I've been referred to this website, hopefully you have the background in set theory to help me out here.
Got two questions, the first is on number systems arising out of set theory and the second is a small one about functions. I've tried to be as clear as possible but please bear with me; the only reason I'm posting this is because I'm so confused.
I'm trying to get my bearings on the set-theoretic construction of the natural numbers. Just as I was confused looking at the the way the reals could be axiomatically defined, or constructed by Cantor or Dedekind, or thinking that Cauchy, Weierstrass etc... all had different constructions so now am I going through that with the various explanations of the natural numbers as originating from set theory.
The Peano axioms can be taken as the starting point, but I found out that they could be taken as a theorem in set theory, so I've been trying to do that. It seems that one way is to look at the idea of an ordinal which is a transitive set whose elements are also transitive. But there's also this idea of equivalence classes that apparently lead to contradictions in ZFC. Also there's the Axiom of Infinity method in the wiki link I've given below. Perhaps there's even more, or these are the same in some way I can't see?
A problem arises for me as I've come to believe that rational numbers are defined as a set of equivalence classes, but Wikipedia says that equivalence classes are ruled out of axiomatic set theory so what replaces Q when dealing with ZFC? When you remove equivalence classes from the picture what defines the rational numbers in ZFC? I'm sure whatever you do will generalize to the reals.
I think the question could best be put as follows:
Starting from the most fundamentals, i.e., the axioms of a specific model of set theory, what is the logical and linear progression of topics that leads to the construction of $\mathbb{N}$, then $\mathbb{Z}$, then $\mathbb{Q}$, then $\mathbb{R}$, then $\mathbb{C}$? It seems to me that there are at least two possible methods this way, one using Von Neumann ordinals (with an axiom of infinity?), while the other uses the ideas of relations/equivalence classes etc... If you know anything about how to do this & understand the distinctions between the various methods of doing this could you just let me know, hopefully with a few analogies!
As an aside, I just read earlier that the difference between the Riemann & Lebesgue integrals can be explained by the analogy of a shop taking in money, the Riemann integral adds the money up as it comes in all through the day while the Lebesgue integral separates the coins out at the end of the day according to a certain scheme (i.e. the idea of measure). I'm sure there exists a nice way to picture the two (or more) schemes for constructing the number systems! (I read this earlier & I had to mention it as it would just be so helpful!).
I think equivalence classes originated with Frege & Russell used them but they run into contradictions which ZFC surmounts but Quine's set theory removes the contradictions & allows the use of equivalence classes. I'm sure there is more than just the use of equivalence classes that distinguishes these two seperate constructions. Is it fair to group this into Frege/Russell/Quine on one side & ZFC-Neumann on the other? What about the axiom of infinity description in the wiki link below? Also, I assume that the construction of the number systems in the "Elementary Set Theory with a Universal Set" link below is just the Frege-Quine idea right, I mean it's not another new one is it?
Some links might be helpful for reference of what I'm talking about:
Definition of ordinal number (from Wikipedia).
A Tour through Mathematical Logic by Robert S. Wolf. On page 77, notice at the top of the page they say that $\in$ is a "well-founded relation". My knowledge of Naive set theory tells me that $\in$ is one of two undefinable constructs that you're supposed to take as given (I could quote the sources of this if needed), but here they call it a relation. Surely they don't mean relation in the subset of the cartesian product way do they? If it is a relation why would naive set theory call it undefinable when NST deals in defining what relations are?
Wikipedia's article on the set-theoretic definition of the natural numbers.
Wikipedia's article on the Axiom of Infinity. Notice here they've used something totally different (I think?) to construct the natural numbers, the axiom of infinity. There is no mention of ordinals in this link.
Elementary Set Theory with a Universal Set (in PDF format), by Holmes and Randall, 1988. (Surely not a different construction, it's the Frege one right?).
So, that is the current state of my thoughts on this topic, it's taken me a quite a while to figure this meagre stuff out & to find good sources which aren't at the graduate level to learn properly. If I hadn't read all this conflicting stuff I'd be happy to just use the following sources as my main material:
Videos on the fundamentals of mathematics by Bernd Schröder.
Foundations of Mathematical Analysis by J.K. Truss (in Google Books).
Basically if you could help me clean up my thoughts as regards the above stuff I've written, i.e. 3 different constructions 1) ZFC, 2) Frege Equivalence Relations, 3) Axiom of Infinity & the ∈ relation statement then I could move on to greener pastures!
My second question is brief, just on the topic of a function. First a function is the idea of $y = f(x)$ in school. Then a function is the same thing with domains and ranges. Then you see that a function is really a special case of a mapping $f\colon A \to B$ defined by $f\colon a \mapsto f(a)$.
But the latest thing I've read is that a function is really a set $(A,B,F)$ which is also written as $((A,B),F)$, & I assume it's to indicate order, i.e. a Kuratowski ordered pair, and $F$ is really $F \subseteq A \times B$. This is obviously a relation & it's dictated by the rule [ $\forall(a\in A)\exists(b\in B)\Bigl( (a,b)\in F\wedge \bigl(f(a)=b\bigr)\Rightarrow (a,b)\in F\Bigr).$
I'll admit that's not exactly clear, if it's possible to clean this up & clarify it please let me know. For example, where has the little $f$ come from? The general idea here makes a lot of sense & seeing as math is always employing set theory I wish they'd just use this notation from the beginning in books.
So the question is, is this the final resting place for the definition of a function? Could you clean this up for me? Could you give me some references where this definition is explained more clearly, I can't find anything other than a mention on wikipedia & the Ali Nesin set theory notes in the link I've given above.
Thanks so much, I know it's asking a lot of your time!