The Emergence of First-Order Logic

118 Gregory H. Moore
The intended meaning was that if the proposition A (x) holds when x is
r'A, then A (x) holds for an arbitrary x, say a. He thus defined (Vx)A(x)
as A(rA) and similarly for (3*M(x) (1923, 157).
Soon Hilbert modified the Transfinite Axiom, changing it into the
e-axiom:
where e was a universal choice function acting on properties. The first
sign of this change occurred in handwritten notes that Hilbert prepared
for his lecture course on logic during the winter semester of 1922-23 ; 15
the new e-axiom appeared in print in (Ackermann 1925) and (Hilbert 1926).
Hilbert had used the Transfinite Axiom, acting on number-theoretic functions,
to quantify over number-theoretic functions and to obtain a proof
of the Axiom of Choice for families of sets of real numbers (1923, 158,
164). Likewise, Ackermann used the e-axiom, acting on number-theoretic
functions, to quantify over such functions while trying to establish the
consistency of number theory (1925, 32).
In 1923 Hilbert stressed not only that his proof theory could give
mathematics a firm foundation by showing the consistency of analysis and
set theory but also that it could settle such classical unsolved problems
of set theory as Cantor's Continuum Problem (1923, 151). In his (1926),
Hilbert attempted to sketch a proof for the Continuum Hypothesis (CH).
Here one of his main tools was the hierarchy of functions that he called
variable-types. The first level of this hierarchy consisted of the functions
from n to M—i.e., the number-theoretic functions. The second level contained
functionals—those functions whose argument was a numbertheoretic
function and whose value was in 1M. In general, the level « +1
consisted of functions whose arguments was a function of level « and
whose value was in M. He permitted quantification over functions of any
level (1926, 183-84). In (1928, 75) he again discussed his argument for CH,
which he explicitly treated as a second-order proposition. This argument
met with little favor from other mathematicians.
From 1917 to about 1928, Hilbert worked in a variant of the ramified
theory of types, one in which functions increasingly played the principal
role. But by 1928 he had rejected Russell's Axiom of Reducibility as
dubious—a significant change from his support for this axiom in (1917).