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.