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August 29, 2010 15:23
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@spencertipping says "Similarly, the Twin Prime conjecture will never be proven independent; | |
it will only remain an unsolved problem. Independent -> true." | |
Let me try to paint a picture of the Twin Prime Conjecture (TPC) having been proven independent | |
of, let's say, ZFC. That means "ZFC + TPC" is consistent, and so is "ZFC + ~TPC". Since both | |
of these are consistent, neither of them is concretely falsifiable. Let's see how: | |
ZFC+TPC: there are infinitely many twin primes. "Pick any number, you will find a twin prime | |
above it." | |
Imagine this: Pick an insanely huge number, and start checking for twin primes above it. What | |
if one never comes? You will just keep checking for eternity, knowing that "the next one is | |
it." This is a situation in which TPC is false despite being assumed true, and no contradiction | |
arises. | |
ZFC+~TPC: there are not infinitely many twin primes. | |
Imagine this: this means that there is an upper bound on all twin primes. So you run a program | |
to find it and the same thing happens: "found one, found one, found one, found one". You will | |
keep finding new ones for eternity, knowing that eventually you will stop. This is a situation | |
in which TPC is true despite being assumed false, and no contradiction arises. | |
So, in this world with an independent TPC, is it true or false? |
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I found a good explanation of Gödel's incompleteness theorem at a Swedish forum:
https://www.flashback.org/sp31523789
Here's a Google translate of it (without any corrections):
Just noticed one word in the last sentence: "kit" should the "theorem".