[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 

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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> FOM at cs.nyu.edu
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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
Room 327 Science and Engineering Offices (SEO)
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Assistant to the director
Jan Nekola: 312-413-3750

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