quote:

**mrimer wrote:**

Whoa. Well, at least it's not a supernatural number.

I'm rather fond of nonstandard natural numbers (more formally, nonstandard models of natural numbers).

The most common example is if you work with the first-order axiom version of Peano Arithmetic. PA is a very simple model for natural numbers, addition, and multiplication: basically an extremely simple foundation for number theory that is defined only for naturals. Here it is:

0. ∀x1∈N. 0 ≠ S(x1)

[a number "

zero"

exists, which does not follow any natural number]

1. ∀x1,x2∈N. S(x1) = S(x2) ⇒ x1 = x2

[if two numbers have the same successor (+1), they're the same number]

2. ∀x1∈N. x1 + 0 = x1

["

zero"

is the identity of addition]

3. ∀x1,x2∈N. x1 + S(x2) = S(x1 + x2)

[definiton of addition (defined recursively on succession)]

4. ∀x1∈N. x1 ⋅ 0 = 0

[any number times "

zero"

is "

zero"

]

5. ∀x1,x2∈N. x1 ⋅ S(x2) = x1 ⋅ x2 + x1

[definition of multiplication (defined recursively on succession, addition)]

There is also a seventh axiom in first-order PA which is a schema: an infinite set of axioms, one for each formula you can define using only the axioms above (and the schema itself). I won't post it here because it's not digestible, but it basically says:

6S. (confusing math omitted)

[for every formula in PA, you can do induction: if the formula holds for "

zero"

, and if, when the formula holds for all previous "

numbers"

, it also holds for the next; then it holds for every "

number"

in PA]

This can be reduced to a single axiom in second-order PA (where we allow things like formulas themselves to be quantified), but there are some benefits to restricting ourselves to first-order PA (where you can only say "

for all"

or "

there exists"

when you're talking about "

numbers"

i.e. objects in the system).

The weirdness is that, while PA provides an excellent model for number theory, first-order PA is not complex enough to limit the model to the one we want. So not only is PA satisfied by the typical system of natural numbers with addition and multiplication, it can also be satisfied with numbers like...

0, 1, 2, ... a, b, c, d, ...

...where the a, b, c, d numbers are an infinite distance from the standard natural numbers. They get much weirder than this, but this "

disconnection"

between sets of numbers is the main idea.

The kicker: if you want to prove something holds for PA, frequently you need to include these odd non-natural numbers in your analysis.

(Now TFMurphy can correct everything I said using the wrong model-theoretic terminology, if he chooses.)

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*[Last edited by Trickster at 04-10-2014 08:51 AM]*