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

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number (a - 1) other than the null-class. Therefore in the simplified theory it is useful to deal with ascending cardinals; but, as
I think, we ought at any rate to set aside the special conventions
of Principia. In the Pure Theory of Types we have to do simply
with homogeneous cardinals. This limitation enables us to get the
theory of cardinals without any supplementary convention.
A. Homogeneous Cardinals.
Let us assume the following abbreviations:
101-0001 'I - = DC
df
101-0002 ~' -
df
103-001 Redl(o) (x'):. x' C w: extens [DC(a) { }.
'd7 a
D ('I): = -: C{x',K}:. extens[DaC()] {w}.
10 -002 Red2(w) = (') (-):: ' C (: extens [DaC(] { }. Psl- -Cls:
df a
a x':{P,( C}. ( Q) Q -: { Q, A }:.extens [DaCJ] (}.
a a
103 003 Red()) =. Red,(co). Red,(co). (HS) C{S. B }.
df
103-004 Red (z,) =. Red (). Red () ).
dJ
We see that Red (w) is the hypothesis of reducibility of subclasses of o, as much as of one-many relations, whose converse
domain is a sub-class of o. We shali deal with reducible classes,
i. e. classes which satisfy Red(ow), using a niethod analogous to that
which we have applied to the problem of extension. As we cannot
prove that there are in the type DaC classes which are not identical with LtCa or Lt-a a the existence of cardinals other than 0,1
and 2 is not assured by any means in this type. As we can prove
that the null- class, as well as the classes containing elements
identical with a unit element, or with one of two given elements,
are reducible classes our dealing with reducible classes is no serious
limitation of the theory of homogeneous cardinals.
I assume the following definition of the relation of similarity
between reducible classes:
103-004 smr -= K: j [. Red (, w). x sm w.]
df