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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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 ...
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2answers
172 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 ...
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1answer
323 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 ...
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0answers
156 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 ...
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0answers
336 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 ...
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1answer
427 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, ...
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2answers
251 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 ...
5
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1answer
231 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 ...
6
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1answer
471 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 ...
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1answer
487 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
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1answer
288 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 ...
3
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1answer
134 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 ...
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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, ...
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1answer
296 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
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1answer
169 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
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2answers
318 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) ...
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1answer
165 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 ...
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0answers
188 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
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1answer
276 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
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0answers
438 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 ...
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3answers
435 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 ...
5
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1answer
168 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 ...
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0answers
31 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
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1answer
262 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. ...
4
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0answers
300 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 ...
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2answers
554 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 ...
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1answer
400 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 / ...
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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 ...
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4answers
515 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
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1answer
126 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
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1answer
163 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 ...
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1answer
159 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 ${\sf AC}$?
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0answers
94 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
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1answer
271 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 ...
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0answers
170 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 ...
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1answer
429 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
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2answers
276 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 ...
3
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1answer
183 views

Reference request: eliminating function symbols in predicate logic

Here is a basic technique in logic which seems well-known in folklore, but which I haven’t managed to find written down anywhere. $\newcommand{\T}{\mathbf{T}}$ Fact. Let $\Sigma$ be a signature (in ...
6
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1answer
190 views

Adding a truth-like predicate to PA

It is well known that adding a truth predicate to arithmetic in the most natural way leads to a contradiction. Suppose as usual that we add a one place relation T to the language of arithmetic, and ...
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1answer
337 views

Every measure on a set $X$ extends to the power set of $X$: Consistent or not with ZF?

Question. Is it consistent with ZF that every (countably additive, non-negative) measure $\mu: \Sigma \to \bf R$, where $\Sigma$ is a sigma-algebra on a given set $X$, extends to a (countably ...
5
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0answers
163 views

Set theory and forcing from the point of view of a formal system $G^+$ of Gentzen type

There are four papers by Vladimir Alfeevich Kuznetsov, which discuss the above titled topic: (1) Some problems in set theory from the standpoint of a formal system G+ of Gentzen type. (Russian) Akad. ...
4
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1answer
223 views

When does an infinite model have a proper class-sized elementary extension?

Suppose that a set of sentences of a 1st order language has an infinite model $M$. Under what conditions is there is a proper class-sized elementary extension of $M$? How does the answer change if ...
6
votes
2answers
156 views

a variant of the Kleene tree

The (a?) Kleene tree is a computable (a.k.a. decidable) sub-tree of the full binary tree with no computable path. It is well-known. I need a variant. (For those in the know, I need a c-bar which is ...
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0answers
351 views

Riemann hypothesis in Zilber's field

Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?
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0answers
152 views

Reducing Consistency of $PA$ [closed]

By godel translation consistency of $PA$ is equivalent to consistency of $HA$. I want to know any similar theorems for $PA$. 1.What is the minimal theory $T\subsetneq PA$ such that the proof of ...
5
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1answer
288 views

Constructive compactness for countable models?

The compactness theorem for countable (Tarski?) models is equivalent to the weak König's lemma by a result of H. Friedman and others as noted here, in the context of classical logic. The weak König's ...
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2answers
303 views

Non-Formal Applications: Higman and Kruskal

After looking through many papers, I noticed that most of the discussions and proofs for Higman's Lemma and Kruskal's Tree Theorem only have formal applications in set theory, logic, and type theory. ...
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1answer
244 views

Is the Martin's axiom number $\mathfrak m$ regular

The Martin's axiom number $\mathfrak m$ is the least cardinal $\kappa$ for which $\text{MA}_\kappa(\text{ccc})$ is false, i.e. the least cardinal such that there exists a ccc poset $P$ and a family ...
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2answers
183 views

Compactness for countable models?

How and where is it proved that WKL$_0$ proves the compactness theorem for countable models? (This is a follow-up to a comment of F. Dorais.)
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1answer
156 views

What restriction(s) of Goedel's primitive recursive functionals is (are) necessary and sufficient to prove the consistency of $PRA$

It is well known that one can use Goedel's primitive recursive functionals of finite type to prove the consistency of $PA$ (Peano Arithmetic). As such, one can certainly use them to prove the ...