Editing Metamathematics (section)

=== ''Principia Mathematica'' ===
{{main|Principia Mathematica}}
''Principia Mathematica'', or "PM" as it is often abbreviated, was an attempt to describe a set of [[axiom]]s and [[inference rule]]s in [[Mathematical logic|symbolic logic]] from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy,<ref name="SEP">{{cite web
|url=http://plato.stanford.edu/entries/principia-mathematica/#SOPM
|title=Principia Mathematica (Stanford Encyclopedia of Philosophy)
|last=Irvine
|first=Andrew D.
|date=1 May 2003
|publisher=Metaphysics Research Lab, CSLI, Stanford University
|access-date=5 August 2009}}</ref> being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, [[Gödel's incompleteness theorem]] proved definitively that PM, and in fact any other attempt, could never achieve this goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, there would in fact be some truths of mathematics which could not be deduced from them.

One of the main inspirations and motivations for ''PM'' was the earlier work of [[Gottlob Frege]] on logic, which Russell discovered allowed for the construction of [[Russell's paradox|paradoxical sets]]. ''PM'' sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with notion of a hierarchy of sets of different '[[system of types|types]]', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of [[Zermelo–Fraenkel set theory]].