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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© Christer Blomqvist 1998.