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Results tagged with set-theory
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user 1384
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
10
votes
3
answers
1k
views
Are inclusions "canonical" injections?
[Background: I asked a vague question the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question]
Summary of question: the inclusions are a particularly …
- 41.4k
3
votes
1
answer
502
views
Can Tychonoff's theorem be applied to topological spaces generated by program output in ZFC?
I am confused about an issue in set theory.
Tychonoff's theorem says that "an arbitrary product of compact topological spaces is compact". We often talk of an index set $I$ and then for each $n\in I$ …
- 41.4k
16
votes
Find a "natural" group that contains the quotient of the infinite symmetric group by the alt...
Let $A$ denote the subgroup of $S_\infty$ consisting of permutations that only move finitely many elements, and have even signature. Then $A$ is a normal subgroup of $S_\infty$, and the quotient $S_\i …
- 41.4k
37
votes
Does anyone still seriously doubt the consistency of $ZFC$?
For decades I was not particularly suspicious about the consistency of ZFC but I was rather surprised about how it had become the standard choice when it contained axioms such as Replacement, which se …
37
votes
Accepted
Is "all categorical reasoning formally contradictory"?
Note: I am not a historian. I'm just guessing as to what prompted the comments.
Here's my guess: if you do set theory naively, in the old-fashioned "anything is a set" way, then you run into Russell' …
- 41.4k
107
votes
9
answers
36k
views
solving $f(f(x))=g(x)$
This question is of course inspired by the question How to solve f(f(x))=cosx
and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a b …
- 41.4k
31
votes
Most 'unintuitive' application of the Axiom of Choice?
The fact that there exist non-measurable sets is highly counter-intuitive; the reason we don't find it so is that we've all been conditioned from day 1 to do measure theory very carefully, and define …
33
votes
What makes dependent type theory more suitable than set theory for proof assistants?
I still find it very surprising that this random talk I gave attracts so much attention, especially as not everything I said was very well thought out. I am more than happy to engage with people in di …
- 41.4k