The Emergence of First-Order Logic

THE EMERGENCE OF FIRST-ORDER LOGIC 113
We can assume that any law of the algebra [of logic] which holds
whatever finite number of elements be involved holds for any number
of elements whatever.... This also resolves our difficulty concerning
the possibility that the number of values of x in x might not be even
denumerable.... (1918, 236)
He made no attempt, however, to justify his assertion, which amounted
to a kind of compactness theorem.
A second logician who introduced an infinitary logic in the context of
Principia Mathematica was Frank Ramsey. In 1925 Ramsey presented a
critique of Principia, arguing that the Axiom of Reducibility should be
abandoned and proposing instead the simple theory of types. He entertained
the possibility that a truth function may have infinitely many
arguments (1925, 367), and he cited Wittgenstein as having recognized that
such truth functions are legitimate (1925, 343). Further, Ramsey argued
that "owing to our inability to write propositions of infinite length, which
is logically a mere accident, ().a cannot, like p.q, be elementarily expressed,
but must be expressed as the logical product of a set of which
it is also a member" (1925, 368-69). It appears that Ramsey was not concerned
with infinitary formulas per se but only with using them in his
heuristic argument for the simple theory of types. Yet his proposal was
sufficiently serious that Godel later cited it when arguing against such infinitary
formulas (Godel 1944, 144-46).
8. Hilbert: Later Foundational Research
Hilbert did not abandon foundational questions after his 1904 lecture
at the Heidelberg congress, though he published nothing further on them
for more than a decade. Rather, he gave lecture courses at Gottingen on
such questions repeatedly—in 1905, 1908, 1910, and 1913. It was the course
given in the winter semester of 1917-18, "Principles of Mathematics and
Logic," that first exhibited his mature conception of logic. 12
That course began shortly after Hilbert delivered a lecture, "Axiomatic
Thinking," at Zurich on 11 September 1917. The Zurich lecture stressed
the role of the axiomatic method in various branches of mathematics and
physics. Returning to an earlier theme, he noted how the consistency of
several axiomatic systems (such as that for geometry) had been reduced
to a more specialized axiom system (such as that for H, reduced in turn
to the axioms for 1M and those for set theory). Hilbert concluded by stating