Nov 26, 2012

THE THIRD CRISIS IN THE FOUNDATION OF MATHEMATICS_Documented by Marsigit





THE CRISIS IN THE FOUNDATION OF MATHEMATICS
[Crisis #3]
Russell’s Problem

Russell noticed that there existed a difficulty in the basic notion that every property has an extension. Some properties are self-referential (they apply to themselves):

For example, the property of being abstract is self-referential: the set of all things that are abstract includes the set of all abstract things. Thus the set is a member of itself. Conversely, a non-self-referential property has an extension which does not include itself. For example, the property of being a building: the set of all things that are buildings (the set of all buildings) does not include itself.

Assume: r r. This implies r ={x | x #x}, so r r. This contradicts with the initial assumption, thus r  r. Yet the fact that r is not a member of itself shows r is indeed a member of itself (the set of things which are not members of itself). Thus the existence of r is contradictory.

As we can see, therefore, the principle of comprehension is internally inconsistent. This is very unfortunate! All of math has been reduced to a few simple notions, but now there is an inconsistency in one of those simple notions! Does this mean that all conceptual thought is ultimately internally inconsistent?

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