The main problem is that it is not obvious that the reals exist. The reals are the most problematic of the "easy" mathematical objects because their existance is quite complex. The first problem is the naturals but they are fairly intuitive. Once you have those integers and rationals are easy.

Okay, but why did you define sqrt 2 as a set of rationals?

Mainly because it's the most common method. You have a couple different ways to do it. The convergent sequences is another.

Then I guess I mean, how is the definition legitimate. Is this something I would learn in an intro to analysis text?

So, you're defining root 2 as the set of all rational numbers below it?

Yeah pretty much any decent intro analysis book will start by constructing the reals. You can also prove they exist through a non constructive way. By showing certain types of completions exist I thonk.

How do we know there isn't another irrational number in between root 2 and the largest rational less than root 2

?

Good question. Turns out its easy enough to show that there is no largest rational less than root 2.

Suppose there is a q such that q is rational and the square of q is less than 2. Than 2 - q^2>0 and moreover it's a rational.

So we show that we can get as close to root 2 as we'd like with rationals, so it's analagous to a continuous limit from basic calculus? But why could I not do the same with a certain set of irrational numbers? Not that I can think of such a set.

Pretty much. You can certainly get a set of irrationals whose limit is sqrt 2 and it doesnt contain sqrt 2. Just take all irrationals <sqrt 2.

The trick isnt

Gah lost a post

The trick is that if we take all sets of rationals A such that A is downward closed (if p is in A and q<p then q is in A) it doesnt have a maximum element and it is not all of Q or the empty set

Then you can define addition and multiplication on them such that they form a complete ordered field such that Q is isomorphic to a subfield of it.

This is exactly what you want from the reals. So you just constructed an isomorphic copy of the reals.

Way beyond my current level, I took introductory abstract algebra and introductory analysis, but I haven't gone beyond, and we didn't learn much of the motivation behind the math, more of just stuff.

I have some books here though, so I'm just going to start reading those. How long have you been studying math? I've got a question about self-studying.

I dont study math anymore but I have roughly 11 years of pure math education(university level).

I studied abstract algebra for my masters then I got another masters in set theory and worked on getting a PhD in set theory before having to quit.

Did you do a lot of self-studying or mostly coursework/course related studying?

Both really. It depende on what I was looking at.