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[watermark]CANTOR’S TWO MISTAKES IN SET THEORY
OUYANG Geng
(Zhangzhou Teachers’ Collge, Zhangzhou, 363000, P.R.China)
Abstract: Through a new analyses on the essential relationship between Bertrand Russell’s Paradox and Geory Ferdinand Ludwig Phillip Cantor’s two important proofs in present set theory, a serious and mysterious error in Cantor’s proof on the uncountability of real number set and the proof on the Cantor’s Theorem of < is discovered.
Key words: proof; logic; real number; diagonal method; set; Russell’s Paradox; Cantor’s mistakes
1.Introduction
In [1~4] we discussed the defects in one of Cantor’s important proofs given in the end of 19th century. Our study disclosed that it is exactly these very same defects in the logic, the mathematical proof theory and the basic theory of set that make Russell construct his famous Russell’s Paradox. In fact, Cantor had unconsciously constructed “Russell’s Paradox” 10 years earlier than Russell, but he didn’t realize that and applied the very same “logic contradiction” in Russell’s Paradox to operate his two important proofs. There is exactly the same nature in Russell’s Paradox and Cantor’s “Real Number Set Uncountability Proof” and “Cantor’s Theorem of < Proof”----the idea of diagonal method. But because of the different purpose, different manifestation, Cantor’s proofs were taken as theorems, praised, protected and were developed into the “basic theories” of set theory while Russell’s proof was taken as a frighten and sorrowful paradox leading to the miserable third mathematical crisis.
2.The diagonal method and Cantor’s two important proofs
2.1. The proof of the uncountability of real number set
Cantor applied the intervals overlapping—disproof method (1874) and diagonal—disproof method (1890) to prove the uncountability of real number set. The two methods have operated with exactly the same idea (the details are discussed in [1~4]). Generally there must be following three steps in this kind of proof:
(1). Assuming that the real number set is countable and we construct a Sequence (1) containing all the real numbers (Actually, there is a very strong condition here: the constructed Sequence (1) in this proof must be a subset of real number set and must not contain those numbers constructed by intervals overlapping method or diagonal method.).
(2). By applying the intervals overlapping method or diagonal method, we show those numbers impossibly existing in the Sequence (1) (In fact, they are purposely excluded from the Sequence (1) in the first step).
(3). By saying that “We find out that Sequence (1) never contain all the real numbers thus the assumption in the first step is wrong” and we disprove that the infinite real number set is uncountable.
In this proof, what form of Sequence (1) should be and why Cantor chose that subset of real number as Sequence (1) is a puzzle. No one in this world till now is able to give a sequence really including all real numbers.
So, Cantor unconsciously hid those real numbers which could be produced by overlapping method or diagonal method, just gave a sequence (the Sequence (1) ) containing only part of the real numbers but assuming that this Sequence (1) contained all the real numbers of the set. Than by applying the overlapping method or diagonal method, he presented the hidden numbers and said that he “found out” that the Sequence (1) never contains all the real numbers in the set. With such a “hiding and then showing” operation Cantor reached his conclusion of “Infinite real number set is uncountable”----in our life, a magician performs his magic on the stage with exactly the same idea and technique. The Sequence (1) forming method, the intervals overlapping method or diagonal method are nothing different, each being one of the subset forming methods. So, in such a proof, Cantor only proved nothing but a fact that Sequence (1) is really only a subset of infinite real number set. This is totally nothing to do with the uncountability of the real number set!
2.2.The proof of Cantor’s Theorem of <
Let’s see the proof given in 1891 by Cantor:
We first prove that . For any x S, let f (x) = {x}. We know that if x1 x2, then
{x1} {x2},that is f (x1) f (x2). So f : S P(S) is an injective map. Thus we get that
(1)
Next we prove that
(2)
If = , then we get an bijective map : S P(S). For any x S, we know that (x) is a subset of S, that is, (x) S. Now does this x belong to (x)? Certainly we know that x (x) or x (x). Now we construct a set S0, which is made up of that x (x), that is
S0={x| x S and x (x) } (3)
Obviously, we know that S0 S, that is, S0 P(S). Thus, as is bijective, there exists an element x0 which belongs to S, such that (x0)= S0. According to the logical law of excluded middle, we get that x0 S0 or x0 S0, and only one case can be true.
If x0 S0, by (3) we know that x0 (x0). As S0= (x0), so x0 S0.
If x0 S0, from S0= (x0) we get x0 (x0). Thus by (3) we have x0 S0.
From above, we know that in both cases we all get contradictions. From these contradictions we know that there doesn’t exist a bijective map between S and P(S). Hence inequality (2) holds.
As both (2) and (1) hold, we have proven that
< (4)
In this proof, Cantor introduced a self-contradictory “set S0” defined by Formula (3) to produce a “set S0 incident”(it is the essential part and a secret weapon in this proof), but the idea and the operation in the proof are wrong. In fact, it is this very same “set S0 incident” that make Russell construct the Russell’s Paradox. Cantor didn’t realize that his proof was based on a paradox ( Russell’s Paradox was constructed 10 years later than Cantor’s proof of < ), so he claimed that he had found a logic contradiction in the “set S0 incident” and with this “set S0 incident” he proved his < theorem smoothly and beautifully. If such a “paradox applying” idea really works in mathematical proof, we can copy Cantor’s operation here to prove many ridiculous conclusions in mathematics.
