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Results tagged with large-cardinals
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user 3106
10
votes
Nonessential use of large cardinals
This may not be quite the sort of thing you were looking for, but some of Harvey Friedman's examples of $\Pi^0_1$ statements unprovable in ZFC but provable using large cardinals can be used to produce …
- 82.6k
9
votes
Are large cardinals about more than just consistency?
Asaf Karagila already wrote an excellent answer at math.SE, but here is a simple point that may be helpful. In any area of math, the natural course of research leads one to ask questions, pose conjec …
- 82.6k
7
votes
Reasons to believe Vopenka's principle/huge cardinals are consistent
The standard heuristic argument for large cardinal axioms (including huge cardinals) is the reflection principle. The intuitive idea is that $V$ is "absolutely infinite" and so cannot be defined as t …
- 82.6k
25
votes
Arguments against large cardinals
You asked:
Why is it so unreasonable to think that the existence of large cardinals contradicts ZFC?
It's not "unreasonable," any more than it's "unreasonable" to disbelieve the Riemann Hypothes …
- 82.6k
11
votes
What "forces" us to accept large cardinal axioms?
This is only a partial answer, but Harvey Friedman has a research program to find concrete $\Pi^0_1$ sentences that are purely combinatorial (i.e., make no reference to concepts from logic such as axi …
- 82.6k