I assume someone has figured this out... If so, would anyone mind explaining it to me?
The ZF set theory is the most common foundation of maths:
The ones that define the thing(s) you are interested in studying.
There are, in fact, many different branches of mathematics, each having its own axioms. And, there are many different choices as to which (equivalent) statements you will take as axioms.
That said, since set theory is about the simplest mathematics you can have, set theory with the Zermelo-Frankel axioms for set theory (the "ZF" arildno referred to) provide a pretty good starting place. Of course, for more complicated branches of mathematics, you will need to add other axioms. And, often, the axioms for one branch contradict the axioms for another.
Set theory covers your arithmetic, cumulative, and associative properties.
the whole is greater than the part.
I prefer to assume ZFC + the axiom of determinacy. It makes proofs much easier, albeit somewhat repetitive.
Quote due to Aristotle, who also gave us that other pillar of mathematics, the principle of non-contradiction.
In the context of the original question I respectfully disagree with the statement "there are many branches of mathematics, each with there own set of axioms", although I can imagine cases in teaching where this would be stated. I suppose the person who said this has in mind something like probability, where we have "axioms" to define a probability space etc, but in this context those definitions serve only as premises. So the theorems of probability theory are ultimately conditional statements A -> B written in set theoretic notation, and the steps of the proofs are all justified in terms of the axioms of set theory.
So having established that all mathematics can be derived from the ZFC axioms, it is worthwhile to say that there is no way to tell if ZFC is consistent, and it's significance derives from being the simplest encompassing framework yet found, it could always be replaced by something simpler (but not too simple, for first-order logic alone falls deeply short of supporting all of mathematics). In fact, since ZF and ZFC are either both consistent or both inconsistent, there is hardly any reason to take one over another other then that ZFC is more powerful (folks have not yet worked out alternate proofs to many important theorems that rely on the axiom of choice).
Yes, that does make proofs easier... since the axiom of determinacy is inconsistent with the axiom of choice!
I assume you either meant ZF, or were making a joke. :P
No, I had in mind things like the axioms of Euclidean geometry versus the axioms of hyperbolic geometry.
You know, I have a related question. Even though I'm an EE, I had some time to kill this summer (between graduation and grad school), and so I embarked on a quest to rigorize everything I know about math in terms of ZFC. I started by proving some very basic set theorems (sans the notion of cardinality), and moved on to constructing the natural numbers, integers, and rationals in terms of equivalence classes. Likewise, I constructed the reals from equivalence classes of Cauchy sequences.
Anyway, I haven't had too much free time as of late, so I've been stuck in real analysis (using Rudin's book). However, partially because of this thread, I realized that I have never actually had to use the Axiom of Choice, which leads me to believe that either I made a mistake somewhere, or that I haven't gotten to the point where I've needed to use it. Is it possible to construct the real numbers without choice, or did I make a grave error somewhere along the way? It would really bum me out if I had to go back and pore through my proofs to find a mistake, because from a philosophical perspective, the only things I trust mathematically are those things which I have proved myself.
That is something you (usually) can do, not something you must do.
One reason why set theory is so useful is that it mimics higher order logic; if you use a first-order axiomization of set theory, then AFAIK, that's (in principle) enough to do any mathematics using only first-order logic.
The other reason I disagree is that arguments that second-order logic is "more powerful" seem to be based on a biased view of the ambient mathematical framework.
For example, one popular argument is that there are many nonisomorphic models of first-order real analysis in ZFC, but there is an essentially unique model of second-order real analysis in ZFC. But the very same idea can be restated as the fact that there is an essentially unique model of first-order real analysis in ZFC whose power set operator coincides with ZFC's power set operator.
(At least, I think that's right)
You don't need the axiom of choice to construct a model of the reals. You don't even need Frankel's axioms or full Zermelo set theory!
Incidentally, it might be worth also trying to construct the reals using Dedekind's construction. The constructions are fundamentally different; for example, while in traditional mathematics it gives the same result as Cauchy's construction, it turns out that in intuitionistic logic, you cannot prove that they are isomorphic!
Thanks for the verification. Incidentally, the axiom of choice wasn't the only ZFC axiom that I didn't use, so I was glad to read your post.
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