Ordinal arithmetic
The transfinite arithmetic is the arithmetic of ordinal numbers. The arithmetic operations between ordinals by transfinite recursion can be introduced as a continuous extension of the finite arithmetic operations or so that their corresponding constraint through appropriate amount compositions on the finite ordinal numbers of the usual arithmetic on the natural numbers. The addition and multiplication of ordinals has been introduced by Cantor (1897 ) by composition, exponentiation, however, functional means of border crossing. The first detailed and systematic study of transfinite arithmetic is by Ernst Jacobsthal ( "On the construction of transfinite arithmetic ", Math Ann. , 1909). It shows that both methods - the functional and compositional method - lead to the same mathematical operations.
Addition
If one of two ordinal numbers is the empty set, then their sum is equal to the ordinal number other. To define the sum of two non-empty ordinals and, one goes as follows: One indicates the elements of order so that the renamed and quantity are disjoint, and " writes the left of", ie one associated with and defined the order so that applies within the previous order and each and every element of is less than every element of. , In this way, the new set is well-ordered and is order to a uniquely determined ordinal number is denoted by. This addition is associative and generalizes the addition of natural numbers.
The first transfinite ordinal number is the ordered set of all natural numbers, it is denoted by. Let us illustrate us the sum: We write the second copy as, then we have
This amount is not, because that is the only number without predecessors, and has two elements without predecessor (and). The amount looks like this:
So we have. is contrast
Unequal, because is the largest element of, but has no greatest. So the addition is not commutative. It is the sum of two ordinals and define functional as follows, where both definitions are equivalent in ZF:
- If, then is,
- If is isolated and is the predecessor of, then be,
- If a limit ordinal, then was.
The addition is monotonic. That is, and. If, then there exists a uniquely determined ordinal so. It is denoted by: . Let and be two ordinals. If the equation has a solution, then it has infinitely many solutions in the case and in the case of exactly one. Has any solutions, then is meant by the smallest among them. In this sense applies to any isolated number. Each transfinite ordinal number can be represented in exactly one way as a sum of a limit ordinal and a finite ordinal. Means of an ordinal remainder, if there is an ordinal number so that. Each ordinal has a finite number of residues.
Multiplication
To multiply two ordinals and, to write back and replaces each element of by another copy of. The result is a well-ordered set which is isomorphic to exactly one ordinal, which is denoted by. Also this operation is associative and generalizes the multiplication of integers.
The ordinal ω · 2 looks like this:
One can see that ω · 2 = ω ω is. In contrast, sees 2 · ω as follows:
And after renaming, we see that 2 · ω = ω is. So the multiplication of ordinals is not commutative.
One of the distributive law applies to ordinals. This can be read directly from the definitions. However, the other distributive law is not general because, for example, is (1 1) ω = 2 · ω = ω, but 1 · ω · ω = ω 1 ω .
The neutral element of addition is 0, the neutral element of the multiplication is the first ordinal None other than 0 has a negative ( an additive inverse element ), thus forming the ordinals with the addition of no group, and certainly no ring. The functional definition of multiplication is:
- If, then is,
- Is for each ordinal
- If a limit ordinal, then was.
Apply the monotony laws:
For any two ordinals and is valid. If, then is called a divisor of links and right divider. It is also said that right-sided and left -sided in multiples of multiples of being. The Limes numbers are the left- multiples of. Each ordinal has finitely many right divider and only finitely many links divider when it is not a limit ordinal. Amounts of positive ordinals have a greatest common right divisor, a greatest common divisor links and a least common multiple left -sided. A rechsseitges common multiple is not always available. Counterexample is. For two ordinals and exist uniquely determined ordinals and so.
General sum
Be a network of ordinals with the ordinal as an index set. were the order relations of copies. The overall sum of all defined as follows:
Multiplication is therefore a special case of the general sum:
For each Ordinalzahlnetz exists exactly one function: with the following three properties:
- For each ordinal
- For each limit ordinal
The value corresponds exactly to the general sum of.
General product
Was for a Ordinalzahlnetz
In which
The name for the canonical projection. We define the relation:
The general product of all elements of is
Defined. The general product thus consists of tuples of length that are ordered antilexikografisch and only finitely many positive component. For each Ordinalzahlnetz exists exactly one function: with the following four properties:
- For each ordinal
- For limit ordinal, if
- For limit ordinal, if
The value corresponds exactly to the general product of
The result
Is an example of a antilexikografische order and represents, according to the definition of a quantity to ordnungsisomorphe dar. It is therefore necessary and! , Which is not surprising, because yes! .
Multiply
The powers are special cases of general products:
Example
One can construct a lot to ordnungisomorphe by considering ( according to the product definition) sequences of natural numbers with a finite number of positive elements:
And this antilexikografisch assigns:
Properties
For ordinals applies:
- .
For two ordinals and apply. It follows from. For two ordinals and exist uniquely determined ordinals: - called logarithm of the base, and positive, so that ( log record). The power rule from the finite arithmetic is not transferable to infinity:
Cantor normal form
For two ordinal and finitely many clearly identified and so that
This representation is known as the Cantor polynomial (or -adic normal form ). They called for Cantor normal representation (or Cantor normal form ). You can use the Cantor normal representation recursive and so represent the ordinals just like in their normal form. When this process ends in finite ordinals after finitely many steps, we obtain an elementary expression for, the natural numbers and symbols for arithmetic operations is made. However, this is not possible for every ordinal. More generally: by finitely many characters can only be countably many ordinals represent - that only a " vanishingly small " part of the whole class. There are ordinals, for Cantor in their normal representation is the same. In this case, the normal image that is does not lead to simplification. The smallest such number designated by you. Using the Cantor Normal display the Hessenberg 's natural operations are defined.