Do you believe in god or math?

[CliveStaples]

Ok, well let's talk just a little bit about set theory and see if we have even remotely the same idea of what it even emtails.

I would like to ask you the following questions just to get an idea of where you're coming from.

1. When do you believe that set theory first became a rigorous mathematical concept.

Hmm. Probably the work of Zermelo, Franco, and Russell in the early 20th century.

2. Who, if anyone, do you associate with introducing this concept formally to the mathematical community. (Feel free to name multiple mathematicians if you like)

Hmm. That's a tough question. I'd probably say Cantor, Weierstrass, or Dedekind.

3. Do you believe that the basic idea of a set was a required intuitive concept prior to this formalization?

Probably not. Greeks did number theory without really thinking of sets, so far as I know.

4. What is the formal mathematical definition of number in general?

If you believe there is more than one formal definition of number please explain as many as you can. (briefly) You can assume a simple cardinal definition here of the "Natural Numbers" or possible include a definition of Real Numbers if you like. You don't need to go off in la la land, speaking to the issues of imaginary numbers, complex numbers, irrational numbers, and so on. Let's try to keep this relatively basic. No need to make a essay out of it. I'm not going to pick at it. I'm just curious of how you think of these basic concepts.

It depends. You can define them algebraically, seeing a "number" as being part of a structure. Weierstrass defined the real numbers in terms of convergent sequences of rational numbers, which were defined in terms of integers, which were defined in terms of natural numbers, which were sort of assumed to exist.


I tend to think the algebraic definition makes the most sense. You don't really care about the numbers themselves, but about how they relate to each other (due to the operations defined on them as a set).

5. What do you believe is the formal definition of the Number One?
(feel free to give more than one definition if you like)

1 is the multiplicative identity (i.e., for any number x, x*1 = 1*x = x), 1 is not equal to 0, and 1 = s(0) (1 is the successor of 0 according to the definition of natural numbers from the Peano axioms).

6. How would you define a "Set"?

A set is a collection of objects that has a well-defined rule for inclusion in the sense that for all x and any set S, either x is in S or x is not in S but not both.