It is easy to address your objection. On a uniform distribution on [n] := {0, 1, 2, ...n}, P(X=42)->0 as n->infty. This is similar to Feynmann's argument.
Those are two separate events. I'm talking about the event n = 42, which makes sense for all [n]. You're talking about the event n >= 42, which is a very different event.
No, I'm not talking about integrating the probability that n >= 42.
What's the chance that a random number in [n] is 42? It's 1/n, if n is at least 42. Sum that over all x in [n] and you get 1, i.e. the probability that there is a number in [n] that is 42.
Richard Feynmann was trying to argue probabilistically about the possible existence of a counterexample to FLT. To do this, he more or less tries to put a probability value on the event "a counterexample exists". The problem is that his kind of reasoning is similar to the probability of an event such as "x = 42". Counterexamples to FLT may, a priori, be as rare as the number 42 is within the integers, i.e. with probability tending to zero as you consider larger and larger sets of integers. But a very small probability of a counterexample, even a probability that tends to zero, does not mean a counterexample doesn't exist anymore than it says that 42 doesn't exist.
See I can't see the analogy from your argument to Feynman's for the following reason: you are computing the probability that x=42 for an individual x and saying that it is going to 0 as x->infinity. Feynman first computes the probability that N is x^n + y^n but then integrates that over all N and n in order to get the probability that there is a solution anywhere. His argument takes into account the fact that there could be just one single lone solution in a infinite sea of non-solutions, since his probability estimate is for "there is a solution anywhere" not "this particular number is a solution". And I think that when you add that part to your estimate, you do get a probability of 1 that 42 exists somewhere, even if individually you get a low probability of it being at any given large x.
This detail doesn't matter. He's still computing a tiny nonzero probability that is only zero in the limit. We came up with other examples of tiny nonzero probabilities that are zero in the limit that can lead to the wrong conclusion, such as saying that 42 doesn't exist.
Feynmann's argument is clumsy and wrong. This kind of approximation is ok for a physicist but it doesn't work for mathematics.
