X$ has a fixed point?

Let $M$ be a non-deterministic Turing machine which recognizes a language $L$, that is, for every input word $u$ there is an accepting computation with input $u$ if and only if $u\in L$. The smallest ...

Recall that a cardinal $\kappa$ is $(\lambda,\infty)$-almost-strongly-compact if every $\kappa$-complete filter can be refined to a $\lambda$-complete ultrafilter. A cardinal $\mu$ has the tree ...

This may be an easy question or it may be related to a well known open problem in Computer Science. Let $\alpha>0$. We say that $\alpha$ is computed in time $T(n)$ if there is a Turing machine ...

Let us consider the first-order logic extended with the least fixed point operator (FO+LFP). That is, together with the usual first-order formulas, we also have formulas of the form: $$\mu X[\...

Say $L$ is a PL1 language and $T$ is an $L$ theory. Then we have: (Gödel) Completeness. $T$ consistent $\Rightarrow$ $T$ has a (set) model Say $T = ZFC$ and $M \subset N$ for two (set) ...

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this ...

Let $\mathcal{L}_{\mathrm{exp}}$ be the language with signature $(0, ^\prime, <, \mathrm{exp})$ (with $0$ interpreted as zero, $^\prime$ as successor, and $\mathrm{exp}(x)$ as $2^x$) and let $\...

The free Boolean algebra on countably many generators is closely related to the classical (two-valued) propositional calculus (after identification of logically equivalent formulas). By the Stone ...

In some related MO questions like The set-theoretic multiverse as a (bi)category it is discussed how one might represent the multiverse (see The set-theoretic multiverse) in a category theoretic way, ...

Gödel’s incompleteness theorem states that for any sufficiently strong formal system $T$ there exists a statement $G$ such that if $T$ is consistent, then $G$ is true but not provable in $T$. But I’m ...

If we assume MA+¬CH, then every boolean algebra with cardinality smaller than the continuum embeds in ℘(ω)/Fin. A proof of this result can be found in Theorem 1.1, Chapter 8 of the book "Hausdorff ...

Let us consider the (metric) theory of Banach algebras. I have a sentence encoding the (possible) openness of multiplication in a given Banach algebra: $$(\forall x) (\forall y) (\forall \varepsilon &...

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 ...

This is a followup question to Does foundation/regularity have any categorical/structural consequences, in ZF? As shown in answers to that question, the axiom of foundation (AF, aka regularity) has ...

In the math.se question Proof of no prime-representing polynomial in 2 variables, Alon Amit asks if Ribenboim's claim that a prime-representing polynomial (a Diophantine polynomial in which the ...

According to Chang and Keisler's "Model Theory", Model Theory = Universal Algebra + Logic. Model theory generalized Universal Algebra in the sense that we allow relations while in Universal Algebra we ...

This is inspired by the Alexander Shen's post here: https://www.facebook.com/groups/mathpuz/permalink/1058782384297603/ (the post is in Russian, but it is easy Russian, and google translate should ...

This is a follow up to my previous question concerning virtual large cardinals, that are generally weaker axioms of infinity obtained from ordinary large cardinals through the so-called virtualization ...

In model theory, a complete first-order theory $T$ is said to be simple if each type does not fork over some subset of its domain of size at most $|T|$. Question (1). What is the significance of ...

Virtual large cardinals belong to a relatively new breed of strong axioms of infinity. They often appear as statements of the following form: Definition. Suppose $A$ is a large cardinal property ...

Question. Does $\text{GCH}$ imply the existence of a non-special $\aleph_2$-Aronszajn tree ? Remark 1. By a result of Jensen, it is consistent that $\text{GCH}$ holds and all $\aleph_1$-Aronszajn ...

A type universe is a type of small types that is closed under the basic type formation operations (dependent product, sum, coproduct etc.), that is to say for example that from $A \colon U_i$ and $x \...

Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P}...

Let the cocone of a Turing degree ${\bf d}$ be the set $cc({\bf d}):\{{\bf c}: {\bf c}\not\ge_T {\bf d}\}$. I'm curious what's known about the various partial orders (isomorphic to ones) of the form $...

In his seminal 1937 paper, Jones [1] proved the following result about Moore spaces: Theorem. (Jones) If $2^{\aleph_0}<2^{\aleph_1}$ then all separable normal Moore spaces are metrizable. Then ...

Let $M$ be a countable transitive model of (enough of) ZFC. I'm looking for notions of forcing $\mathbb{P}$ such that if $G$ is $M$-generic for $\mathbb{P}$, then $c$ is a Cohen real over $M$ if and ...

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