3 ms·
If Peano arithmetic (PA) is consistent, the following statements are true but not provable in PA: 1) Consistency of PA 2) http://en.wikipedia.org/wiki/Paris%E
by cousin_it 13y ago
If Peano arithmetic (PA) is consistent, the following statements are true but not provable in PA:
1) Consistency of PA
2) http://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem http://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theore...
3) http://en.wikipedia.org/wiki/Goodstein%27s_theorem http://en.wikipedia.org/wiki/Goodstein%27s_theorem
- cwzwarich 13y agoBut that's not much of a practical problem, since ZF set theory proves them both.
- wfn 13y agoAh, but the consistency of ZFC itself is undecidable from within ZFC! [1] I see what you're saying; it's quite the rabbit hole, however. [1]: https://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC https://en.wikipedia.org/wiki/List_of_statements_undecidable...