FOM: Tarki's elementary geometry
Harvey Friedman
friedman at math.ohio-state.edu
Mon Feb 22 08:59:27 EST 1999
This is a response to Simpson, 5:27PM 2/17/99, Tarski's elementary geometry.
Forgive me, but I think that Steve missed the main point of my FOM posting
of 4:52AM 2/1/99, More Axiomatization of Geometry.
Steve discusses the axiomatization of geometry from Tarski's landmark paper
`What is Elementary Geometry?', Symposium on the axiomatic method, with
special reference to geometry and physics, Amsterdam, 1959, pp. 16-29.
However, the axiomatization Steve discusses involves first order schemes.
In that sense, it is not purely geometric.
In contrast, what I do in my posting is give simple purely geometric
axioms. One uses only conjunctions of atomic formulas without equality,
involving only equidistance. Thus the axiomatizations are purely
diagrammatic.
There is a question of the role of degeneracies in diagrams; that matter is
also addressed in that posting.
Here is some of the relevant quotes from that posting:
"Fundamental to all aspects of this theory is the concept of **diagrammatic
condition**. This is a formula in the language of (R^2,0,1,i,E) of the
following special form: a conjunction of one or more atomic formulas
without equality.
We say that this is diagrammatic for the following reason. Suppose the
variables in the formula are x_1,...,x_k. Then the formula indicates that
we have points x_1,...,x_k where the distances between specified pairs,
including 0,1,i, are equal. This is like having a diagram with k labeled
points plus 0,1,i, with some line segments drawn in and marked indicating
certain equalities among the lengths. Of course, one is entirely
noncommittal about degeneracies; e.g., about which of the x's are equal or
nonequal, which line segments cross or don't cross, which triples of points
are or are not colinear, etcetera."
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