degen@faui44.UUCP (Wolfgang Degen) (05/04/88)
Fermat's Last Theorem is true
J.W. Degen, 4-May-88
Assuming ZFC + there is a strongly inaccessible cardinal, we construct
the consistency of a certain extension T of ZF by iterated forcing.
T proves, among other things, the following:
1) There is an infinite Dedekind-finite set.
2) The Dedekind-finite sets are linearily ordered by injectivity.
3) Every family of finite sets has a choice function.
The main argument runs as follows:
p p p
If x + y = z , p an odd prime, has a solution, then it has a
solution in infinite Dedekind-finite sets (This is a trivial
observation). It is also trivial thet we can find an infinitely
descending chain of solutions. The main lemma, which is a theorem
of T, is the following:
p p p
If x + y = z has a solution. Then we can construct a chain
z > z' > z'' > ... of infinite length of solutions such that
z - z', z' - z'', ... etc are finite (not only Dedekind-finite).
Because of 3) above, we can define a countably infinite subset
of z. This is a contradiction, because z was assumed to be
Dedekind-finite.
J.W. Degen (University of Erlangen, West Germany)