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> In 1940 the famous logician Kurt Gödel proved that, under the commonly accepted rules of set theory, it’s impossible to prove that an infinity exists between
by mabbo 4y ago
> In 1940 the famous logician Kurt Gödel proved that, under the commonly accepted rules of set theory, it’s impossible to prove that an infinity exists between that of the natural numbers and that of the reals. That might seem like a big step toward proving that the continuum hypothesis is true, but two decades later the mathematician Paul Cohen proved that it’s impossible to prove that such an infinity doesn’t exist! It turns out the continuum hypothesis can’t be proved one way or the other.
This, to me, is the exciting bit of the whole article.
There are questions that cannot be answered definitively. There is no way to say that it's true or false. Not simply that we haven't figured it out yet, but that no one, no matter how clever, will ever be able to prove the answer one way or another. If we meet an extraterrestrial alien civilization that is a million years of math theory ahead of us, they will not have an answer.
- sorokod 4y agoI think that a question has been answered definitely. Continuum hypothesis is not provable within the theory along with the other axioms the the theory is derived from. Similar to parallel postulate really https://en.m.wikipedia.org/wiki/Parallel_postulate https://en.m.wikipedia.org/wiki/Parallel_postulate
- scapp 4y ago> along with the other axioms the the theory is derived from Axioms are trivially provable in any system. Unless you mean prove them without using them, in which case you're actually talking about a system where they aren't axioms.
- karmakurtisaani 4y agoIf I remember correctly, this means that CH is independent of the other axioms of set theory. So in theory you could come up with another set of meaningful axioms (I.e. ones that produce and capture interesting mathematics), and in which you could prove CH. So, an extraterrestrial civilization could in fact have an answer!
- BeetleB 4y agoIt's certainly cool, but don't make more of it than it is. It's merely pointing out that with a given set of axioms, there will be questions that cannot be answered. It's not a statement about the Real World Truth, but about the models we use.
- Viliam1234 4y agoThe actual meaning of "continuum hypothesis can’t be proved one way or the other" is that the axioms we use in set theory are not sufficient to uniquely define what set is. They allow multiple possible interpretations. Under some of those interpretations, an infinity between N and R exists, under other interpretations it does not. As a simple example, if I make the following set of axioms -- "each natural number is either big or small", "10 is a small number", "20 is a big number", "it cannot be simultaneously true that X is big and X+1 is small", -- then I can prove that 5 is small, or that 30 is big, but I can neither prove nor disprove that 15 is big. Because this set of axioms allow multiple possibilities for what "big" and "small" mean.