## 4. 5 Necessity of the Set Theory

All current mathematical theories are constructed on the set theory.
However, as it is shown in an appendix, it seems possible to write the
natural number theory without the set concept. Because it needs fewer axioms,
is it not a more honest way of writing? Is it not enough to substitute
the set with the proposition to write [A(x)]instead of [x belongs to a
set A]?

It is impossible to think that the set concept is essential to the natural
number theory. It may be easier to think things as sets; however, that
is a mathematician's practice, instead of real problems of the mathematics
system. Therefore, to see how far current mathematics is able to go without
the set, we tried to write the theory of groups in the appendix without
the set concept.

If there is a necessity for sets, that is when equal signs are guided
from the logical equivalence. For example, the following formulas can be
constructed for the set theory.

(All A) (All B) (Pset (A) and Pset (B) and (All x) (Bel (A) (x) eqv
Bel (B) (x)) imp [A=B]);
Here, Pset points to the proposition saying "...are sets".
Therefore, among the sets, equal signs are produced from the logical equivalence.
This is important in some respects. That is, in THEAX, equal signs are
treated in a very narrow sense and structurizing "A =(A)" can
not be stated unconditionally. If the equal signs are formed, substitutions
can be carried out freely. Because of these reasons, the formulas above
maybe needed to state the equal signs.

The theoretical density that ranks infinitely can be rehashed from the
set to the logical proposition. However, even though the density of the
set A is sometimes written as Card (A), it is not certain if these functions
can be defined without being dependent on the axiom. The proposition Card
- Eqv (A,B) saying " the density of the sets A and B are equal"
can be easily defined.

We often hear "avoid inconsistencies by limiting objects as sets"
as the reason for another set being necessary. However, the idea of inconsistencies
being avoided by placing an axiom (of the set) is strange. New inconsistencies
can be born by placing an axiom; however, contradictions that were already
there do not disappear. Mathematicians of the primary theory who protect
the set theory should first clearly indicate the mathematical system before
the axiom, then show such things as: its inconsistencies, the fact that
any mathematical theory (natural number theory and the theory of groups)
can not be described as they are, and as soon as adding the axiom of the
set theory, describing ability is acquired.