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Results tagged with set-theory
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user 39521
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.
7
votes
3
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229
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Ordinal-indexed transitive antichain of sets with urelements
Operate in ZFC. Can we find a function-class $\phi$ whose domain is the class of ordinals such that the following properties hold?
If $x \in \phi(\alpha)$, then either $x \in \mathbb{N}$ or there ex …
- 236k
46
votes
Accepted
What's the bijection between reals and infinite sequences of integers?
[Note: this answer uses the convention where $\mathbb{N} := \{ 0, 1, 2, \dots \}$ contains zero.]
There's an elegant explicit order-preserving bijection between the Baire space $\mathbb{N}^{\mathbb{N} …
- 12.4k
6
votes
Set theories without "junk" theorems?
The problems you mention occur as a result of two related reasons:
Objects such as the set of real numbers, which do not intrinsically belong to set theory, are 'encoded' as a set, so we can ask mea …
- 35.5k
1
vote
Can we add set complements on top of ZF?
If you take your description and rename $\textrm{set} \mapsto \textrm{class}$ and $\textrm{small set} \mapsto \textrm{set}$, and add some further axioms beyond the ones you mention (such as global cho …
- 12.4k
5
votes
Automorphism of the transfinite rooted binary tree
Although this question already has an accepted answer, which is correct for the question as stated, I posit that the surreal number tree is best viewed as a tree in the order-theoretic rather than the …
- 12.4k
2
votes
Are there non-commutative models of arithmetic which have a prime number structure?
Commutativity is not necessary for the notion of primes. For instance, consider the Hurwitz integers, namely quaternions whose components are either all integers or all half-integers:
$$ H = \{ a + b …
- 12.4k
5
votes
Ordinal-indexed transitive antichain of sets with urelements
Assuming Vopenka's principle (a large cardinal axiom), we can show there is no such $\phi$. In particular, a corollary of Vopenka's principle is that every proper class of directed graphs contains som …
- 12.4k
36
votes
Does an existence of large cardinals have implications in number theory or combinatorics?
There's an extremely elementary theorem whose only known proof relies on the existence of a rank-into-rank cardinal (basically the strongest large cardinal axiom not known to contradict ZFC).
Let $R_ …
- 12.4k