The theory of constructive types (principles of logic and mathematics). By Leon Chwistek.

The Theory of constructive Types.
(Principies of Logic and Mathematios).
Part 1.
General P.rinciples of Logic: Theory of Classes anu Relations,
By
Leon Chwistek.
Introduction,
The purpose of the present paper is to s how ow we canz build
up a system of Logic and Mathematics, assuming no other prinitive ideas and propositions than those of the Logical Calculus. It
is to be remarked that, for foundation of Mathematics, there is
hardly any other method to be found. Suppose we assume any sy"
stem of mathematical axioms: we then must prove, that this system
contains no contradiction. To prove anything, we must have some
primitive ideas and propositions. These in their turn must contain
the primitive ideas and propositions of the Logical Calculus. There
is no means of building up a system of Mathematies, without assuming the primitive ideas and propositions of the Logical Calculus,
or their equivalents. Therefore any system of Mathematics mus
contain the primitive ideas and propositions of the Logical Calculus.
We shall see that numbers are classes, and classes are propositional fuintions. Therefore, Mathematies is a part of the theory of
propositional functions. Now, the logical calculus being a part of
the theory of propositional functions, it seems obvious that we can
get at least a part of Mathematics without assuming any other prinitive propositions than those which belong to the Logical Calculus.
This part of Mathematics appears to be the most solidly founded.
Other parts of Mathematics, — the theories based e. g. on the axiom
of iTfiuity or on Zermelo's axiom - are to be considered as con