Set Theory and Logic


A Thing Exists If It Is A Grouping Defining What Is Contained Within: Application to The Russell Paradox and Godel's Incompleteness Theorem

Authors: Roger Granet

    The Russell Paradox (1) considers the set, R, of all sets that are not members of themselves.   On its surface, it seems like R belongs to itself only if it doesn't belong to itself.  This is where the paradox come from.  Here, a solution is proposed that is similar to Russell's method based on his theory of types (1,2) but is instead based on the definition of why things exist as described in previous work (3).  In that work, it was proposed that a thing exists if it is a grouping defining what is contained within. A corollary is that a thing, such as a set, does not exist until what is contained within is defined. A second corollary is that after a grouping defining what is contained within is present, and the thing exists, if one then alters the definition of what is contained within, the first existent entity is destroyed and a different existent entity is created. Based on this, set R of the Russell Paradox does not even exist until after the list of the elements it contains (e.g. the list of all sets that aren't members of themselves) is defined.  Once this list of elements is completely defined, R then springs into existence. Therefore, because it doesn't exist until after its list of elements is defined, R obviously can't be in this list of elements and, thus, cannot be a member of itself; so, the paradox is resolved.  Additionally, one can't then put R back into its list of elements after the fact because if this were done, it would be a different list of elements, and it would no longer be the original set R, but some new set. This same type of reasoning is then applied to the Godel Incompleteness Theorem, which roughly states that there will always be some statements within a formal system of arithmetic (system P) that are true but that can't be proven to be true. Briefly, this reasoning suggests that arguments such as the Godel sentence and diagonalization arguments confuse references to future, not yet existent statements with a current and existent statement saying that the future statements are unprovable. Current and existent statements are different existent entities than future, not yet existent statements and should not be conflated. In conclusion, a new resolution of the Russell Paradox and some issues with the Godel Incompleteness Theorem are described.

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[v1] 2017-01-07 23:13:35

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