Constructively proving that the integers are countable

[Kuciwalker]

Solver, he's trolling you.

[KrazyHorse]

I know he's trolling, but it's a good opportunity to demonstrate that math concepts are not necessarily scary or non-intuitive.

[Solver]

I never know if people are serious when displaying an inability to deal with numerical concepts or intuitive mathematical concepts. Given how many people I know who can't reliably calculate percentages, eh...

[Solver]

Originally posted by KrazyHorse
I know he's trolling, but it's a good opportunity to demonstrate that math concepts are not necessarily scary or non-intuitive.

It's just a lucky coincidence that we're talking countability here, for which you can demonstrate that it's intuitive. If you can show that the fact that i is a well-defined constant isn't scary, then my hat is off to you.

[Dauphin]

I've recently come across literature that says "There is a one in three chance of you winning, that's almost 30%!".

Technically correct, but I think not in the intended fashion.

[Krill]

Nice troll Elok

[Solver]

Originally posted by Dauphin
I've recently come across literature that says "There is a one in three chance of you winning, that's almost 30%!".

Technically correct, but I think not in the intended fashion.

I've seen stuff that is just as bad, related to Civ4's game speeds. At Normal speed, you build at 100% of speed, at Epic, you build at 150% of speed. So people say that building on Epic is one half longer than on Normal. Okay. But then they say that Normal is half the build speed of Epic. Gaaaaah!

[Elok]

Technically I wasn't trolling, since I figured KH knew me a little too well to think I was not only that stupid, but also sufficiently willing to expose my own ignorance as to argue with an expert about his own subject.

Mostly I just wanted to see how long I could go on deliberately misreading KH while still appearing to be reading his posts. So you might say I was testing his ability to simplify math. Or maybe I was just being a doofus for fun. Yeah, probably the latter.

[tinyp3nis]

It worked out like the "your profession is stupid I have common sense" troll Asher uses succesfully on Agathon

[Elok]

Except Agathon's profession IS stupid. While mathematics is a relatively alien subject to me, and I don't claim to understand its mindset, I can respect the contributions it makes to human society, via computers and physics if nothing else.

[Jon Miller]

Originally posted by KrazyHorse
LulThyme is the only serious mathematician here. I am a distant second to him.

Kuci, Ramo and (dare I say it) even JM are behind me.

I haven't done much math in years... so I am sure.

Despite the fact that I have a math degree, I remember our math backgrounds were very similar when we graduated college.

JM

[Felch]

I skipped most of this thread because I knew I wouldn't be smart enough to understand it, but why isn't it just axiomatic that you can count something that is discrete, like integers? What good does it do? It's less useful than memorizing a hundred digits of pi, because that at least impresses some people. Being able to prove that you can count integers is just a math-wank.

It's been done before, so you're not proving anything that isn't already understood. It's self-evident, so you're not going to impress the laymen. And it doesn't seem to have any application, so it's not like you wind up with some sweet laser or doomsday device or anything.

Pure math like this should not be on tests. It should be done for the joy of working with numbers.

[KrazyHorse]

Originally posted by Jon Miller


I haven't done much math in years... so I am sure.

Despite the fact that I have a math degree, I remember our math backgrounds were very similar when we graduated college.

JM

Cool. I just never see you post on math stuff.

[Lul Thyme]

Since the set of integers Z = Zn + Zp, it's a union of two sets. Zn and Zp have been proven to be countable, so a union of them is also a countable set. I don't think including a proof of that would be needed?

Disclaimer: this is not official advice

I know this thread has probably veered way off-topic by now

It's true that a finite union of countables sets is countable but, on the other hand, the proof of this statement is essentially the proof that the integers are countable.

If they are developing the basic theory in class, they probably would only allow things that have already been proved in proofs so they would need to prove this "from scratch".

[Kuciwalker]

I skipped most of this thread because I knew I wouldn't be smart enough to understand it, but why isn't it just axiomatic that you can count something that is discrete, like integers? What good does it do? It's less useful than memorizing a hundred digits of pi, because that at least impresses some people. Being able to prove that you can count integers is just a math-wank.

What's non-obvious, and was in fact extremely controversial when it was first postulated, is the idea that there are are 'as many' integers as naturals. Intuitively there should be infinitely more integers than naturals.

Furthermore, it turns out there are as many ordered pairs (x,y) as there are integers. That is, there are only as many discrete points on the Cartesian plane as there are on its x axis. Intuitively that's nonsense.