Timeline for Is there any math conjecture that would cause a lot of damage if disproven?
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| Nov 15, 2019 at 17:51 | comment | added | Radovan Garabík | Even more interesting would be a proof that PA is ω-inconsistent. That means nonstandard numbers exists, period. Foundational consequences would be immense, though probably no practical changes. | |
| Nov 15, 2019 at 11:40 | comment | added | John Coleman | @Mark There are even weaker theories of arithmetic that people like Edward Nelson worked on. He wrote an entire book called Predicative Arithmetic which developed one in detail. Nelson would of course agree that Peano Arithmetic is a formalization of ordinary integer arithmetic, but would disagree that it is the formalization of ordinary arithmetic. From an ultrafinitist point of view, PA is in a sense a lazy theory that doesn't carefully spell out what finite beings can do when they do arithmetic, but instead engages in a sort of infinite hand-waving (full-fledged induction schema). | |
| Nov 15, 2019 at 3:51 | comment | added | Mark | Peano Arithmetic is a formalization of ordinary integer arithmetic. If it's inconsistent, the problems extend far beyond the realm of pure mathematics. | |
| Nov 13, 2019 at 22:21 | history | answered | John Coleman | CC BY-SA 4.0 |