first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, ...

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9
votes
1answer
322 views

Coherence and rewriting

In category theory there are numerous coherence theorems (https://ncatlab.org/nlab/show/coherence+theorem). One example is the Mac Lane's coherence theorem for monoidal categories. This and probably ...
3
votes
1answer
103 views

How to get $\omega$-regular expression from buchi automaton

Is there an algorithm or a trick on how to get $\omega$-regular expressions from Buchi automatons? If yes, is there also some way to do create minimal such regular expressions? It is extremely ...
4
votes
0answers
221 views

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).
5
votes
1answer
224 views

A button for individual reals

Hamkins introduced the notion of a "button" in forcing. This is a set-theoretic statement that can be forced, and can never be made false by further forcing. An example is $V \not= L$. Another ...
8
votes
1answer
268 views

Is there a modern account of Veblen functions of *several* variables?

Veblen $\phi$ functions extend the $\xi \mapsto \phi(\xi) := \omega^\xi$ and the $\xi \mapsto \phi(1,\xi) := \varepsilon_\xi$ functions on the ordinals by repeatedly taking fixed points (I won't ...
3
votes
0answers
173 views

Borel equivalence relations in models of determinacy

The following appears as fact 3.1 in the slides from Hjorth's 2010 Tarski Lectures. Assume ${L(\mathbb R)} \models \mathrm{AD}$. Fact 3.1 For E and F Borel equivalence relations one has $$E \...
14
votes
1answer
492 views

Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?

Background. The field of Borel equivalence relation theory provides a robust, unifying theory that organizes most of the classification problems of classical mathematics into a hierarchy, allowing us ...
10
votes
0answers
295 views

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 ...
2
votes
1answer
155 views

Non-completeness of the Borel-Lebesgue measure and countable choice

Is it possible to prove the non-completeness of the Borel-Lebesgue measure on $\mathbb{R}$ (restricted to the Borel $\sigma$-algebra) without the full axiom of choice, but still with Countable Choice ?...
10
votes
0answers
229 views

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 $j(\omega_1) = \omega_n^V$ ...
4
votes
1answer
218 views

Reference request, proof about well-ordering of ordinals

I am looking for a second order proof that ordinals are well-ordered up to $\epsilon_0$. So, starting with encoding those ordinals in numbers, defining the < relation on those encoded ordinals and ...
3
votes
0answers
333 views

A question about the Chaitin constant of a theory

Chaitin's famous incompleteness theorem states that for every r.e. theory $T\supseteq Q$ in the language of arithmetic, there is a constant $d_T$ such that for any $m\geq d_T$ and any $x$, $T$ can ...
12
votes
1answer
408 views

Changing cofinalities above supercompact cardinals

Question. Suppose $\kappa$ is a supercompact cardinal and $\lambda > \kappa$ is measurable (or even larger large cardinal if necessary). Is there a set generic extension of the universe in ...
1
vote
1answer
154 views

A question on the provability predicate of Q

I am not familiar with Robinson's construction as I do not have access to his text or to precise accounts of this, but I have come to understand that the proof predicate of Robinson arithmetic is non-...
4
votes
2answers
179 views

Which Heyting algbras arise out of some elementary topos which satisfies the ultrafilter principle?

It is known that AC implies LEM constructively, and also that AC implies the ultrafilter principle. Is there a similar relationship between the ultrafilter principle and classical logic? In other ...
11
votes
1answer
339 views

Singular in $V$ regular in $HOD$

Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have: (1) $\kappa$ is singular in $V$ of cofinality $\omega,$ (2) $\kappa$ is regular (and in fact ...
1
vote
0answers
167 views

“Kolmogorov complexity” of models of computation

This question was partly inspired by my learning about John Tromp's binary lambda calculus and similar minimal languages such as Jot. A more detailed discussion of some of these ideas is in Michael ...
2
votes
0answers
340 views

A question on the consistency of a (seemingly) very weak set theory

I have preoccupied myself some with very weak set theories that suffice to interpret Robinson Arithmetic, as in this question Is Extensionality needed for the incompleteness of very weak set theories?....
6
votes
1answer
442 views

Are there 'finitistic' nonrecursive functions (assuming Church's Thesis is false)?

