3 ms·
tl;dr: Godel's incompleteness theorem. When you try to create completely self-consistent axioms, it just creates an infinite hierarchy of further unprofitable
by neuroelectron 2y ago
tl;dr: Godel's incompleteness theorem. When you try to create completely self-consistent axioms, it just creates an infinite hierarchy of further unprofitable axioms. Numbers don't really exist, so that was your first mistake.
tl;dr^2: Mentally move the hyphen one word to the right.
- bubblyworld 2y agoThis is a somewhat different situation. Gödel's theorem gives you independent statements but they are highly contrived (containing expressions encoding provability and consistency in an ad-hoc fashion). This is talking about the cardinalities of very basic sets, and it was (is? haha) extremely unclear to what extent statements about them are independent, if at all. I would personally be shocked if you can prove independence of CH directly from the incompleteness theorem, though I would love to be wrong! It required a lot more delicate machinery to be developed (Cohen's forcing method). Numbers certainly exist in a useful sense - they model real properties of the universe. But I agree that higher set theories may be moving away from that...
- neuroelectron 2y agoLots of heuristics are useful.
- bubblyworld 2y agoI suggest asking yourself what you think you mean by "exists".