"Most logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking."7

Cantor's Paradox

  Cantor's paradox arises from the following. Let there be an infinite Universal Set of all Sets, U, which means this set contains all sets, subsets, and power sets. U = {sets, subsets, power sets}. Let the infinite power set be the Universal set's power set. Since the power set is a set, then it must be contained within the infinite Universal set, because the infinite Universal set contains all sets. From this it can be concluded that the power set is less than or equal to the Universal set. But Cantor's theorem demonstrates that the power set is always larger than the set. And herein lies the paradox. The infinite |P(C)| ≤ the infinite |C|. The cardinality of the infinite power set is less than or equal to the cardinality of the infinite Set of all Sets. But by Cantor's theorem, |P(C)| ˃|C|. The cardinality of the power set is greater than the cardinality of the set. And thus, the paradox. The power set is larger than its set, but the power set is equal to or smaller than its set.
   By the definition of Cantor's Theorem and by the rules of Tspace, it will be accepted as truth, by logical, inductive reasoning, that the cardinality of the finite set will always be smaller than the cardinality of its finite power set.    And it will also be accepted that in Ispace the infinite Set of all Sets will always be equal to or greater than its power set.
   So, in Ispace, the infinite set and the power set are the same size; and in Tspace, the finite set will always be smaller than the power set. Since both conditions of the paradox are being accepted as truth, it seems to be a paradox.
    But it is not.
  
Before this paradox can be solved, the question has to be asked if Cantor's paradox can be solved and transferred to Tspace. It can only exist in Tspace if all the premises are true, specifically, does the Set of all Sets exist in Tspace?
   In Ispace, the infinite Set of all Sets can be divided into two sets: those that cannot be translated to Tspace and those that can. Once the division has taken place, both sets are still infinite. (When infinity is divided in half, both halves are infinite.) So now, there is an infinite Set of all Sets in Tspace.

           
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