つづく

PXV
Large Cardinals

PXVII
The set-theoretic universe V viewed as the cumulative hierarchy
∪αVα is open-ended and under-determined by the set-theoretic axioms, and invites further
postulations based on reflection and generalization. In 1946 remarks G¨ odel
[90: 151] suggested reflection in terms of a set-theoretic proposition being provable
in “the next higher system above set theory”, which proof being replaceable
by one from “an axiom of infinity”. This ties in with V cast as Cantor’s Absolute
being mathematically incomprehendable, so that any property ascribable to it
must already hold in some sufficiently large Vα, some properties leading directly
to large cardinal hypotheses. In a 1966 footnote G¨ odel [90: 260ff] acknowledged
“strong axioms of infinity of an entirely new kind”, generalizations of properties
of ℵ0 “supported by strong arguments from analogy”. This ties in with Cantor’s
unitary view of the finite and transfinite, with properties like inaccessibility and
measurability technically satisfied by ℵ0 being too accidental were they not also
ascribable to higher cardinals. Both reflection and generalization are latent in the
eternal return of successive domains as envisioned by Zermelo [30]. Whatever
the heuristics, the theory of large cardinals like other mathematical investigations
was to be driven by open problems and growing structural elucidations. (These
matters are taken up in §3. Other heuristic arguments are described in Maddy
[88, 88a].)
(引用終り)
以上