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?


1 Answer 1


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .