Objections to set theory as a foundation for mathematics
From set theory's inception, some mathematicians
have objected to it as a
foundation for mathematics. The most common objection to set theory, one
Kronecker voiced in set theory's earliest years, starts from the
constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in
naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by
Errett Bishop's influential book
Foundations of Constructive Analysis.
[13]
A different objection put forth by
Henri Poincaré is that defining sets using the axiom schemas of
specification and
replacement, as well as the
axiom of power set, introduces
impredicativity, a type of
circularity, into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo-Fraenkel theory, is much greater than that of constructive mathematics, to the point that
Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]".
[14]
Ludwig Wittgenstein condemned set theory philosophically for its connotations of
Mathematical platonism.
[15] He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers".
[16] Wittgenstein identified mathematics with algorithmic human deduction;
[17] the need for a secure foundation for mathematics seemed, to him, nonsensical.
[18] Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical
constructivism and
finitism. Meta-mathematical statements — which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory — are not mathematics.
[19] Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in
Remarks on the Foundations of Mathematics: Wittgenstein attempted to refute
Gödel's incompleteness theorems after having only read the abstract. As reviewers
Kreisel,
Bernays,
Dummett, and
Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as
Crispin Wright begun to rehabilitate Wittgenstein's arguments.
[20]
Category theorists have proposed
topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as
constructivism, finite set theory, and
computable set theory.
[21][22] Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for
pointless topology and
Stone spaces.
[23]
An active area of research is the
univalent foundations and related to it
homotopy type theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with
universal properties of sets arising from the inductive and recursive properties of
higher inductive types. Principles such as the
axiom of choice and the
law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results.
[24][25]