Fringe Math

[Categories+Sheaves]

[Tiberius]

We were taught about countably infinite and uncountably infinite sets in my first year at university. It's hard to wrap your head around at first, but it does make sense. The way we were taught was:

The set of all natural numbers (i.e. all the positive whole numbers from 0 to infinity) are countable, since you can theoretically count them one by one forever. There is a specific start value (0) and the next number in the sequence is well defined (1).

The set of real numbers (i.e. any non-imaginary / non-complex number) are not countably infinite, since there is no specific start value (negative infinity is not a number, but a limit), and even if we were to define a start number (say, 0), the number that comes after 0 is not defined. In fact, for any number you can come up with, there exists a number which fits between it and 0. For instance, if we were to pick 1 as the next number, we find that 0.5 fits between them, so 1 cannot be the next number in our count. If we use 0.5, we find that 0.25 fits between them...and so on.

Using standard notation to try and represent the first number that comes after 0 gets us into even more trouble, since 0.000....1 is an invalid number (you can't have an infinite amount of 0's followed by a 1). This renders the entire set of real numbers uncountable, and ultimately leads to all the lovely "impossible" examples of mathematics you posted about.

[Faith No More]

Interesting, when i was taught that concept we were told that the infinite set of real numbers is a larger infinity than the set of integers. Which was apparently to complicated to prove to us, but we just had to take her word for it that there big and small infinities. Never heard the countable and uncountable stuff before, but it makes sense.

[Categories+Sheaves]

[Modular Moog V]

These questions should cut into the teleology of math; what deserves study/acceptance, and why? Do we consider ourselves citizens of the world we construct, or the world we intuit?

You sound like a fan of Constructive mathematics, as am I. But I am merely guessing here. Is it true?

[Categories+Sheaves]

You sound like a fan of Constructive mathematics, as am I. But I am merely guessing here. Is it true?

I mean, I prefer constructive results to non-constructive ones, but I was hoping to defend the other side of the discussion once I pushed people far enough out of their math-comfort zones and gathered some interesting opposition.

[Modular Moog V]

[Categories+Sheaves]

No, I meant this:http://en.wikipedia.org/wiki/Constructiv...ematics%29

I think that's what I meant too? I'm sure constructionists would scoff at the steps taken to prove that every non-unit of a ring is contained in a maximal ideal (it relies very strongly on Zorn's Lemma). But I would hardly want to do algebra without that result.
I like the nonconstructive mathematics. But I don't want the opposition to be a straw argument.

[CliveStaples]

I don't find constructivism very appealing. What would a constructivist make of the following argument:

n = 2 if the Goldbach conjecture is true, 3 otherwise

Is n prime?

[Darth]

I have magnetic coloured letters (usually arranged into humorous rude words), a calendar and some phone numbers on my fridge.

Now I feel sad because my fridge isn't as smart as everybody else's =(