3.Diagonal method and Russell’s Paradox
When talking about the third mathematical crisis, people often mention Burali-Forti Paradox,
Cantor’s Paradox and Russell’s Paradox. Among these three paradoxes, the remarkable Russell’s Paradox was from the careful study of the other two. Generally, people consider Russell’s Paradox is a deepening of the other two and it contains all the essence of the relating paradoxes leading to the third mathematical crisis.
Let’s see Russell’s Paradox:
Let T be a set of elements which don’t belong to themselves, that is, T: ={x | x x}, now does T belong to T?
If T T, then T is an element of T. As elements of T have this property: they don’t belong to themselves, we get that T T. This contradicts to the hypothesis of T T.
If T T, then T is not an element of T. From the definition of T we know T consists of elements which don’t belong to themselves, so we get T T. This also contradicts to the hypothesis of T T.
From above we can see that in either case T T or T T, we both get contradictions. Simultaneously, according to the logical law of excluded middle we know that T T is true or T T is true. Thus we get a paradox, as is called Russell’s Paradox.
Russell’s Paradox was generally acknowledged as a remarkable proof directly applying the diagonal method to construct a “set T” (just like the ones we met in Cantor’s above two proofs) to expose the secret weapon—logic contradiction in paradox. (This very same logic contradiction was used in Cantor’s above two important proofs.). The paradox is so simple but so powerful. That is why Russell’s Paradox not only made people sad but also won praises.
4.Conclusion
Through the comparison of Russell’s Paradox and Cantor’s above two proofs, we can see clearly that their ideas and operations are nothing different: just constructing a self-contradictory set of x f(x) than launching an attack.
In Russell’s proof, he applied the diagonal method to expose the logic contradiction to tell the theoretical defects in science to awake people to work out solutions to improve theories. This is a kind of healthy and positive work ( so we have Z-F System and Type Theory, etc.). But Cantor applied the same diagonal method to make paradoxes then made use of the logic contradiction of the paradoxes as well as some other defects in science, unpurposely drew some wrong conclusions. Cantor’s work in his above proofs is a kind of unhealthy and negative one, it brought mysterious and chain-mistakes into mathematics, because the mistaken conclusions of his above two proofs are taken as basic theory to produce many wrong contents in science.
Russell’s Paradox and Cantor’s above two important proofs survive with exactly the same background, same foundation and same existing reason. If there is no room for Russell’s Paradox in science then there is no reason for the survival of Cantor’s above two proofs!
It is the fatal defects in the classical infinite theory and the fatal defects in present classical limit theory that cause such mysterious errors in set theory[5~12].
REFERENCES AND PUBLICATIONS
[1]. OUYANG Geng, An Unsolved Crisis in the Foundation of Modern Mathematics. Journal of
Kashgar Teachers’ College. 2005, 26(3): 84-87
[2]. OUYANG Geng, A Mysterious Error in Set Theory. Journal of Northwest University (Natural Science). 2000,30(4): 8-11.(special issue)
[3]. OUYANG Geng, A Fresh Understanding on the Third Mathematical Crisis. Journal of Kashgar Teachers’ College (Natural Science). 2000, 20(2): 69-73
[4]. OUYANG Geng, The Non-defined Actual Infinity Concept is the Very Source of The Third Mathematical Crises. Journal of Yichun University (Natural Science). 2005,27 (4): 26-29
[5]. OUYANG Geng, On Unsolved Problems in Mathematical Analysis. Ji’an Teachers’ College (Natural Science). 1995, 16 (5): 29-34
[6]. OUYANG Geng, On Three New Number Forms in Mathematics. Journal of Kashgar Teachers’ College .2003,24(3): 31-37
[7]. OUYANG Geng, A New Way Out of Pending Problems in Mathematics. Journal of Heilongjiang Hydraulic Engineering College. 1998, 25(3): 105-112
[8]. OUYANG Geng, On Actual Infinity And Potential Infinity in Mathematics. Journal of Ji’an Teachers’ College (Natural Science). 1998, 19 (6): 26-28
[9]. OUYANG Geng, Another Modern Version of Zeno’s Paradox-----the Paradox of Harmonious Series. Journal of Kashgar Teachers’ College. 2003,24 (6): 25-28
[10]. OUYANG Geng, A Gold Queue: Analogy---Equality---Basitheoryology. Journal of Yichun University (Natural Science). 2004,26 (4): 23-25
[11]. OUYANG Geng, The End of the Classical Infinite Theory in Human Sience. Journal of Hebei Polytechnic University (Social Science Edition). 2006,6 (2): 5-7
[12]. OUYANG Geng, Two Fatal Defects in Present Limit Theory. Journal of Kashgar Teachers’ College. 2006,27 (3): 23-25
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发表于 2006-6-11 23:39
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