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Paradoxes mainly include logic paradox, probability paradox, geometry paradox, statistics paradox and time paradox.

Russell's paradox shocked the whole field of mathematics with its simplicity and clarity, which led to the third mathematical crisis. However, Russell paradox is not the first paradox. Needless to say, not long before Russell, Cantor and Blary had discovered the contradiction in set theory in their forties. After the publication of Russell's paradox, a series of logical paradoxes appeared. These paradoxes remind me of the ancient liar paradox. That is, "I'm lying" and "this sentence is a lie". The combination of these paradoxes has caused great problems, prompting everyone to care about how to solve these paradoxes.

The first published paradox is Blary's Forty Paradox, which means that ordinal numbers form an ordered set according to their natural order. By definition, this well-ordered set also has an ordinal ω, which should belong to this well-ordered set by definition. However, according to the definition of ordinal number, the ordinal number of any segment in ordinal number sequence is greater than any ordinal number in that segment, so ω should be greater than any ordinal number, so it does not belong to ω. This was put forward by Blary Forti in an article read at the Balomo Mathematics Conference on March 28th, 1997. This is the first published modern paradox, which aroused the interest of the mathematical community and led to heated discussions for many years. There are dozens of articles discussing paradox, which greatly promotes the re-examination of the basis of set theory.

Blary Foday himself thinks that this contradiction proves that the natural order of this ordinal number is only a partial order, which contradicts the result ordinal number set proved by Cantor a few months ago. Later, Blary Foday didn't do this work either.

In his Principles of Mathematics, Russell thinks that although the ordinal set is fully ordered, it is not well ordered, but this statement is unreliable because the first paragraph of any given ordinal number is well ordered. French logician Jourdain found a way out. He distinguished between compatible sets and incompatible sets. This distinction has actually been used privately by Cantor for many years. Soon after, Russell questioned the existence of ordinal set in an article in 1905, and Zemelo also had the same idea, and later many people in this field held the same idea.

Paradox of classical mathematics

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There are many famous paradoxes in ancient and modern China and abroad, which have impacted the foundation of logic and mathematics, stimulated people's knowledge and precise thinking, and attracted the attention of many thinkers and enthusiasts throughout the ages. Solving paradoxes requires creative thinking, and the solution of paradoxes can often bring people new ideas.

In this paper, paradox is roughly divided into six types, which are divided into three parts: upper, middle and lower. This is the first part: the paradox caused by the concept of self-reference and the paradox brought by the introduction of infinity.

(A) the paradox caused by self-reference

In the following example, there is a problem of concept self-reference or autocorrelation: if we start from a positive proposition, we will get its negative proposition; If we start with a negative proposition, we will get its positive proposition.

1- 1 liar paradox

In the 6th century BC, the philosopher epimenides, a Crete, said, "All Cretes are lying, and one of the poets also said so." This is the origin of this famous paradox.

It is mentioned in the Bible: "A local prophet of Park Yung-soo said,' The Celts often lie, but they are evil beasts, greedy and lazy'" (Titus 1). It can be seen that this paradox is famous, but Paul is not interested in its logical solution.

People will ask: Is Epiminides lying? The simplest form of this paradox is:

1-2 "I'm lying"

If he is lying, then "I am lying" is a lie, so he is telling the truth; But if this is true, he is lying again. Contradictions are inevitable. A copy of it:

1-3 "This sentence is incorrect"

A standard form of this paradox is: if event A occurs, non-A is deduced; if non-A occurs, non-A is deduced, which is a self-contradictory infinite logic cycle. One-sided body in topology is the expression of image.

