Axiom 3

Using the axiom "It is 0 in Z such that A 0 = a for every a in Z," I want to apply it to games, to equal 0 to the empty set. However, I have to first prove that I can do and that 0 is in fact equal to the empty set. Please help!

0 is not equal to null set.
0 is an element of the set Z.
The empty set is a subset of A to Z and it has zero elements, not even the element 0.

I think I understand what you're getting ...

But first to clarify, saying: "SET NULL = 0" is not quite correct.
For your example using A to Z, 0 is an element of A to Z, ie it has "measure" of a ... ie it is a singleton, and therefore can not be the empty set. So, arguing that 0 is equal to the empty set is out of question.


Example *****:

Let S be the set of places:

S = (A, B)

A = All four squares of
square B = set of all non-4 sides

A set is invalid if it has a cardinality of 0.
Since there is no non-four seats facing, | B | = 0, then the set B = null.

So if I understand you want to make the analogy so that you can say:

"There is a set N in S such that QUN = Q for each Q in S

***** Note: U = Union

For this example, N = B = empty set. Then AUB = A

The problem is that the null set "N" need not be included in S, since its 'measure' is 0

S = (A, B) = (A) U (B) = (U) (A) = (A)

However, using (Z, +) as an example:

Z = (..., -4, -3, -2, -1, 0, 1, 2, 3, 4,...)

We can not exclude the additive identity, 0.

Z is not equal to: (..., -4, -3, -2, -1, 1, 2, 3, 4,...)

We hope you see the difference?
-------------------

Now back to show:

"There is a set N in S such that QUN = Q for each Q in S

What we actually want to show is "QUN = Q" where N is the empty set.
I do not know if there is a good way to show it is obvious / trivial.

Let Q be a non-empty and N is the empty set.
Then QUN = QU () = Q.
Thus QUN = Q.
----------------

Hope this helps.

In the construction of natural numbers, 0 * defined as the empty set.

It is true that the AU () = A, and it is not difficult to show.

but you would not use facts about Z to prove things about games. you start with the axioms of set theory, then build from A to Z and prove all its properties.