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edited Jan 13 at 18:20

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The question is what to modify and how much one is willing to modify. As everybody who seriously thought about elementary geometry knows, Euclid's axioms are inadequate. There are several "standard" alternative axiomatics, most common is Hilbert's. Marvin Greenberg in Appendix A to Chapter 10 of his book

Greenberg, Marvin Jay, Euclidean and non-Euclidean geometries. Development and history, New York, NY: W. H. Freeman and Company (ISBN 978-0-7167-9948-1). xxix, 637 p. (2008). ZBL1127.51001.

modifies Hilbert's primitive notion of betweenness in Euclidean geometry (he replaces it with a primitive notion of separation). - Greenberg explains why this modification is needed if one were to get the elliptic geometry. With this modification, the geometry of the projective plane (aka the "elliptic geometry") is then obtained via a "negation" of the Playfair's axiom (the negation is that "no parallels exist"). You can find details by reading Greenberg's book which you should be able to find in a nearby library. I am unaware of a similar modification of Birkhoff axioms and Tarski axioms (and I do not see how it can be done without a complete revision of their axiomatics). One

You can then regard Greenberg's modified form of Hilbert's axioms, minus the Playfair's axiom, as a list of axioms of an absolute geometry that "underlies" hyperbolic, euclidean and elliptic geometry. One can, in principle, also get the spherical geometry, but that requires dropping an additional axiom. Maybe this is what you are interested in.

The question is what to modify and how much one is willing to modify. As everybody who seriously thought about elementary geometry knows, Euclid's axioms are inadequate. There are several "standard" alternative axiomatics, most common is Hilbert's. Marvin Greenberg in Appendix A to Chapter 10 of his book

Greenberg, Marvin Jay, Euclidean and non-Euclidean geometries. Development and history, New York, NY: W. H. Freeman and Company (ISBN 978-0-7167-9948-1). xxix, 637 p. (2008). ZBL1127.51001.

modifies Hilbert's primitive notion of betweenness in Euclidean geometry (he replaces it with a primitive notion of separation). - Greenberg explains why this modification is needed if one were to get the elliptic geometry. With this modification, the geometry of the projective plane (aka the "elliptic geometry") is then obtained via a "negation" of the Playfair's axiom (the negation is that "no parallels exist"). You can find details by reading Greenberg's book which you should be able to find in a nearby library. I am unaware of a similar modification of Birkhoff axioms and Tarski axioms (and I do not see how it can be done without a complete revision of their axiomatics). One can, in principle, also get the spherical geometry, but that requires dropping an additional axiom.

The question is what to modify and how much one is willing to modify. As everybody who seriously thought about elementary geometry knows, Euclid's axioms are inadequate. There are several "standard" alternative axiomatics, most common is Hilbert's. Marvin Greenberg in Appendix A to Chapter 10 of his book

Greenberg, Marvin Jay, Euclidean and non-Euclidean geometries. Development and history, New York, NY: W. H. Freeman and Company (ISBN 978-0-7167-9948-1). xxix, 637 p. (2008). ZBL1127.51001.

modifies Hilbert's primitive notion of betweenness in Euclidean geometry (he replaces it with a primitive notion of separation). - Greenberg explains why this modification is needed if one were to get the elliptic geometry. With this modification, the geometry of the projective plane (aka the "elliptic geometry") is then obtained via a "negation" of the Playfair's axiom (the negation is that "no parallels exist"). You can find details by reading Greenberg's book which you should be able to find in a nearby library. I am unaware of a similar modification of Birkhoff axioms and Tarski axioms (and I do not see how it can be done without a complete revision of their axiomatics).

You can then regard Greenberg's modified form of Hilbert's axioms, minus the Playfair's axiom, as a list of axioms of an absolute geometry that "underlies" hyperbolic, euclidean and elliptic geometry. One can, in principle, also get the spherical geometry, but that requires dropping an additional axiom. Maybe this is what you are interested in.

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created Jan 13 at 16:36

12.3k
1
36
59

The question is what to modify and how much one is willing to modify. As everybody who seriously thought about elementary geometry knows, Euclid's axioms are inadequate. There are several "standard" alternative axiomatics, most common is Hilbert's. Marvin Greenberg in Appendix A to Chapter 10 of his book

Greenberg, Marvin Jay, Euclidean and non-Euclidean geometries. Development and history, New York, NY: W. H. Freeman and Company (ISBN 978-0-7167-9948-1). xxix, 637 p. (2008). ZBL1127.51001.

modifies Hilbert's primitive notion of betweenness in Euclidean geometry (he replaces it with a primitive notion of separation). - Greenberg explains why this modification is needed if one were to get the elliptic geometry. With this modification, the geometry of the projective plane (aka the "elliptic geometry") is then obtained via a "negation" of the Playfair's axiom (the negation is that "no parallels exist"). You can find details by reading Greenberg's book which you should be able to find in a nearby library. I am unaware of a similar modification of Birkhoff axioms and Tarski axioms (and I do not see how it can be done without a complete revision of their axiomatics). One can, in principle, also get the spherical geometry, but that requires dropping an additional axiom.