June 12th, 2017, 05:11 PM  #1 
Newbie Joined: Jun 2017 From: CT Posts: 1 Thanks: 0  Axiomatic Systems
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.
Last edited by skipjack; June 12th, 2017 at 11:42 PM. 
June 12th, 2017, 05:24 PM  #2 
Senior Member Joined: Aug 2012 Posts: 1,618 Thanks: 411 
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 https://math.stackexchange.com/quest...euclidsaxioms Likewise, the ancient parallel postulate has now been replaced by Playfair's axiom.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 ZermeloFraenkel axioms of set theory, called ZF. Sometimes it's called ZFC in honor of Mrs. Fraenkel, who was prochoice. That's a lame joke. https://en.wikipedia.org/wiki/Zermel...kel_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. Last edited by skipjack; July 15th, 2017 at 10:02 AM. 

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