Reading about Tarski's axioms of geometry https://en.wikipedia.org/wiki/Tarski%27s_axioms I was puzzeling how they would look for 3 dimensional geometry.

the big problem then is the upper dimension Axiom

For 2 dimensions it is (copied from wikipedia)

$(xu \equiv xv \land yu \equiv yv \land zu \equiv zv \land u \ne v) \rightarrow (Bxyz \lor Byzx \lor Bzxy)$

in 3 dimensional geometry it should be something like:

For every points P , Q , X, Y and Z if PX = QX, PY=QY and PZ= QZ then there is a plane A and X , Y and Z all belong to plane A .

But how to formulate plane A in Tarski's axioms?


The upper n dimension axiom can be found page 182 of http://citeseerx.ist.psu.edu/viewdoc/download?doi=

The Upper n-Dimensional Axiom for n = 2, 3, . . . asserts that any three points a, b, c which are equidistant from each of $n$ distinct points $p_1$, $p_2$, . . . , $p_n$ must be collinear.

The axiom does not directly involve the concept of plane.


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