# Tagged Questions

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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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?
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### 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 ...