All Questions
Tagged with etcs or elementary-theory-category-of-sets
8 questions
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Is there a foundational approach that takes "structure" as primitive?
As per the title, I'd be curious to know if there have been attempts at constructing a foundation of mathematics taking, somehow, purely the notion of "structure" as primitive, maybe via a system of ...
- 23.3k
38
votes
4
answers
6k
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Could groups be used instead of sets as a foundation of mathematics?
Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The ...
- 3,137
11
votes
0
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342
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Categorial foundations via "categories of algebras"
There are categorical foundations for mathematics axiomatizing the category of sets (Lawvere's ETCS), cartesian closed categories (type theory), and the category of spaces (homotopy type theory). ...
user30211
5
votes
1
answer
492
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Is ETCS well-founded?
I can't find a statement about the axiom of regularity anywhere in treatments of ETCS. Perhaps this is due to the unfortunate clash of terminology with 'foundations'.
- 1,083
36
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6
answers
6k
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Who needs Replacement anyway?
The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in \...
- 35.4k
2
votes
1
answer
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Can we define geometric morphisms (between ETCS categories) elementarily?
The ETCS axioms give conditions on a category for it to be a category of sets. These axioms can be written out in first order language, resulting in a finite axiomatisation of the category of sets. ...
- 35.4k
7
votes
1
answer
513
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How do we compare models of ETCS?
The elementary theory of the category of sets (nLab) gives axioms on a category such that it is a category of sets. In answering this MO question I realised that we might have trouble comparing ...
- 35.4k
17
votes
10
answers
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Set theory and alternative foundations
Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set ...
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