Editing Reductionism (section)

=== In mathematics ===
In [[mathematics]], reductionism can be interpreted as the philosophy that all mathematics can (or ought to) be based on a common foundation, which for modern mathematics is usually [[axiomatic set theory]]. [[Ernst Zermelo]] was one of the major advocates of such an opinion; he also developed much of axiomatic set theory. It has been argued that the generally accepted method of justifying mathematical [[axioms]] by their usefulness in common practice can potentially weaken Zermelo's reductionist claim.<ref>{{cite journal |doi=10.1305/ndjfl/1093633905 |first=R. Gregory |last=Taylor |title=Zermelo, Reductionism, and the Philosophy of Mathematics |journal=Notre Dame Journal of Formal Logic |volume=34 |issue=4 |year=1993 |pages=539–563 |doi-access=free }}</ref>

Jouko Väänänen has argued for [[second-order logic]] as a foundation for mathematics instead of set theory,<ref>{{cite journal |first=J. |last=Väänänen |title=Second-Order Logic and Foundations of Mathematics |journal=Bulletin of Symbolic Logic |volume=7 |issue=4 |pages=504–520 |year=2001 |doi=10.2307/2687796 |jstor=2687796 |s2cid=7465054 }}</ref> whereas others have argued for [[category theory]] as a foundation for certain aspects of mathematics.<ref>{{cite journal |first=S. |last=Awodey |title=Structure in Mathematics and Logic: A Categorical Perspective |journal=Philos. Math. |series=Series III |volume=4 |issue=3 |year=1996 |pages=209–237 |doi=10.1093/philmat/4.3.209 }}</ref><ref>{{cite book |first=F. W. |last=Lawvere |chapter=The Category of Categories as a Foundation for Mathematics |title=Proceedings of the Conference on Categorical Algebra (La Jolla, Calif., 1965) |pages=1–20 |publisher=Springer-Verlag |location=New York |year=1966 }}</ref>

The [[Gödel's incompleteness theorems|incompleteness theorems]] of [[Kurt Gödel]], published in 1931, caused doubt about the attainability of an axiomatic foundation for all of mathematics. Any such foundation would have to include axioms powerful enough to describe the arithmetic of the natural numbers (a subset of all mathematics). Yet Gödel proved that, for any ''consistent'' recursively enumerable axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are (model-theoretically) ''true'' propositions about the natural numbers that cannot be proved from the axioms. Such propositions are known as formally [[Undecidable problem|undecidable propositions]]. For example, the [[continuum hypothesis]] is undecidable in the [[Zermelo–Fraenkel set theory]] as shown by [[Forcing (mathematics)|Cohen]].