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.
352 questions
8
votes
1
answer
317
views
Is van Dalen’s “open problem” about $\bf{CT}$ and indecomposability actually open?
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 ...
- 1,073
16
votes
2
answers
989
views
Defining Lebesgue non-measurable sets with countable information
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?}
$...
- 237
4
votes
2
answers
431
views
Can custom functions be defined in Peano Arithmetic or the language of First-order Arithmetic?
Can we specify custom recursively-defined functions in the language of First-order Arithmetic?
I know that we can define functions in Second-order Arithmetic ($Z_2$). For example, we could define ...
- 183
33
votes
1
answer
2k
views
What is the consistency strength of Russell & Whitehead's ‘Principia Mathematica’?
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 ...
- 38.7k
8
votes
1
answer
589
views
Measurable cardinals and $HOD$
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 ...
- 183
13
votes
2
answers
794
views
Universal delta-functors and ZFC
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 ...
- 42.1k
8
votes
1
answer
684
views
Are Mike Shulman's "stack semantics" complete?
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 ...
- 467
-2
votes
1
answer
537
views
Is this theory of bottomless hierarchy, consistent?
Language: mono-sorted ${\sf FOL}(=,\in,S)$, where $S$ is a unary predicate standing for ".. is a stage".
Axioms:
Extensionality: $\forall z \, (z \in x \leftrightarrow z \in y) \to x=y$
...
- 13.1k
-4
votes
2
answers
371
views
Formal theory of structure [closed]
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 ...
- 1,289
8
votes
1
answer
553
views
In praise of Replacement, type-theoretically
A type-theoretic version of replacement says that given a set $A$ in a universe $U$, and another set $B$ with no universe constraints, the image of any function $A \to B$ is essentially in $U$. (Let ...
- 2,166
20
votes
3
answers
2k
views
Ordinary mathematics intrinsically requiring unbounded replacement/specification?
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 ...
- 3,158
2
votes
0
answers
154
views
A special name for Hilbert's Axiom I.7?
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 ...
- 44.4k
2
votes
0
answers
105
views
Can this salvage Cyclic Stratified Comprehension?
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 ...
- 13.1k
6
votes
1
answer
609
views
A form of reverse mathematics that works with hereditarily finite sets instead of numbers
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
Why not begin studying sets with Hierarchy Theory?
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: $\...
- 13.1k