This was brought on by the multiplication question I posted earlier (that I am still trying to find an answer to).
So what are they? I assume that there must only be a very few and that they are simple to understand, like the Euclidian geometric axioms.
Modern math is based on set theory. There are only a few axioms, it’s true. They’re mostly simple, too–although it would take some effort to understand the controversies surrounding the axiom of choice and the axiom of regularity.
Look here for a list of the Zermelo-Fraenkel axioms. ZF is the set theory that most people work from, although there are others–most notably Kelly-Morse set theory, von Neumann-Bernays-Gödel set theory, and Russell’s theory of types.
However, there are some mathematicians who study disciplines which could supplant set theory as the basis of math. These are category theory and topos theory, and probably other, related theories.
About a year ago, a poster named MrDeath did a thread in MPSIMS discussing the basic axioms in set theory. It’s a good read, and should be interesting to you.
[sub]On preview, I see that ultrafilter provided most of the info I give below. But I wrote it, so I may as well post it.[/sub]
My understanding is that there are several different ways to set up the foundations of mathematics. The primary two I’ve heard of are Set Theory and Category Theory. I don’t know much at all about how Category Theory is used, but the standard formal axiomatization of Set Theory is provided by the Zermelo-Fraenkel axioms. (Click on the links of the axioms to read their plain-language translation)
There are two reasons these axioms might not be as satisfying for you as the Euclidean axioms. First, they are not all immediately intuitively obvious. It requires some thought to see why they must be true. Even after you understand them, some may seem to be phrased in an unnatural way, but in fact they have been phrased very carefully to avoid certain known paradoxes, e.g. Russell’s Paradox.
The second reason these axioms may seem unsatisfying is that a lot of work of developing mathematics from the theory of sets is in the definitions. Even defining numbers in terms of sets requires a little cleverness. This page gives the method due to von Neumann.
1 + 1 = 2 is always important, as is the rest of the basic addition table (1 + 2 = 3, etc), which is the basis of most math.
Do you have a cite, Qwertyasdfg? I admit that I’m a bit unfamiliar with this stuff, but it seems that facts like those would result from the Axiom of Infiinity, not be axiomatic themselves.
Actually that looks like a really simplified version of the Peano axioms:
[ul][li]There is a natural number 0. [/li][li]Every natural number a has a successor, denoted by a + 1. [/li][li]There is no natural number whose successor is 0. [/li][li]Distinct natural numbers have distinct successors: if a <> b, then a + 1 <> b + 1. [/li]If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers.[/ul]
You can prove the Peano axioms from the ZF axioms. There is a slight technicality–the 5th Peano axiom refers intuitively to every property of the natural numbers, and there’s an uncountable number of those. However, in ZF, you can only talk about properties that can be expressed by a finite well-formed formula, and there are only a countable number of those.