[FOM] Tarski's real projective geometry
John Baldwin
jbaldwin at uic.edu
Thu Nov 8 09:32:42 EST 2007
On Wed, 7 Nov 2007, Neil Tennant wrote:
> In an abstract in JSL 1949, at p.78, Tarski mentions
> that he had shown that the theory of real projective
> geometry is decidable.
>
> Can any fom-er cite a reference for the detailed proof
> of this result? Or tell me exactly what axioms Tarski
> took the theory to have?
1) an ahistorcial viewpoint: The standard (Hilbert) interpretation of a
field in a projective plane will give an ordered field. Now translate back
the schema that every odd order polynomial has a root to a geometric
problem.
2) problem: Hilbert in his foundations doesn't actually interpret
subtraction, the addition is only a cancellative semigroup (since
multiplication is defined on congruence classes of segments). So this is
really the structure on the positive elements of a field.
It is easy enough to define subtraction in the normal way (using the
order on the line which is implicit in Hilbert's betweeness axioms) and
then to extend multiplication to the whole field by fiat.
But is there a uniform definition of both operations on the entire field
or a precise meaning for uniform (definitions must be conjunctions of
atomics ???) under which one can prove there is no uniform interpretation.
>
> Neil Tennant
>
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John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
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Jan Nekola: 312-413-3750
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