Some set theoretical aspects.

Created 980505. Last change 981106.

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Some set theoretical aspects.

The axioms of the Peano's arithmetic can be shown to be theorems under any strong enough set theory, for example the
Zermelo-Fraenkel's system (ZF). In this system we uses some suitable system of predicate logic (why not NAQ),
and a few axioms. We shall state these in ordinary language and dive deeper in them later.

In the following will we use two undefined terms, 'set' and 'member of '. Using these we can define the common operations
of the set algebra.

1 : Axiom of extensibility : Two sets with equal members are equal.

This states that sets are uniquely defined by their members.

2 : Axiom of the empty set : There is a set that contains nothing. This is called the empty set..

This set is written or {}.

3 : Axiom of unordered pairs : For any two sets there is a third set that contains those two sets and only those
two sets.

If A and B are two different sets then we can construct a third set {A,B} containing A and B and only A and B.

4 : Axiom of union : For every set A there is a set B that contains the union of the member of the sets that are
members of A.

In ZF will all members be sets. Now suppose A={C,D,E}, then the axiom of union states that we can construct
a set C containing all members of C,D and E.
We write this as (A).

5 : Axiom of infinity : There exist at least one infinite set. The axiom does actually show you the form of this set.
It states that there exist a set such that {} belongs to this set, and such that if x belongs to this set so does x{x}.

More about this later.

6 : Axiom of replacement : If F(x,y) is a formula such that for any x in A, there is a unique y, then there is a set B such
that y belongs to it if and only if there is a x in A such that F(x,y) is true.

One could say that the codomain B of the function F over domain A is a set, or, in other words, if you do something to
each element of a set, the result is a set.

7 : Axiom of the power set : For any set A there is a set B that includes every subset of A.

This set is called the power set of A and is written P(A).

Using these we can show that the Peano's axioms holds under a suitable interpretation of 'successor' and 'natural number'.

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