Re: Help in answering news story on refutation of fermat's last theorem

In article <vcbr7gdw1f4.fsf@xxxxxxxxxxxxxxxx>,
Torkel Franzen <torkel@xxxxxxxxxx> writes:
>"Mark Nudelman" <markn@xxxxxxxxxxxxxxxxxxxxx> writes:
>
>> If the statement 0=1 is an axiom, then the symbols 0, 1, and = cannot be
>> interpreted as they are in normal arithmetic.
>
> Sure they can. 0=1 simply becomes a false axiom.
>
>> Symbols can't
>> be interpreted unless you know how they're used in the axiomatic system
>> which they're part of.
>
> So what axiomatic system are the symbols in your statement above a
>part of?

The following problem is on one of my assignment sheets for field theory.

Define a `paddock' to be a set with operations +, ., satisfying the
axioms for a field, except that the axiom 0 != 1 is replaced by 0 = 1.
Give an example of a paddock, and prove that this is essentially the
only example.

If a student asked me "is it always true that 1 != 0 ?", I would say, "no,
not in the ring with one element".

I prefer to use the the term `axioms' to refer to a collection of statements
defining some mathematical structure. So a collection of axioms can be
inconsistent, but I would never say that a single axiom was false.

Derek Holt.
.

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