[Note: In what follows, I will be using the same type of argument Laszlo Kalmar did in his paper "An Argument Against the Plausibility of Church's Thesis" found in Constructivity in Mathematics, (...
9
votes
2answers
277 views

Bounded Arithmetic vs Complexity Theory

In this post, when I talk about bounded arithmetic theories, I mean the theories of arithmetic according to "Logical Foundations of Proof Complexity", which capture the complexity classes between $AC^...
5
votes
1answer
236 views

Is the set of subsequences of branches through a tree Borel?

Let $T$ be pruned subtree of $\omega^{<\omega}$. For my cases of interest, we may assume that $T$ is infinitely branching at every node, and consists of increasing sequences. Let $A=\{x\in\omega^{\...
6
votes
1answer
479 views

Godel's proof of Completeness

Where could I find a detailed exposition in English of Godel's proof (not Henkin's) of Completeness Theorem for first order logic? The wikipedia article omits certain details that I am not clear about,...
2
votes
1answer
519 views

Can Turing machines clarify mathematical, philosophical, and physical existence?

From Harvey Friedman's manuscript on "Order Invariant Relations and Incompleteness": DEFINITION 4.4. A $\Pi_1^0$ sentence is a sentence asserting that some given Turing machine never halts at the ...
10
votes
1answer
334 views

Uniform elimination of imaginaries

Does the following principle follow from uniform elimination of imaginaries? For every formula $\varphi(x;y)$ there is a formula $\vartheta(x;z)$ such that $$\forall y\;\exists^{=1}z\;\forall x\;\...
3
votes
1answer
166 views

Theories of arithmetic from recursively inseparable sets

Edit: all sets / theories considered below are supposed to be recursively enumerable, although I'd also be interested in any possible generalizations to non-enumerable theories. In the comments on ...
2
votes
6answers
2k views

Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...
7
votes
1answer
305 views

Relationship between first and second incompleteness theorems

By my understanding, Gödel's first incompleteness theorem says that any theory with sufficient1 interpretability strength is essentially incomplete, that is, any consistent recursively enumerable ...
1
vote
1answer
178 views

In what sense is the “descending chain principle” for ordinals less than $\epsilon_0$ 'infinitary?

In the introduction to his paper "Assignment of Ordinals to Terms for Primitive Recursive Functionals of Finite Type", W.A. Howard writes: Gentzen...showed that the consistency of first order (...
3
votes
2answers
348 views

Determinacy of (infinite, possibly loopy) combinatorial games

I am looking for references and hopefully enlightening proofs of the following statement(s) concerning the determinacy of not-necessarily-well-founded (i.e., possibly infinite, possibly loopy) ...
5
votes
1answer
169 views

Deciding isomorphism between structures which interpret in the pure set

I am interested in the following decision problem: Given descriptions of two relational structures $A,B$ which interpret in the pure set $\mathbb N=(\{0,1,2,\ldots\},=)$, decide whether $A$ and $B$...
8
votes
0answers
191 views

ω-categorical, ω-stable structure with trivial geometry not definable in the pure set

Briefly, my question is the following. does every countable ω-categorical, ω-stable structure with disintegrated strongly minimal sets interpret in the countable pure set? By countable pure set I ...
6
votes
1answer
280 views

$\omega$-colorings of $\kappa^2$

Let $\kappa\le 2^{\aleph_0}$ be an infinite cardinal. We have a collection of functions $\{f_i|i<\kappa\}$ such that $f_i:i\rightarrow \omega$ and the collection is "triangle-free", i.e. there are ...
3
votes
0answers
451 views

What's Reeb's take on naive integers?

Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models ...
7
votes
3answers
498 views

Is there research on Machine Learning techniques to discover conjectures (theorems) in a wide range of mathematics beyond mathematical logic?

