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Set theory

2007-11-27T05:02:32Z

Conservaccount:

Set theory a branch of mathematics dealing with collections of objects.

*The language of set theory is based on a single fundamental relation, called membership. We say that A is a member of B (in symbols A ∈ B), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. [http://plato.stanford.edu/entries/set-theory/]

==History of set theory==

It was developed in the late 1800s, primarly by the German mathematician [[Georg Cantor]]. This initial attempt became known as "naive set theory" because mathematicians found flaws in it. It was replaced by "axiomatic set theory" in the early 1900s.

One paradox in naive set theory was announced by [[Bertrand Russell]] in 1901, and is known as [[Russell's Paradox]].

Like all sufficiently strong mathematical theories, set theory is incomplete, as shown by [[Kurt Godel]]. However, set theory is the received axiomatization of mathematics today, with subjects like analysis, algebra, topology, and geometry using set theory and its language for their own foundation.

[[Category: set theory]]

Kurt Gödel

2007-11-27T04:57:26Z

Conservaccount: Cleaned up--removed "folly", a word too harsh if one considers that before Godel, even extraordinary mathematicians like Hilbert thought complete, consistent axioms could be found.

[[Image:Goel.jpg|thumb|right|Kurt Godel at Institute for Advanced Study]]
'''Kurt Gödel''' (1906-1978) was an Austrian mathematician who did pioneering work in logic and the foundations of mathematics. His Incompleteness Theorem demonstrated some limitations of the program that would have placed all of mathematics on a complete axiomatic basis.

Gödel published his remarkable proof in 1931. He showed that in any consistent (first-order) axiomatic mathematical system there are always propositions that cannot be proved or disproved using the axioms of the system. He additionally showed that it is impossible to prove the consistency of the axioms from those same axioms. This was the famous incompleteness theorem: any axiomatic system powerful enough to describe arithmetic on natural numbers cannot be both consistent and complete. Moreover, the consistency of the axioms cannot be proven within the system.

Godel's work abruptly ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of first-order axioms for all of mathematics that is both provably consistent as well as complete. [[Bertrand Russell]] had already published, in Principia Mathematica (1910-13), a massive attempt to axiomatize mathematics in a consistent, complete way. Gödel's proof also showed that the formalist approach of [[David Hilbert]] was bound to fail to prove consistency, the key being that Hilbert needed weaker theories (Peano Arithmetic) to prove the consistency of stronger theories (set theory).

The [[Gödel's incompleteness theorems|incompleteness theorems]] also imply that there is no mechanical procedure which would determine, for all sentences of mathematics S, whether or not S was a theorem of the axioms for mathematics.

Gödel's proof was a landmark for mathematics, and demonstrated that it can never be a finished project as many mathematicians had believed.

Among Godel's other remarkable achievements: the first to discover a solution to Einstein's equation (for general relatively) in which there are closed, time-like curves. This means it is mathematically possible for there to be universes in which one can go back in time (provided one has enough fuel and time--something probably not physically possible).

[[Category:Mathematicians|Gödel, Kurt]]

Godel's Incompleteness Theorems

2007-11-27T04:44:02Z

Conservaccount: Made changes to make this more precise.

'''Gödel's Incompleteness Theorems''' are two theorems published in 1931 by [[Kurt Gödel]] that show that all sufficiently strong first-order theories can never yield answers to all mathematical questions--they are incomplete.

'''Godel's First Incompleteness Theorem''': If T is any consistent, sound, recursively enumerable first-order theory containing the axioms of [[Peano Arithmetic]], there is a sentence G_T such that T cannot prove G_T and such that T cannot prove ~G_T. This means that T is not complete: there is some statement (G_T) that cannot be proved or refuted in the theory T.

The statement G_T says "G_T is not provable in T". This statement is encoded in number theory--symbols of first-order logic are assigned numbers in such a way so that statements in logic can be viewed as statements about numbers. Further, this statement refers to itself; that this is possible is the key to Godel's argument.

Recursively enumerable (r.e.) theories are theories that can be written down in a mechanical way. The restriction to recursively enumerable theories is required, since it is easy to construct non-r.e. theories that are complete--for example, the theory that contains exactly all true statements of number theory is complete, but cannot be written down in a mechanical way.

The first incompleteness theorem is itself a theorem of Peano Arithemetic. A corollary of this fact, noted independently by [[John von Neumann]] and Godel, is the:

'''Godel's Second Incompleteness Theorem''': If T is any consistent, sound, recursively enumerable first-order theory containing the axioms of Peano Arithmetic, then T cannot prove its own consistency. In particular, Peano Arithmetic cannot prove its own consistency. This means that the consistency of the axioms of Peano Arithmetic and stronger mathematical theories must be justified in other ways, usually by appealing to the fact that the axioms are obviously true or by appealing to the fact that no one has yet derived an inconsistency.

[[category:mathematics]]