Note: Before reading my comment, I don’t know jack shit about mathematics and much else. I’m still struggling with basic algebra. But I’ve got the same questions as you and compiling a lot of stuff to educate myself. Here are some ideas and links.
One of the most important concepts for the foundations of mathematics is recursion. Naturally, you’ll have to fall back on some circular definition if you seek answers without dissent, progress ad infinitum or assumption. That is, you have to rely on a recursive axiomatic definition (e.g. factorial n = if n == 0 then 1 else n * factorial (n − 1)). I was told that learning a functional programming language like Haskell can be very helpful.
Addition Recursively:
Let n’ be the successor of n, that is the number following n in the natural numbers, so 0′=1, 1′=2. Define a + 0 = a. Define the general sum recursively by a + (b’) = (a + b)‘. Hence 1+1=1+0’=(1+0)‘=1’=2.
Addition Set Theoretic:
Let N(S) be the cardinality of a set S. Take two disjoint sets A and B, with N(A) = a and N(B) = b. Then a + b is defined as N(A U B).
Some sentence I found really interesting is the following by Eliezer Yudkowsky:
A decision procedure is a finite specification of all truths of euclidean geometry; I can use that finite fact anywhere I could use any truth of geometry.
If you go outside of mathematics into decision theory and programming you’ll be able to create finite statements as a kind of infinite workaround to Gödel’s incompleteness theorems.
So I have no idea about measure theory but it sounds like some top-level concept, not what you want right now but something you’ll have to get back to later to prove the consistency of other concepts.
Well, I’ve no clue but that sounds right right now ;-)
Note: Before reading my comment, I don’t know jack shit about mathematics and much else. I’m still struggling with basic algebra. But I’ve got the same questions as you and compiling a lot of stuff to educate myself. Here are some ideas and links.
http://en.wikipedia.org/wiki/Mathematics
http://en.wikipedia.org/wiki/Foundations_of_mathematics
One of the most important concepts for the foundations of mathematics is recursion. Naturally, you’ll have to fall back on some circular definition if you seek answers without dissent, progress ad infinitum or assumption. That is, you have to rely on a recursive axiomatic definition (e.g. factorial n = if n == 0 then 1 else n * factorial (n − 1)). I was told that learning a functional programming language like Haskell can be very helpful.
Addition Recursively:
Addition Set Theoretic:
That’s from Wiki: http://en.wikipedia.org/wiki/Addition#Addition_of_natural_and_real_numbers
Some sentence I found really interesting is the following by Eliezer Yudkowsky:
If you go outside of mathematics into decision theory and programming you’ll be able to create finite statements as a kind of infinite workaround to Gödel’s incompleteness theorems.
So I have no idea about measure theory but it sounds like some top-level concept, not what you want right now but something you’ll have to get back to later to prove the consistency of other concepts.
Well, I’ve no clue but that sounds right right now ;-)
More links:
A Measure Theory Tutorial (Measure Theory for Dummies)
Topology Without Tears
(I’ve links and stuff about a lot of fields, just ask if you need something.)