We understand from our science and mathematics history, the fundamental defect in present classical set theory disclosed by the members of Russell’s Paradox Family in different periods of time is: looking for something belongs to an infinite set but is impossible to be found inside this infinite set--------no logic in our science can solve such paradox family as all the members of Russell’s Paradox Family are produced by the confusion of “potential infinite” and “actual infinite”.
Even more, some members of Russell’s Paradox Family are taken as paradoxes while some are taken as “great theorems and axioms” .
Some people really believe that Russell's theory of types or ZFC really can solve Russell’s Paradox, but following case (two members of Russell’s Paradox Family) tells that neither Russell's theory of types nor ZFC works:
Applying exactly the same mechnisim of Russell’s Paradox------looking for something belongs to an infinite set but is impossible to be found inside this infinite set, Cantor proved two inportant results of “power set theorem” and “Real number set is uncontable”.
Philadelphia, PA
Dear Ouyang,
One way to look at the defect of naive set theory revealed by the Russel paradox is that it rests in the assumption that there is a a genuine set corresponding to any condition on set membership which we can state.
So, e.g., if there is a set of all sets which are members of themselves and equally a set of all sets not members of themselves. then it makes sense to ask whether the set of all sets not members of themselves is a member of itself or not? We come to a contradiction. The various reformulations of set theory, taking the Russell paradox into consideration, then introduce restrictions on conditions of set membership. In spite of that, it is not possible to prove (Godel's proof) that the restricted, reformulations of set theory do not still involve some hidden contradiction.
The point is of some considerable importance since it casts doubt on a priori conceptions of mathematical truth.
H.G. Callaway
I think Russell's Paradox Family proves that just because a mathematical concept is logically consistent, that does not mean it can be translated successfully into the real world. The actual infinite is logically consistent, but good philosophical arguments exist against the real existence of an actual infinite. In short, logical possibility does not guarantee metaphysical possibility.
Thank you very much dear Dr. Valentin Bura, H.G. Callaway, Kirk MacGregor，
Some people really believe that Russell's theory of types or ZFC really can solve Russell’s Paradox, but following case (two members of Russell’s Paradox Family) tells that neither Russell's theory of types nor ZFC works:
Applying exactly the same mechnisim of Russell’s Paradox------looking for something belongs to an infinite set but is impossible to be found inside this infinite set, Cantor proved two inportant results of “power set theorem” and “Real number set is uncontable”.
Yous,
Geng Ouyang
It is of most significance to consider the proposition as a "test of fitness" for set theory as the suitable logical construct for partitioning of the principle elements under study. What is often ignored then is the presence of the contradiction and it's cause. One is a cause of incompleteness, and this is not primarily an accounting of the elements under analysis. It is more a test of what are the perceived partitioning conditions of the elements that are grouped and accounted into sets. I see the contradiction and proposal as a poke at perceived boundaries that usually are of no consequence when applying set theory. Usually there is no contradiction, for an object oriented model. But when you apply set theory to wave functions, the boundaries place you into a contradictory perception , where there are no objects or boundaries.
Thank you dear Dr. Mark Aaron Simpson,
It is impossible to run away from (or solve) any members of Russell’s Paradox Family in present classical set theory basing on “potential infinite--actual infinite” related classical infinite theory system.
Paradox families in set theory and those paradox families in analysis form the thousands years old black cloud of “infinite related paradoxes symptom complex” over the sky of mathematics and science.
Yous,
Geng Ouyang
The real issue is self-reference. It is in the same class of paradox as: What happens when Pinocchio says "My nose will grow!" ?
Thank you dear Dr. Colin Naturman,
Our science history has proved that the existence of paradoxes in certain area of human science is not only the touchstone for the defects in the very science area (such as the suspended paradoxes in set theory) but also a strong “metabolism signal for the basic theory”, and the “‘infinite paradox’ symptom complex” tells in its special way to scientists clearly that something must be done to solve those fundamental defects disclosed by the paradoxes and get rid of the “paradoxes generating fundamental factors” in that very science area.
Yous,
Geng Ouyang
Five of my published papers have been up loaded onto RG to answer such questions:
1，On the Quantitative Cognitions to “Infinite Things” (I)
2，On the Quantitative Cognitions to “Infinite Things” (II)
3，On the Quantitative Cognitions to “Infinite Things” (III)
4 On the Quantitative Cognitions to “Infinite Things” (IV)
5 On the Quantitative Cognitions to “Infinite Things” (V)
Russell’s Paradox shows that it is impossible for the mathematical numbers to be also objects of themselves which makes the set theort scholastic and calls to see that mathematics is enpirical science.
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