[Set Theory] 03-ordinal set and ordinal number

1. Potential

In the previous article, I mentioned the dual nature of natural numbers "quantity" and "order". If I think twice, "quantity" is actually produced and determined by "order, to sort Finite Elements in a certain order is the process of determining their quantity. Is there such a relationship between infinite sets, "quantity" and "order? How should we define the "quantity" and "order" of an infinite set? Since they are derived from natural numbers, the answer is naturally in the expansion of natural numbers. For the Quantity \ (n \) of a finite set, it can be regarded as a one-to-one correspondence between the elements of a finite set and the elements of \ (n. This intuitive method is also applicable to infinite sets. If you can find a ruler and match the elements of an infinite set with those of the ruler, you can get the "quantity" of an infinite set ".

No matter what the ruler is, we need to verify the rationality of this method first. At least "equal" can be defined. Intuitively, we often use the size or inclusion relationship to determine whether the set is as large as possible. However, this intuition is unreliable and is not a benign definition, A good definition is required in mathematics. For two sets with one-to-one ing \ (A \) and \ (B \), it is calledBalance(Equinumerous), recorded as \ (A \ approx B \), it is easy to prove that the Equipotential can be defined as "equal amount. Sometimes some functions can be found so that the local and the whole can be mapped one by one. For example, \ (2N \) maps the natural numbers and even numbers, \ (\ cot \ pi x \) maps \ (0, 1) \) and real numbers \ (\ BBB r \), so they are also "as big.

"Greater than" and "less than" can also be defined in a similar way: if there is a single shot from the set \ (A \) to the set \ (B \), it is called \ (\)SubjectIn \ (B \), remember as \ (A \ preccurlyeq B \). If \ (A \) and \ (B \) are unequal, \ (A \) is strictly limited by \ (B \), as \ (A \ prec B \). The benign nature of the subject needs to be guaranteed, which is as follows:SB Theorem(SCHR öder-Bernstein ). In the proof, assume that there are single-shot \ (F: A \ To B \) and \ (G: B \ To A \), and \ (A \) must be constructed \), \ (B \), so that \ (f (A_1) = B _1 \), \ (G (B _2) = A_2 \). For \ (x \ In A \), note \ (x ^ * = a-g (B-f (x) \), from \ (\) gradually scale down \ (x \), and ensure \ (x ^ * \ subseteq x \), the smallest (matching \ (x) is \ (A_1 \). The Sb theorem is a powerful weapon for judging the Equipotential of a set. It can be used to easily prove that any range is equal to the real number set \ (\ BBB r. A restricted set is called a comparable one. It is a "good" definition of the set size, but the problem is that all sets can be larger? This issue needs to be put on hold for the time being and will be raised later.

\ [A \ preccurlyeq B \ wedge B \ preccurlyeq A \ rightarrow A \ approx B \]

Using the Equipotential, "finite" and "infinite" can be precisely defined: a set of equipotential and a natural number is called a finite set; otherwise, it is called an infinite set. This definition also needs to be proved to be benign, that is, to prove the difference of Natural Numbers (The need for inductive principles), so that the finite set \ (A \) is only equal to a natural number, this natural number is also called a set.Potential, As \ (N ()\). The induction principle can prove that the real subset \ (B \) of finite set \ (A \) is also finite set, and there are \ (N (a)> N (B )\). Conversely, if the set is equal to its real subset, it must be an infinite set, which can also be defined as an infinite set. Infinite sets are most special to \ (\ Omega \) and are called by \ (\ Omega \).NUMBER SETEach of its elements can be "counted. Even number sets, integer sets \ (\ bbb z \), and so on are all number sets (easy to prove), and can also prove that \ (\ Omega \ times \ Omega \) is only a handful, there can be only one number of distinct sets. This conclusion can be directly pushed to two or three dimensions... there are only a few Integer Points in a number-dimension space. All rational numbers can be expressed as scores, and scores can be considered as subsets of a two-dimensional space. Therefore, rational number set \ (\ bbb q \) is also a few.

