How to state Pythagorean theorem in a neutral synthetic geometry?In some lists of statements equivalent to the parallel postulate (such as Which statements are equivalent to the parallel postulate?), one can find the Pythagorean theorem. To prove this equivalence one has first to state the pythagorean theorem in neutral geometry (I name 'neutral geometry' a geometry in which parallel lines do exist but with the parallel postulate removed).If one start with an axiom system like Birkhoff's postulates which assume reals numbers and ruler and protractor from the beginning then there is no problem stating the Pythagorean theorem.My question is how one can one state the Pythagorean theorem in a neutral synthetic geometry based on axioms such as Hilbert's axioms group I II III or Tarski's axioms A1−A9 ?It is possible to define segment length in neutral Tarski's or Hilbert's geometries as an equivalence class using the congruence (≡) relation. It is also possible to define the congruence of triangles.However, the geometric definition of multiplication as given by Hilbert assume the parallel postulate. The existence of a square is equivalent to the parallel postulate.

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How to state Pythagorean theorem in a neutral synthetic geometry?In some lists of statements equivalent to the parallel postulate (such as Which statements are equivalent to the parallel postulate?), one can find the Pythagorean theorem. To prove this equivalence one has first to state the pythagorean theorem in neutral geometry (I name &#039;neutral geometry&#039; a geometry in which parallel lines do exist but with the parallel postulate removed).If one start with an axiom system like Birkhoff&#039;s postulates which assume reals numbers and ruler and protractor from the beginning then there is no problem stating the Pythagorean theorem.My question is how one can one state the Pythagorean theorem in a neutral synthetic geometry based on axioms such as Hilbert&#039;s axioms group I II III or Tarski&#039;s axioms A1−A9 ?It is possible to define segment length in neutral Tarski&#039;s or Hilbert&#039;s geometries as an equivalence class using the congruence (≡) relation. It is also possible to define the congruence of triangles.However, the geometric definition of multiplication as given by Hilbert assume the parallel postulate. The existence of a square is equivalent to the parallel postulate.

Lucille Lucas · Accepted Answer

Within synthetic approach for geometry (such as Hilbert&#039;s axioms) one can define the notion of segment measure:Definition. Denote the set of segments by S. We say that a function   μ  :      S    →      R   is a segment measure whenever:1.   μ  (  a  b  )  &amp;gt;  0  μ  (  a  b  )  &amp;gt;  02.   μ  (  a  b  )  =  μ  (      a    1        b    1    )3.   μ  (  a  b  )  +  μ  (  b  c  )  =  μ  (  a  c  )B(abc) stands for &quot;point b lies between a and c&quot;, which is a primitive notion.The definition involves real numbers but they are regarded independently from the theory of geometry.The following two theorems serve to answer the question about the Pythagorean theorem:Theorem 1. If   μ  ,      μ    1   are segment measures, then there exists   λ  &amp;gt;  0 such that   μ  =  λ      μ    1  Theorem 2. There exists a segment measure   μ (In fact it can be chosen in such way that   μ  (  a  b  )  =  x for arbitrary segment ab and x&amp;gt;0).Asssume we are given a triangle   △  a  b  c. Combining these two theorems we see that we can state  μ  (  a  b      )    2    =  μ  (  b  c      )    2    +  μ  (  a  c      )    2  and prove that its logical value does not depend on the choice of the measure since assuming above sentence is true for some measure   μ, then for arbitrary measure       μ    1   we have   μ  =  λ      μ    1   for some   λ  &amp;gt;  0 and      λ    2        μ    1    (  a  b      )    2    =      λ    2        μ    1    (  b  c      )    2    +      λ    2        μ    1    (  a  c      )    2        μ    1    (  a  b      )    2    =      μ    1    (  b  c      )    2    +      μ    1    (  a  c      )    2  For the proof of theorems 1 and 2 see &quot;Foundations of geometry&quot; by Borsuk and Szmielew.

How to state Pythagorean theorem in a neutral synthetic geometry? In some lists of statements equiv

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