The philosopher Russell once seriously thought about this paradox and tried to find a solution. He said in the seventh chapter "Mathematical Principles" of "The Development of My Philosophy": "Since Aristotle, logicians of any school seem to be able to deduce some contradictions from their recognized premises. This shows that there is a problem, but it cannot point out the way to correct it. 1903 In the spring, a contradictory discovery interrupted the logical honeymoon I was enjoying. "

He said: The liar paradox simply sums up the contradiction he found: "The liar said,' Everything I said is false'. In fact, this is what he said, but this sentence refers to all he said. Only by including this sentence in that crowd will there be a paradox. " (same as above)

Russell tried to solve the problem through hierarchical propositions: "The first-level propositions can be said to be those that do not involve the whole proposition; Second-level propositions are those that involve the whole first-level proposition; The rest is like this, even infinite. " But this method has not achieved results. "During the whole period of 1903 and 1904, I almost devoted myself to this matter, but I was completely unsuccessful." (same as above)

Mathematical principles try to deduce the whole pure mathematics on the premise of pure logic, explain concepts in logical terms, and avoid the ambiguity of natural language. But in the preface of this book, he called it "publishing a book that contains so many unresolved disputes." It can be seen that it is not easy to completely solve this paradox from the logic of mathematical basis.

Then he pointed out that in all logical paradoxes, there is a kind of "reflexive self-reference", that is, "it contains something about that whole, and this kind of thing is a part of the whole." This view is easy to understand. If this paradox is said by someone Park Jung-soo thinks, it will be automatically eliminated. But in set theory, the problem is not so simple.

1-4 barber paradox

In Saville village, the barber put up a sign: "I only cut the hair of those people in the village who don't cut their own hair." Someone asked him, "Do you cut your hair?" The barber was speechless at once.

This is a paradox: a barber who doesn't cut his hair belongs to the kind of person on the signboard. As promised, he should give himself a haircut. On the other hand, if the barber cuts his own hair, according to the brand, he only cuts the hair of people in the village who don't cut their own hair, and he can't cut it himself.

So no matter how the barber answers, he can't rule out the internal contradictions. This paradox was put forward by Russell in 1902, so it is also called "Russell paradox". This is a popular and story-telling expression of the paradox of set theory. Obviously, there is also an unavoidable problem of "self-reference".

The 1-5 Paradox of Set Theory

"R is the set of all sets that do not contain themselves."

People will also ask: "Does R include R itself?" If not, according to the definition of R, R should belong to R. If R contains itself, R does not belong to R..

Kurt G?del (Czech, 1906- 1978) put forward an "incomplete theorem" in 193 1 year, which broke the mathematician's "all mathematical systems are This theorem points out that any postulate system is incomplete, and there must be propositions that can neither be affirmed nor denied. For example, the negation of the "axiom of parallel lines" in Euclidean geometry has produced several non-Euclidean geometries; Russell's paradox also shows that the axiomatic system of set theory is incomplete.

1-6 bibliography paradox

A library compiled a dictionary of titles, which listed all the books in the library without their own titles. So will it list its own title?

This paradox is basically consistent with Barber's paradox.

1-7 Socrates paradox

Socrates (470-399 BC), an Athenian, is known as "Confucius in the West" and a great philosopher in ancient Greece. He was once opposed to the famous sophists Prut Golas and Gogis. He established a "definition" to deal with the confusing rhetoric of sophists, thus finding out hundreds of miscellaneous theories. But his moral concept was not accepted by the Greeks, and he was regarded as the representative of sophistry when he was seventy years old. Twelve years after expelling Prut Golas and burning books, Socrates was also sentenced to death, but his theory was inherited by Plato and Aristotle.

Socrates famously said, "I only know one thing, and that is nothing."

This is a paradox, and we can't infer from this sentence whether Socrates doesn't know the matter itself. There are similar examples in ancient China:

1-7 "Words are full of contradictions"

This is what Zhuangzi said in Zhuangzi's Theory of Everything. Later Mohism retorted: If "everything is against the truth", isn't Zhuangzi's statement against the truth? We often say:

1-7 "There is no absolute truth in the world"

We don't know whether this sentence itself is "absolute truth".

1-8 "absurd truth"

Some dictionaries define paradox as "absurd truth", and this contradiction modification is also a kind of "compressed paradox". Paradox comes from the Greek word "para+dokein", which means "think more".

All these examples show that logically, they can't get rid of the vicious circle brought by the concept of self-reference. Is there a further solution? We will continue our discussion in the last part of the next section.