Is there an infinitely infinite set? Conway himself gave a negative answer, which is the famous"Diagonal argument". Here we will make a slight modification. Consider the range \ ([0, 1) \). If it is a handful, we will use the binary number to represent them. Then construct a number \ (x \). Its \ (I \) decimal places are opposite to the \ (I \) decimal places of \ (a_ I, then \ (x \) cannot be counted, which is a conflict. Generally, the idempotence of any set is the same as that of the Set (\ (x \ prec \ mathscr p (x. Assume that there is a one-to-one ing \ (F: X \ To \ mathscr p (x) \), construct \ (B =\{ x \ In x | x \ In F (x) \} \), then \ (B \) has no original image (counterproof), conflict. Because \ (\ mathscr p (x) \ approx 2 ^ x \), this definition can also be written as \ (x \ prec 2 ^ x \), thus, any set has a set that is "larger" than it.

The \ (\ Omega \) potential is generally represented by \ (\ Aleph _ 0 \), and The \ (\ BBB r \) trend is generally represented by \ (C, \ ([0, 1] \) binary representation can be considered as a bit representation of a natural number subset, SO \ (\ BBB r \ approx 2 ^ {\ Omega }\) and \ (C = 2 ^ {\ Aleph _ 0 }\). \ (\ Aleph _ 0 \) is the smallest infinite potential. If the next infinite potential is recorded as \ (\ Aleph _ 1 \), will it \ (C? Generally, Is there \ (2 ^ {\ Aleph _ {\ Alpha }}=\ Aleph _ {\ Alpha + 1 }\)? This is the famousContinuity hypothesis(CH, continuum hypothesis, \ (0, 1) \) is called continuity) and generalized continuity hypothesis. It was proposed and proved by Conway, but it was not completed. In 1900, Hilbert proposed 23 important mathematical problems to be solved in the 20th century. The continuous hypothesis was listed first. Later, Godel proved its compatibility with zfc, and Cohen proved its independence by force forcing. That is to say, the continuous hypothesis cannot be proved in the zfc system, however, no suitable principle has been found for its establishment.

Hilbert (1862-1943) Cohen (1934-2007)

2. Good order

There is a common proof method. It selects a representative element from each of the given cluster sets. Whether these elements can form a set cannot be deduced by the previous principle. Cemero specifically proposed the choice principle (AC) for this method. Future Generations prove the compatibility and independence between AC and ZF. When the number of sets is infinite, such operations cannot be completed through limited-step reasoning. Therefore, this method has been resisted by constructor. However, many important conclusions in mathematics cannot bypass AC, and even the opponents themselves are unconsciously using it. Then the opposition sounds naturally weaker. AC can draw many useful conclusions from previous concepts. For example, if there is a full shot \ (F: A \ To B \), there is a single shot \ (G: B \ To A \), and then, for example, a subset of the potential of \ (\ Omega \) can be extracted from an infinite set, thus, an infinite set is always equal to its real subset (an infinite set can be defined ).

【ZFC-8]Select Principle(Atom of choice): pair set cluster \ (a_ I) _ {I \ In I} \), with cluster \ (a_ I) _ {I \ In I }\).

Now let's continue to discuss how to extend the natural number to get our ruler, which is an orderly scale. The relationship that satisfies self-inversion, anti-symmetry, and transmission is calledPartial Order(Partial order). In the partial order, symbols such as \ (\ leqslant \) and \ (<\) can be easily introduced. They form a mesh of elements. Of course, the ruler is linear, which requires each element to be comparable. This partial order is calledFull Order(Total order) or linear Order (linear order ). To expand a natural number, it must also satisfy the minimum number principle (any subset has a minimum element ).Good Sequence(Well order), a good order set contains almost all the features of a natural number. For the full order of a line, the left part of the cut is calledTruncationAnd \ (S (a) =\{ x \ In A | x <A \} \) is called \ (A \) in \ (\)Front end(Initial segment), different from truncation, there is a "Successor" in the previous section ". From the principle of least number, it is easy to know that the truncation of good order is the front segment. This is an important feature that distinguishes it from the general full order. the following principle of over-limit induction and over-limit recursion can be obtained:

　　Principle of over-limit Induction(Transfinite induction principle): If \ (A \) is a subset of good order \ (w \), and \ (\ forall A \ in W (S () \ subset A \ rightarrow A \ In A) \), then \ (A = W \).

