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Results tagged with large-cardinals
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user 39521
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
7
votes
3
answers
229
views
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 …
- 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