Re: primes of the form n^2+1



In article <g4th48$1je9$1@xxxxxxxxxxxxxxxxxxxxxxxxx>,
<tchow@xxxxxxxxxxxxx> wrote:

Joe Shipman has posted a thorough reply already, but perhaps the following
less formal comments will also be helpful.

In article <g4m30h$7tr$1@xxxxxxxxxxxxxxxxx>,
Kevin Buzzard <buzzard@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Sorry for the naive question. Is it theoretically possible that
a statement such as "there are infinitely many primes of the form n^2+1"
could be true, but not provable, in ZFC?

Yes.

the idea was that if FLT really were false then there would
exist a counterexample, and "writing down the counterexample" would be
a proof that FLT was false.

Yes. You will not go far astray with the following rule of thumb: Any
"finite computation," if correct, is provable in ZFC, and in fact in far
weaker systems. Other than that, just from the *form* of a statement, we
can't tell a priori whether it could be true but not provable in ZFC.

The *form* can tell us something about this...
Logicians call it Pi-1 I think. (Or is it Sigma-1)

If "There is an odd perfect number" is true, then it is provable.
If "There is an even number > 6 not the sum of two primes" is true,
then it is provable.
But, as you say, as far as we know the negations of these could
possibly be true but unprovable.


So for example, "There is no proof of a contradiction from ZFC with
length < 10^10^10," if true, is certainly provable in ZFC. We can say
this even though we don't know whether the statement is in fact true.
Ditto for "There is no odd perfect number less than 10^10^10."

On the other hand, it's entirely possible that the Riemann hypothesis
or the Goldbach conjecture or P = NP or almost any "interesting" conjecture
is true but not provable in ZFC.

One thing that sometimes confuses the issue is that we currently know only
a small number of ways to prove unprovability. Hence you will sometimes
encounter assertions such as, "large cardinals cannot settle the continuum
hypothesis" or "forcing cannot prove the independence of absolute
statements." Such assertions, if interpreted narrowly, are theorems, but
can still sometimes cause arguments because they can also be stated more
broadly and imprecisely so that they are not quite theorems any more. For
your present purposes, you should not be distracted by such debates,
because they are concerned only with the limits of current technology and
say nothing about whether future new techniques will be able to establish
the unprovability of your favorite conjecture.
.

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