Abstract:
Many diagnoses remain on offer in
recent philosophy of why Frege’s system of
Grundgesetze
collapsed in contradiction. They include his recourse to unrestricted
objectual
quantification and — not the same thing — to impredicative
first-order
quantification, the simultaneous impredicativity of the higher-order
logic of
Begriffschrift, and his oversight of the
ontologically
‘inflationary’
character
of the course-of-values operator as characterized by Law V. There are
issues
about whether (and which of) these diagnoses are fundamentally in
tension. In
any case, the focus of the present paper will be on another one, less
clear and
even harder to assess: that Frege’s problems were owing to his
having not even
a “
glimmering of a
suspicion of the
existence of indefinitely extensible
concepts” (Dummett, my italics).
The very idea that
there is any
coherent notion of “indefinite extensibility” has been
received skeptically by
some authorities (Boolos, Burgess). My purpose here is to address this
skepticism. I will offer a new
characterization of indefinite extensibility, outline its connection
with
paradox, and try to explain wherein Frege’s oversight of it
consisted and the
obstacle posed to anything worth regarding as a logicist foundation for
the
mathematics of indefinitely extensible domains.