What is gödel’s theorem?
In: 0
There are various unproven conjectures in arithmetic, like Goldbach’s conjecture that “Every even integer greater than 2 can be expressed as the sum of two primes”. We’ve checked a bunch of integers and haven’t found any counterexamples to that conjecture, but we haven’t been able to check *all* the even integers because there are infinitely many of them.
Gödel proved that at least some of these unproven conjectures (though not that particular one) *can’t* be proven. We will never be able to generate a counterexample *nor* prove that there could never be a counterexample.
The way his proof works is basically that it assigns a number to each possible mathematical statement. For example, we might represent “3 + 5 = 8” as “11 4 101 8 1000”. Once we’ve done that, we can characterize a “proof” mathematically: it’s a series of numbers, each of which bears a certain kind of relationship to the preceding ones.
Which meant that Gödel could construct an arithmetic claim that amounted to “No series of numbers has the property of encoding a proof of this claim”. Assuming that it’s not possible to prove false stuff, this claim is true but unprovable.
Of course, if we could *prove* that it’s not possible to prove false stuff, then we’d be able to prove that the claim is true. So that means “No series of numbers has the property of encoding a proof of a false statement” is *also* unprovable.
Latest Answers