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.
354 questions
14
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Alternative to Grothendieck universes which is a conservative extension of ZFC?
Using Grothendieck universes as the foundations of (higher) category theory is problematic:
The existence of Grothendieck universes relies on the existence of inaccessible cardinals. However, one can ...
- 377
4
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1
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648
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Is there a natural topos where the Riemann hypothesis is provable or disprovable?
While constructive logic is compatible with classical logic and is sufficient to develop almost all important theorems from classical complex analysis, constructive is also compatible with axioms that ...
- 2,069
8
votes
1
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331
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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
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2
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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?}
$...
- 237
4
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2
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443
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Can custom functions be defined in Peano Arithmetic or the language of First-order Arithmetic?
Can we specify custom recursively-defined functions in the language of First-order Arithmetic?
I know that we can define functions in Second-order Arithmetic ($Z_2$). For example, we could define ...
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33
votes
1
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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 ...
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8
votes
1
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594
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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
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2
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805
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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
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1
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684
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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
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1
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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: $\forall z \, (z \in x \leftrightarrow z \in y) \to x=y$
...
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-4
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2
answers
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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
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1
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555
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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
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2k
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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 ...
- 3,164
2
votes
0
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155
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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 ...
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2
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0
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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 ...
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