Hello All, First of all, I have to warn that I'm not a mathematician (but an engineer and physicist) so I might not know some elementary results, terminology etc... I remember long ago having had a course on formal logic and Goedel's theorem and so on, and recently I read a small book on the subject, which brought back some souvenirs. If I remember well the structure of Goedel's proof, he embeds the metamathematical statements such as "there exists a proof" etc... within an axiomatic system of the natural numbers, and expresses these metamathematical statements as properties to be satisfied by natural numbers. He then carefully constructs a statement about natural numbers, G, which is constructed from the metamathematical statement "there exists no proof of the statement G or of ~G". So G is a formula in the axiomatic system of which we know metamathematically that there exists no proof of it, nor of its contrary, so G is undecidable within that axiomatic system. So people then usually claim that although G cannot be proven within the axiomatic system, we know it is true. This by itself is not a contradiction, because our metamathematical reasoning is based on other axioms, and hence another Goedel numbering, which itself must also produce a formula G', but which is not G. But I now wonder: if no proof of G nor of ~G can be given, we can just as well add ~G to the axioms (and not G, as is usually done). So now we have a strange system of axioms of the natural numbers, which contains a statement of which we somehow "know it is false". Nevertheless, this system is consistent. In what way do we get now different mathematics for the natural numbers ? I mean, let's for the sake of argument say that Fermat's last theorem is such a formula G. So if Fermat's last theorem is undecidable, we can just as well add the negation to the axioms: "there exists numbers n>2 and x, y and z such that x^n + y^n = z^n". Then we can take the smallest ones, say n1, x1, y1 and z1. And from there on, we can probably deduce a lot of theorems using n1, x1 etc... These must be very strange theorems. How should one think then of this system of natural numbers which is consistent, contains all the "basic" ingredients of the natural numbers, and which also contains a "false axiom" ? cheers, patrick.