The Banach-Tarski Paradox, non-measurable sets, continuous functions that map measurable sets to non-measurable sets, impossible-to-implement strategies with even more impossible results (the blog's link to the paper is broken, but this one works). And then there's the result that many well-orderings on the real numbers exist, but none of them can be constructed.

Do we take our math seriously enough to accept the things above? Why or why not? Where do we draw the line(s)?

(For those keeping score: all the things above use the axiom of choice)

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.

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.

Quote:We were taught about countably infinite and uncountably infinite sets in my first year at university

This is a weaker point than what I'm aiming for. Cantor's diagonalization proof doesn't need the axiom of choice; we suppose the bijection f: N -> P(N) exists and is well-defined, and we define the set S in relation to the "diagonal." To answer the question "Is the element a in N an element of S?" we need only look at the f we defined, and see whether a is in f(a). This process can be performed without regard to whether any b in N (such that b != a ) is in S; our 'counterexample set' S can be constructed without AC. And we need to already understand the difference between countable and uncountable sets in order for us to interpret what that 'countably additive measure' stuff is supposed to even mean... (see earlier links)

Quote:Using standard notation to try and represent the first number that comes after 0 gets us into even more trouble...

If this is in relation to the statement about a well-ordering on the reals: no. The standard order relation on the reals is not well ordered. But by the axiom of choice, there is some well-ordering. Such a well-ordering must disregard the standard order relation on the reals. And again, it 'exists' but cannot be constructed using finite strings in a finite language.

Quote:since 0.000....1 is an invalid number

Regardless of what we're going to take 'valid number' to mean, many nonstandard formulations of analysis posit the existence of nonzero infinitesimals (the most readable one I know of is here). You're 100% right in the standard construction of the reals but... this is thread is supposed to be about the fringe, not the canon.

Quote:...and ultimately leads to all the lovely "impossible" examples of mathematics...

As I was saying before, there's a gap in the required axioms. Some things from the list, esp. B-T, can be done with a weaker formulation of AC, but... well, you know what I mean.

Since you seem to know your math; what about large cardinals, i.e. the cardinality of a proper class, i.e. the cardinality of the set of all sets? There are some formulations of class-set theory (adding the continuum hypothesis as an axiom and a fistful of other reasonable-sounding things too) that are able to disprove the existence of 'large cardinals'. But the Hardin & Taylor paper (about the infinite hat problem) states some results that require the existence of large cardinals. Does this 'split' matter to us? Don't we still want there to be some sort of 'truth' on the matter?

As far as 'truth' goes, large cardinals either exist or don't exist. We have that business about the excluded middle to thank for that. But the truthfulness of these sorts of statements may be completely unrelated to the "natural" or "intuitive" machinery of math (e.g. the continuum hypothesis is independent of ZFC). Are these curiosities some sort of 'nightmare' in our mathematical imagination that we need to dispense with and free ourselves from? Or are these things that should be embraced, and studied as objects in-themselves?

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?

(January 26, 2012 at 12:38 am)Categories+Sheaves Wrote: 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?

(January 28, 2012 at 12:58 am)Pendragon Wrote: 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.

(January 28, 2012 at 6:04 am)Categories+Sheaves Wrote:

(January 28, 2012 at 12:58 am)Pendragon Wrote: 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.

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

(January 28, 2012 at 10:35 pm)Pendragon Wrote: 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.

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?

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 =(