Conservapedia - User contributions [en]

Bourbaki

2009-10-12T22:49:25Z

PhilMath: Created page with ''''Nicolas Bourbaki''' was a fictional mathematician invented by a group of French academics in the 1930s. Over a period of about fifty years, the group published a series of de...'

'''Nicolas Bourbaki''' was a fictional mathematician invented by a group of French academics in the 1930s. Over a period of about fifty years, the group published a series of densely worded books supposedly written by Bourbaki. Their aim in doing so was to make mathematics as abstract and divorced from physical intuition as possible. Although little read today, the writings of Bourbaki are still strongly influential in university classes and parts of leftist Europe. More significantly, perhaps, the teachings of Bourbaki have been used to justify such educational initiatives as the [[new math]] program championed by liberals in American schools in the 1960s and 1970s. Many people have also criticised the Bourbaki group for destroying the healthy partnership between math and physics that had inspired such English-speaking mathematicians as Isaac Newton, and for moving college mathematics teaching away from the traditional geometry of Euclid and Pythagoras.

Set theory

2009-10-07T21:47:49Z

PhilMath:

'''Set theory''' is 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. <ref>http://plato.stanford.edu/entries/set-theory/</ref>

==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. The most commonly used such axiomatization is [[Zermelo-Fraenkel]] set theory.

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.

==References==

<References/>

[[Category: set theory]]

Zermelo-Fraenkel

2009-10-07T13:46:42Z

PhilMath: Corrected the claim that AD is a weaker axiom than AC; they are incompatible

In [[mathematics]], '''Zermelo-Fraenkel set theory''' ('''ZFC''') is the standard formal [[axiomatization]] of axiomatic [[set theory]]. It is commonly considered the foundation of [[modern mathematics]].<ref>[http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html Mathworld]</ref> It was formulated by two [[logician]]s, [[Zermelo]] and [[Fraenkel]].

The nine axioms in Zermelo-Fraenkel set theory are:

* [[Axiom of Empty Set]]
* [[Axiom of Extensionality]]
* [[Axiom of Unordered Pairing]]
* [[Axiom of Subset]]
* [[Axiom of Superset]]
* [[Axiom of Power]]
* [[Axiom of Infinity]]
* [[Axiom of Replacing]]
* [[Axiom of Foundation]]
* [[Axiom of Choice]]

Many authors take "Zermelo-Fraenkel" (ZF) to refer to these axioms without the axiom of choice, indicating the inclusion of choice by writing "ZFC".
The [[Axiom of Choice]] is independent of (i.e. neither provable nor disprovable from) the Zermelo-Fraenkel axioms. Sometimes alternate axioms are added to ZF, for example the [[Axiom of Determinacy]], which is incompatible with the [[Axiom of Choice]].

====References====

{{reflist}}

[[Category:Logic]][[Category:Advanced Mathematics]]