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Questions tagged [foundations]

Mathematical logic, Set theory, Peano arithmetic, Model theory, Proof theory, Recursion theory, Computability theory, Univalent foundations, Reverse mathematics, Frege foundation of arithmetic, Goedel's incompleteness and Mathematics, Structural set theory, Category theory, Type theory.

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8 votes
1 answer
323 views

In the paper "How connected is the intuitionistic continuum", D. van Dalen proves that in intuitionistic mathematics, the set $\mathbb{R} \setminus \mathbb{Q}$ is indecomposable, which means ...
16 votes
2 answers
1k views

Is there a formula $\phi$ in the language of set theory such that $$ \text{ZFC proves } \exists x \in \mathbb{R}:\text{ the set }A_x​:=\{y\in\mathbb{R}:\phi(x,y)\} \text{ is not Lebesgue measurable?} $...
4 votes
2 answers
436 views

Can we specify custom recursively-defined functions in the language of First-order Arithmetic? I know that we can define functions in Second-order Arithmetic ($Z_2$). For example, we could define ...
33 votes
1 answer
2k views

Russell and Whitehead's Principia Mathematica is of mostly historical interest (e.g., in that Gödel's incompleteness theorem was originally formulated against it), and I must admit never having read ...
8 votes
1 answer
593 views

Let $\kappa$ be some measurable cardinal and let $j:V \rightarrow M \cong Ult_U(V)$ be the canonical embedding with critical point $\kappa$ for some $\kappa$-complete non principal normal ultrafilter ...
13 votes
2 answers
798 views

I am certainly not an expert in foundations, although when I see some mathematics I usually feel like I would be able to formally write it down in theory in a formal system like ZFC or Lean's ...
8 votes
1 answer
684 views

In Shulman's Stack semantics and the comparison of material and structural set theories, he defines the stack semantics for a Heyting pretopos. He notes that (1) the stack semantics validate the ...
-2 votes
1 answer
537 views

Language: mono-sorted ${\sf FOL}(=,\in,S),ドル where $S$ is a unary predicate standing for ".. is a stage". Axioms: Extensionality: $\forall z ,円 (z \in x \leftrightarrow z \in y) \to x=y$ ...
-4 votes
2 answers
371 views

I use the concept of structure in my physics research. In particular, I would say things like "We probe structure with functors into a local structured system as a category", or "the ...
8 votes
1 answer
554 views

A type-theoretic version of replacement says that given a set $A$ in a universe $U,ドル and another set $B$ with no universe constraints, the image of any function $A \to B$ is essentially in $U$. (Let ...
20 votes
3 answers
2k views

Encouraged by some users on MO, I'm going to ask this question that I have had for years. I have always felt that the iterative conception of sets makes some sense for justifying BZFC (i.e. ZFC with ...
2 votes
0 answers
154 views

The famous Hilbert's Axioms of Geometry include the Axiom I.7: If two planes have a common point, then they have another common point. Question 1. Was David Hilbert the first mathematician who ...
2 votes
0 answers
105 views

This is an endeavor to salvage the approach presented at earlier posting. Is there a clear inconsistency with this axiom schema? Cyclic Stratified Comprehension: if $\varphi$ is a stratified formula ...
6 votes
1 answer
609 views

Does anyone know of any texts where reverse mathematics is developed using hereditarily finite sets and subsets of $V_\omega$? Reverse mathematics is typically carried out in the framework of second-...
1 vote
1 answer
523 views

By $\sf HT^\psi$ I mean the Hierarchy Theory of $\psi$ height. This is a set theory written in mono-sorted first order logic with equality and membership, with the following axioms: Specification: $\...

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