6
votes
4answers
736 views

which axiom(s) are behind the Pythagorean Theorem

There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the ...
6
votes
2answers
377 views

Gödel's (in)completeness theorems and the axiomatization of Euclidean geometry

In David Hilbert's 1899 Grundlagen der Geometrie, Hilbert gives a rigorous axiomatization of Euclidean geometry. As I understand it, some of Hilbert's axioms must be expressed in second order logic ...
0
votes
1answer
109 views

Problems with axioms and their potential uses in real life.

Okay, so I am looking for more examples that revolve around this topic. My topic is: "Is the assumption of basic truth needed in order to create a system of mathematics that is reliable?" Here, when ...
0
votes
0answers
61 views

Proof of congruent angles from Hilbert's axioms

I'm looking for a proof, using Hilbert's plane axioms (compiled, for instance, here), of the congruence of the four blue angles. where the lines $CE$ and $HF$ are parallel. This is a well known ...
3
votes
1answer
289 views

Is Euclid's Fourth Postulate Redundant?

Euclid's Elements start with five Postulates, including the fifth one, the famous Parallel Postulate. Less well, known however is the Postulate that forms the basis for motivation behind the fifth: ...
4
votes
1answer
646 views

What are the differences between Hilbert's axioms and Euclid's axioms?

Euclid had his axioms. Why would we need Hilbert's modern axiomatization of Euclidean geometry? What are key differences between the two sets of axioms?
1
vote
3answers
255 views

Postulates inter-dependency OR why the reluctance in removing Euclid's 5th postulate?

So I'm reading about the history of hyperbolic geometry and something like this came up: "two thousand years later, people gave up on trying to derive the fifth postulate from the other 4 and begun ...
15
votes
5answers
856 views

What is the modern axiomatization of (Euclidean) plane geometry?

I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
10
votes
5answers
554 views

What are the postulates that can be used to derive geometry?

What are the various sets of postulates that can used to derive Euclidean geometry? It might be nice to have several different approaches together for comparison purposes and for ready reference. It ...