0^0

The code I gave translates easily into Agda, a computerized proof checker, and as such more formal than most mathematics. However, here's the same thing in a modernized version of Peano arithmetic. We assume all the usual properties of equality: reflexivity, symmetry, transitivity, and substitution.

    -- Axioms (only 1 and 2 are relevant)
    1. 0 ∈ N
    2. ∀ x∈N. S(x) ∈ N
    3. ∀ x∈N. 0 ≠ S(x)
    4. ∀ x∈N, y∈N. S(x) = S(y) ⊃ x = y
    5. P(0) ∧ (∀ x∈N. P(x) ⊃ P(S(x))) ⊃ ∀ x∈N. P(x)

    -- Definition of addition
    6. ∀ a∈N. a + 0 = a
    7. ∀ a,b ∈ N. a + S(b) = S(a+b)

    -- Proof that S(0) + S(0) = S(S(0))
    9. S(0) ∈ N                     [from 2 and 1]
    10. S(0) + S(0) = S(S(0) + 0)   [from 7 and 9]
    11. S(0) + 0 = S(0)             [from 6 and 9]
    12. S(S(0) + 0) = S(S(0))       [substitution of equals, from 11]
    13. S(0) + S(0) = S(S(0))       [transitivity from 10 and 12]

Edit: Ah, I see I was beaten to it by pavelrub.

It's arbitrary when we create a system out of thin air -- given different axioms (and findings over time) "math" would be extremely different.

Think about it this way -- there's no particular reason computers HAVE to be on a binary system. It's convenient for many a reasons, but there was an era where computers were analog and continuous, and it's feasible to engineer systems using a higher base and be discreet (and many have researched exactly this.) Quantum computers work even more differently too.

The fact that you count something as an object or see the world in discreet terms is also similarly arbitrary. You could see "objects" as something that's more interconnected and thus would count them differently. If we were far smaller and "looking" at things on an atomic level, putting that grouping together would not be quite as likely. And even still, you're focusing on the "discreet" positive space versus the negative space.

There are many ways to view the world that also would create its own system of abstraction and eventually "come out to be true." You're just used to one particular variety and it's all you know, so you call it the truth.

Oi vey, engineers. Everything is so 1D.

I HIGHLY recommend reading this to understand a bit more about how differently the world can be understood given just your culture alone (read: its all arbitrary): http://www.amazon.com/Women-Dangerous-Things-George-Lakoff/d...

A lecture on category theory and ZFC ignores my point.

To run with your example: why must a definition of natural numbers have to involve sets of certain cardinality? You're telling me that because some people came up with a more general definition for natural numbers after the fact, that makes one definition of natural numbers more 'natural' than another.

I claim you're just reinforcing my point: the definitions were chosen to make natural numbers a specific instance. Everything here is chosen in some way to further some goal (usually mathematical aesthetics), and eventually you get down to the bottom and what do you have? A bunch of definitions that one may choose to use or not.

So whether I want to impose 0^0 = 1 by fiat is equivalent to whether I want to assume enough foundations of category theory to prove 0^0 = 1 in that system. You (and mathgrad) are just under the spell that because there is a whole lot more mathematics floating around (and big words and important people working in those fields) it somehow makes the latter less arbitrarily motivated by a desire to make theorems work out nicely. It's certainly pleasing that it does, but that doesn't make it somehow deeper or more natural than a different formalization in which 0^0 = 0. It just serves a different purpose.

This is the entire point of the OP and the original comment: you can generalize the meaning of 0 and 1 and a^b to make it suit your need to express certain theorems and patterns, and that is a key part of the power of mathematics. Whether it's a "truth" or an "axiom" just depends on how far down the rabbit hole you're willing to go, and the distinction is irrelevant because as far as deciding what 0^0 should be they're equivalent.

The expression 52x/x has a restricted domain as well (x /= 0). But that's normally not what you mean when you write it. It isn't often you really care about expressions like 52x/x; they are generally just intermediate steps in getting to a real solution.

For example: I have done a lot of work on some equation that is interesting to me, and finally I have reduced it to 5+yx=52x+5. Now obviously the rules of algebra let me subtract 5 from each side and be left with yx=52x, and this subtraction also has no effect on the domains for which our variables may be defined. All is well.

But dividing out the x is what we are concerned with now. Surely y=52 is a solution to the equation - why can this not be true for all values of x?

Well, for nonzero x we have y=52 and nobody will complain. For x=0, though, solving for y is problematic. Note that if x=0, y could be 1, or 33, or any number. If there is some function f such that y=f(x), then it follows that f(x) holds a unique value y for each input of x/=0, but for x=0 we cannot know what y might be; this is what we mean by undefined. Thus we say the domain of f(x) is the set of all real numbers x, such that x is not equal to zero.

If you have been told otherwise, or even gotten away with doing algebra or calculus under the assumption that the domain of our function f may include zero, you are taking a mathematical shortcut rather than performing formal analysis. It is not calculus nor algebra that is broken by saying f is undefined for x=0, but rather your (albeit practically useful) misconception of these systems.

I'll finish with some formal rules of algebra, to hammer this in:

- [P6] Existence of a multiplicative identity: a * 1 = 1 * a = a ; 1 /= 0.

- [P7] Existence of multiplicative inverses: a * a^(-1) = a^(-1) * a = 1, for a /= 0.

These are taken from page 9 of Spivak's Calculus, 3rd edition. He goes on to build the foundations of all of calculus from rules like these. Surely he would not present this as a fundamental axiom of his system, only to immediately (and silently) reject it and build a flawed calculus instead!

Indeed, on pg. 41, when defining functions, Spivak later writes (emphasis his):

> It is usually understood that a definition such as "k(x) = (1/x) + 1/(x-1), x /= 0, 1" can be shortened to "k(x) = (1/x) + 1/(x-1)"; in other words, unless the domain is explicitly restricted further, it is understood to consist of all numbers for which the definition makes any sense at all.