　　Principle of over-limit Recursion(Transfinite Recursive Theorem): There is a function that satisfies the recursive definition \ (u_a = f (u \ restriction S (.

The good order of the same (homogeneous) structure can be considered as "equal". For accuracy, for the good order \ (x, \ leqslant _ x) \) and \ (Y, \ leqslant _ y) \). If dual-shot \ (F: X \ to Y \) causes \ (A \ leqslant _ XB \ leftrightarrow F () \ leqslant _ YF (B) \), which is \ (x \), \ (Y \)Similar(Isomorphic), recorded as \ (x \ simeq Y \). As you can intuitively imagine, the similar ing on the full sequence set can move left and right, but the partial order with no head or tail can only move to the right, in this way, the similarity ing between the good order and its subset always has \ (x \ leqslant f (x) \), and thus the uniqueness of the similar ing between the good order and its truncation similarity and the good order can be obtained. For the comparison of any two good orders, it is natural to choose the Head Alignment to see who is longer. The principle of over-limit recursion is easy to prove that they are either similar or one is similar to the other.

The framework of the ruler (good sequence) already exists. A natural question is, can any element in the Set correspond to the scale of the ruler? In other words, can any set be sorted? Conway raised this question (Ordering principle), But it fails to be solved. cemer' proposed the right of choice and provided a proof. The process of proof is completely based on the selection principle and the over-limit induction principle is repeatedly used. However, the induction principle only applies to the good sequence, and some common skills and processing are required. With the ordering principle, you can answer questions about whether any set can be more appropriate, because any set can be sorted in good order, and the good order set can be more.Any set can be largerThat is, \ (A \ approx B \), \ (A \ prec B \), and \ (B \ prec A \) have exactly one of them (three differences ).

The principle of good order can draw another basic conclusion, that isZORN Theorem: If any chain of the sequence has an upper bound, it has a maximum value. In the proof, \ (M \) can be sequenced to construct the desired polar chain to obtain the maximum value. When Zorn's theorem is not selected, we can prove the choice of them. It is equivalent to proving that there is a function with the same definite domain in a relation, it can be solved by composing all functions in an inclusive order. At this point, the principle of selecting principle, good sequence, and Zorn can be deduced from each other. They can be considered as equivalent.

3. ordinal number and base number

The framework and feasibility of the scale have been solved. The next step is to scale the scale. According to the definition of natural numbers, we naturally think that the scales should be \ (1, 2, 3, \ cdots, \ Omega, \ Omega ^ +, \ cdots \), however, to sum them up in one sentence, there must be a strict definition: Meet the condition \ (\ forall x \ In \ alpha (S (x) = x )\) \ Alpha \ is calledOrdinal number. It is easy to prove that all natural numbers are ordinal numbers. If \ (\ Alpha \) is ordinal numbers, \ (\ Alpha ^ + \) is also ordinal numbers, so \ (\ Omega, \ Omega ^ +, \ cdots \) are ordinal numbers. It is easy to prove that any ordinal number contains \ (0 \), and its elements are also ordinal numbers. The over-limit induction principle can also prove that similar ordinal numbers must be equal, thus, the "uniqueness" of the ordinal number is guaranteed. Ordinal numbers are good ordinal sets, and they also satisfy the three difference, so they can be exclusive, and the "ordinal number" of each ordinal number is its successor.

