Axiomatic Systems

Jun 2017
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CT
Hello, this is my first post. I'm working through Euclid's Elements right now and I really love the idea of beginning with basic, intuitive axioms, and build in a systematic way to an entire structure, with a solid foundation. I've been wondering if there are other systems like this? I've been looking but I haven't found much. For instance, has someone attempted to continue Euclid's Elements to include trigonometry? Or has anyone built an axiomatic system of algebra? I'd really be interested to know if there's anything out there.
 
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Aug 2012
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I'm very far from knowing much about this. Evidently people who have looked at Euclid have realized that his axioms are far more murky than realized. Hilbert has an axiomatization of Euclidean geometry that's regarded as better. This is a direction for you to go in.

https://math.stackexchange.com/questions/328028/what-are-the-differences-between-hilberts-axioms-and-euclids-axioms

Likewise, the ancient parallel postulate has now been replaced by Playfair's axiom.

Like I say, I don't know anything about any of this; these are just some things I've run across that will be of interest to you.

I definitely commend you for reading Euclid.

All of modern math has been axiomatized by the Zermelo-Fraenkel axioms of set theory, called ZF. Sometimes it's called ZFC in honor of Mrs. Fraenkel, who was pro-choice. That's a lame joke. https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory

All of the object of modern algebra are axiomatized. Look up groups, rings, fields, monoids, semigroups, vector spaces, and modules.

There are axioms for the real numbers. There are the Peano axioms for the natural numbers.

There's an axiomatic description for pretty much everything these days.
 
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