Although there already exists active research area, so-called, automated theorem proving, mostly work on logic and elementary geometry. Rather than only logic and elementary geometry, are there ...
6
votes
1answer
170 views

Heyting algebras originating from directed graphs

The category RefGph of reflexive directed graphs is the functor category $\hat{∆}_1=\mbox{Fun}(∆^◦_1,$Set), where $∆_1$ is the simplex category truncated at level 1. Hence the poset Sub(X) of ...
0
votes
0answers
35 views

How to convert an expression to conjunctive normal form for Maximum-2-satisfiability?

I have a simplified Boolean expression almost ripe for maximum-2-satisfiability: $(A\lor \neg B)\land(A\land C)\land(D\lor \neg A)$ In other words, I want to find the assignment of variables so that ...
3
votes
1answer
264 views

Simplest PA theorem whose proof requires encoding of sequences even though the statement itself doesn't

What is the simplest number-theoretic theorem whose proof requires exponentiation or finite sequences/sets (so any proof in Peano Arithmetic would need to use encodings of such things using e.g. Gödel'...
4
votes
0answers
307 views

About the “semi-classical” view of Prof. Weaver and Prof. Feferman [closed]

In the thread "Is platonism regarding arithmetic consistent with the multiverse view in set theory?", Prof. Hamkins writes: The view you are suggesting is something close to what is held by ...
12
votes
2answers
607 views

Is there a consistent arithmetically definable extension of PA that proves its own consistency?

I asked this on stackexchange with no answer. The negation would be the obvious generalization of Gödel's second incompleteness from r.e. extensions of PA to any arithmetically definable extension of ...
10
votes
1answer
408 views

“Set theory” founded on lists rather than sets

On a computer, sets are often represented rather "indirectly / implicitly", e.g. in terms of some properties that they or their members satisfy. But some sets can be represented more "directly / ...
0
votes
1answer
198 views

Does every ultrafilter contain sets of sup-measure $0$?

Let $\mathbb{N}$ be the set of positive integers and for $A\subseteq {\mathbb{N}}$ set $$m(A) = \text{lim sup}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$ Does every ultrafilter ${\cal U}$ on $\...
6
votes
4answers
534 views

Direct axiomatization of ordinal and cardinal numbers

Again, this question is related (**) to a previous one: in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
2
votes
1answer
129 views

Can tests for the convergence and divergence of series be used to create undecidable sentences?

Let f(k) be a recursive function which maps the set of positive integers into itself. Let T be a formalized theory which is axiomatizable and contains Peano's Arithmetic as a sub-theory. For example, ...
2
votes
1answer
170 views

Some very weak statements on choice

This is a follow-up question to Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$? Consider the statements $(\text{S}1)$ For any infinite set $X$ there ...
2
votes
1answer
163 views

Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?

Consider the statement For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$. Does this imply the ${\sf AC}$?
4
votes
0answers
95 views

Ref request: modelling regular theories as an injectivity condition

The background to the following observation is standard material in categorical logic, and I thought this was was too — I don’t remember learning it, but I don’t think it is original — but I can’t now ...
2
votes
1answer
286 views

A derivation in Tait calculus

I have seen in at least two different places (here, p. 183; and here, last slide) the Tait calculus defined the following way. Here $\Gamma$ denotes a set of formulas $\{A_1, \ldots, A_k\}$, which is ...
1
vote
0answers
174 views

Reference request: Models of isomorphic languages result into isomorphic categories

This is basically a reference request by someone who has not been educated as a logician and would like to be rigorous about certain preliminary aspects of model theory. Fix an uncountable universe $\...
7
votes
1answer
448 views

Taller models of ZFC

This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy. Using forcing techniques, at least the ones I know of, one starts from a ...
7
votes
2answers
314 views

How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$?

$\newcommand{\omegaoneck}{\omega_1^{\text{CK}}}$ Pardon the extremely basic question - this isn't quite my area - but I'm confused about the definition of proof theoretic ordinals. The proof ...