So there is another question: Do all sets (or good-order sets) have their corresponding ordinal numbers? Before answering this question, we need to introduce another principle, which is the replacement principle below. The replacement principle recognizes the existence of a class of sets without restriction sets, because these sets are subject to known sets and their existence is reasonable. However, the replacement principle is not independent of other ones. It can completely replace the subset principle and prove the even set principle. The zfc justice system also has the last slightly redundant Regular Expression justice, which avoids the generation of a large set, but the other nine ones cannot actually construct such a set.

【ZFC-9]Replacement Principle(Atom schema of replacement): if any element of a given set has a unique set corresponding to it, these sets can form a set.

\ [\ Forall x \ In A, \ forall Y_1 \ forall Y_2 (\ varphi (x, y_1) \ wedge \ varphi (x, y_2) \ rightarrow Y_1 = Y_2) \ rightarrow \ exists B =\{ y | \ exists X (\ varphi (x, y) \}\]

【ZFC-10]Regularization principle(Atom of regularity): For any non-empty set \ (A \), an element \ (x \) makes \ (x \ cap A = \ varnothing \).

\ [\ Forall A \ exists x (x \ In A \ wedge x \ cap A = \ varnothing) \]

With the replacement principle, we can construct the ordinal number of a good ordinal set, and evaluate the anterior part of the ordered number in a good ordinal set. The sum (replacement principle) of these ordinal numbers is the ordinal number to be searched, this is what we wantCounting Principle: Any good sequence set \ (x \) is similar to the unique sequence number \ (\ Alpha \) and is recorded as \ (\ alpha = {\ RM ord} (x )\). Ordinal numbers can be used as extensions of natural numbers.Exceeding limitYou can use the following method to define the addition and multiplication of ordinal numbers. They satisfy most calculation laws, but multiplication does not satisfy the Exchange Law and the right allocation law.

(1) \ (\ Alpha + \ Beta = {\ RM ord} (\ hat \ Alpha \ cup \ hat \ beta )\), where \ (\ hat \ Alpha =\{ (x, 0) | x \ In \ Alpha \} \), \ (\ hat \ beta =\{ (1, Y) | Y \ In \ beta \}\);

(2) \ (\ Alpha \ cdot \ beta ={\ RM ord} (\ Alpha \ times \ beta )\).

The ordinal number can only expand the nature of the Natural Number "ordinal", but it cannot reflect the nature of the "quantity", because different ordinal numbers can be equipotential. The ordinal number of these equipotential has the smallest value. It is called a measure of "quantity ".Base(Cardinal number), recorded as \ ({\ RM card} (x )\). The base is the quantitative description of the potential. The basis of a non-natural number isOverlimit BaseObviously \ (\ Aleph _ 0 \) is the minimum over-limit base. According to the ordinal principle, we can see that each ordinal set has a corresponding base. This makes the SB theorem very obvious, but we need to know that the ordinal theorem is based on the selection principle, and the SB theorem itself does not rely on the selection principle. The addition, multiplication, and power of the Base are easy to define, and the general arithmetic law is easy to prove. I will not repeat it here. It is worth mentioning the following arithmetic law, they can be proved by Zorn guidance and reverse evidence.

(1)Addition absorption law: \ (B \) is the over-limit base and \ (A \ leqslant B \), then \ (a + B = B \);

(2)Multiplication Law: \ (B \) is the over-limit base and \ (1 \ leqslant A \ leqslant B \), then \ (A \ cdot B = B \);

(3)Power-down Law: \ (B \) is the over-limit base and \ (2 \ leqslant A \ leqslant B \), then \ (a ^ B = 2 ^ B \).

It seems that we are already on the road, but this is just the beginning. When we go along \ (\ Omega, 2 \ Omega, \ cdots, {\ Omega ^ 2 }, {\ Omega ^ 3}, \ cdots, {\ Omega ^ \ Omega}, {\ Omega ^ \ Omega}, \ cdots \) keep thinking and you will be lost ......

 

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[Set Theory] 03-ordinal set and ordinal number