What is ZFC axioms and how are they different than normal axioms?

Question

I'm in algebra, I'm not yet at the level of writing proofs but I'm curious what is the difference between a ZFC axiom vs a normal axioms? Obviously the ZFC is named after the authors of it, but what is so special about it that it would differ from an axiom in geometry?

Answer 1

In math, an "axiom" is anything you assume to be true in order to form the basis for a body of deductions. Depending on which axioms you choose, you can get completely different mathematical results.



In geometry (in high school geometry, anyway) we take Euclid's 5 axioms and go from there. These axioms are:



1. There is a line connecting any 2 points

2. Any line can be extended infinitely

3. There is a unique circle with a given center and radius

4. Any two right angles are equal

5. The parallel postulate



Based on only these 5 statements, we can deduce everything we know about "Euclidean" geometry (for example: sum of a triangle's angles is 180˚, the triangle postulates, and so forth). If we change that 5th statement, the result is a totally different kind of geometry. The systems of geometry that use a different 5th axioms are called "non-Euclidean geometry". These systems of geometry can usually be thought of as the geometry of a warped surface; for example, triangles on the surface of a globe have angles adding to more than 180˚.



The point is, what exactly you choose as your axioms has a significant consequence on the deductions (the "theorems") that result.



So those axioms deal with geometry. The ZFC axioms, on the other hand, deal with set theory. Since we use set theory to count, these axioms essentially tell us what numbers are and how to use them. Based on these axioms, one may deduce everything we know about mathematics as we use it today. The axiom of choice (which corresponds to the C in ZFC) is a little bit like the parallel postulate in that you need it for a lot of things, but a lot of people think it's ugly, so they don't like it and try to get around using it if possible. So, sometimes people talk about using "ZF" instead of "ZFC". There are other sets of axioms that can be used, but these are by far the most popular.



So what's the difference? The axioms talk about different things. Where your "normal" axioms (Euclid's axioms) talk about points and lines, ZFC talks about sets and their subsets and elements.



Hope that helps.

Answer 2

Zfc Axioms

Answer 3

ZFC is not a particular type of axiom, rather it refers to a collection of axioms in set theory. ZFC is Zermelo-Fraenkel (reference 1) along with the Axiom of Choice. Other axioms are at the foundation of other systems - for instance the 5 Euclidean axioms are what Euclidean geometry is based on.



It is unlikely that you would need to understand the ZFC axioms at your level. You would only need them to deal with unusual cases such as Russell's Paradox (reference 2).