Was sollen wir tun? Was dürfen wir glauben? - bei DuEPublico ...

WHEN IS IT RATIONAL TO BELIEVE A MATHEMATICAL STATEMENT 187
the real numbers that, when compared to all the reals, has Lebesgue measure zero). The idea
here is quite intuitive: Anything, regardless of how we measure it, should have a measure of
zero if, for every positive measure ε (no matter how small), the thing to be measured is
smaller than ε. In other words, if any given positive measure, regardless of how small we
make it, is still too big, then the thing to be measured has size zero. Turning this into a
definition, we get:
Definition Let (a, b) be an open interval in R. Define l(a, b), called the length of
(a, b), as b − a. An interval of real numbers [a, b] has measure zero if, given any ε > 0,
there exists a (finite or countable) family of open intervals 〈l i 〉 i∈I such that [a, b] ⊂
⋃〈l i 〉 i∈I and ∑ l(l i ) ≤ ε.
Theorem The set of naturals N has measure zero within the reals R.
Proof. Given ε > 0, let
Then
and
l n = n −
ε ε
, n +
2n+1 2n+1 , n = 1,2, …
∞
l(l n ) = ε 2 n , N ⊂ l n,
∞
∞
n=1
l(l n ) = ε = ε.
2n n=1
The concept of zero measure set translates into probability theory quite straightforwardly.
Given that the concept of measure we employed here implies that the full set of reals has
measure one, and given that the larger an interval, the bigger its measure, it’s not a big leap to
simply use the size of a set as the probability of us randomly drawing a number from it. So the
probability of drawing a real number from the set of reals is one, and the probability of
drawing any member of a zero measure set is zero. And indeed, this is the way measure
theory typically handles probabilities over the reals. 2
So the probability of drawing an integer from the full set of reals is zero. But of course this
means that our rational conviction that we will not, on randomly drawing a real number,
draw an integer, should be absolute.
Taking all of this together, we come to the conclusion that it would be completely irrational to
expect Goldbach’s conjecture to be anything but true, based on the Oracle’s statement. Our
degree of belief in the conjecture ought to be one. Even though it is not logically impossible
that the number we draw is a natural number, just as it is not logically impossible that a solid
block of iron passes through a solid surface via quantum tunnelling, the probability in both
cases is so utterly minuscule that even the suggestion that we should remain open to the
possibility of it actually happening is preposterous.
Let’s reiterate this. Based on the fact that the Oracle never lies, and the fact that it has
essentially told us that the probability of Goldbach’s conjecture being false is zero, we should
by all means be completely convinced that it is true. This might not be very satisfying
knowledge, since we still have no way of knowing why it is true, but we should be convinced
nonetheless.
n=1
2
See for instance Athreya & Lahiri 2006.