Path: utzoo!mnetor!uunet!mcvax!unido!fauern!faui44!degen
From: degen@faui44.UUCP (Wolfgang Degen)
Newsgroups: sci.logic
Subject: Fermats Last Theorem
Message-ID: <270@faui14.UUCP>
Date: 4 May 88 11:26:48 GMT
Organization: CSD., University of Erlangen, W-Germany
Lines: 42
Keywords: Dedekind-finite sets



        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)