Although Zermelo-Fraenkel set theory (ZFC) is generally accepted

as the appropriate foundation for modern mathematics, proof theorists

have known for decades that virtually all mainstream mathematics can

actually be formalized in much weaker systems which are essentially

number-theoretic in nature. [...] not only is it possible to formalize

core mathematics in these weaker systems, they are in important ways

better suited to the task than ZFC [...] most if not all of the

already rare examples of mainstream theorems whose proofs are

currently thought to require metaphysically substantial set-theoretic

principles actually do not; and set theory itself, as it is actually

practiced, is best understood in formalist, not platonic, terms, so

that in a real sense set theory is not even indispensable for set

theory. [...] set theory should not be considered central to

mathematics.

Probably most mathematicians are more willing to be platonists

about number theory than about set theory, in the “truth platonism”

sense that they firmly believe every sentence of first order number

theory has a definite truth value, but are less certain this is the

case for set theory. Those mathematicians who are unwilling to affirm

that the twin primes conjecture, for example, is objectively true or

false are undoubtedly in a small minority; in contrast, suspicion that

questions like the continuum hypothesis or the existence of measurable

cardinals may have no genuine truth value seems fairly widespread.

Some possible reasons for this difference in attitudes towards

number theory and set theory are (1) a sense that natural numbers are

evident and accessible in a way that arbitrary sets are not; (2)

suspicion that sets are philosophically dubious in a way that numbers

are not; (3) the existence of truly basic set-theoretic questions such

as the continuum hypothesis which are known to be undecidable on the

basis of the standard axioms of set theory, and the absence of

comparable cases in number theory; and (4) the fact that naive set

theory is inconsistent. The classical paradoxes of naive set theory

particularly cast doubt on the idea of a well-defined canonical

universe of sets in which all set-theoretic questions have definite

answers.

One philosophically important way in which numbers and sets, as

they are naively understood, differ is that numbers are physically

instantiated in a way that sets are not. Five apples are an instance

of the number 5 and a pair of shoes is an instance of the number 2,

but there is nothing obvious that we can analogously point to as an

instance of, say, the set {{0/}}. [...]

Unfortunately, the philosophical difficulties with set-theoretic

objects platonism are extremely severe. First, there is the

ontological problem of saying just what sets are.[...]

Perhaps the most influential philosophical defense of set theory is

the Quine-Putnam indispensability argument. According to this

argument, mathematics is indispensable for various established

scientific theories, and therefore any evidence that confirms these

theories also confirms the received foundation for mathematics, namely

set theory. But as a result of work of many people going back to

Hermann Weyl, we now know that the kind of mathematics that is used in

scientific applications is not inherently set-theoretic, and indeed

can be developed along purely number-theoretic lines. This point has

been especially emphasized by Feferman. Consequently, contrary to

Quine and Putnam, the confirmation of present-day scientific theories

provides no special support for set theory. [...]

This raises the possibility that the use of set theory as a

foundation for mathematics may be an historical aberration. We may

ultimately find that ZFC really has no compelling justification and is

completely irrelevant to ordinary mathematical practice.

[NIK WEAVER: "IS SET THEORY INDISPENSABLE?"]

http://www.math.wustl.edu/~nweaver/indisp.pdf

Den Hinweis auf diesen lesenswerten Artikel verdanke ich Albrecht

Storz.

Gruß, WM

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