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

The Theory of Constructive Types.
(Principles of Logic and Mathematics).
Part II1.
Cardinal Arithmetic.
By
Leon Chwistek.
V. Complements of Part I.
A. Extension and Intension.
The Theory of Types, as explained in Part I, may be called
the Pure Theory of Types, as it is based on the most general
idea of logical types, and as it does not assume any other propositions, than the axioms of the Logical Calculus. This method enables us to get a system of Mathematics which appears to be a part
of Logic, and as such may be called Pan-Mathematics. This system
is more general than Classical Mathematics, as it does not enable us
to prove that there is a class of inductive numbers other than the
null-class, which does not contain the greatest element Nevertheless,
if we assume the axiom of infinity as a hypothesis, we get a special system, which is as a matter of fact the same thing as what
is called Classical Mathematics,- Cantor's theory appears then as a
hypothetical system that we can get, if we assume the existence of alephsConformably to the hypotheses which we assume, we can get many
special systems of Mathematics. As the Pure Theory of Types does
not assume any existence - axiom and does not lead to Richard's
paradox, it is a natural base for rational Semeiotics, a science whose
importance can scarcely be denied. Note that the simplified theory
of types, as expounded on p. 12 of Part I, may be used in Mathematics without any risk of getting a contradiction. To avoid such