Operations on transfinite numbers

In conclusion, I will now consider the numbers of the second number-class (II) and the operations that can be carried out with them, but on this occasion I will limit myself to what is most obvious, and reserve the publication of detailed investigations on them for later.

I defined the operations of adding and multiplying in general terms in § 3 and showed that for infinite integers they are generally not subject to the commutative rule, but are subject to the associative rule; this also applies in particular to the numbers in the second number-class. With regard to the distributive rule, it is only generally valid in the following form:

(α + β ) y = αy + βy

where α + β, α, and β appear as multipliers, as can be seen directly from inner intuition.

Subtraction can be viewed in two ways. If α and β are any two integers, α < β, it is easy to convince oneself that the equation:

α + ξ = β

always allows one and only one solution for ξ, where, if α and β are numbers from (II), then ξ will be a number from (I) or (II). This number ξ is set equal to β - α. On the other hand, consider the following equation:

ξ + α = β

where it often turns out that it is not solvable for ξ. For example such a case occurs in the following equation:

ξ + w = w + l .

But in those cases where the equation ξ + α = β, even when ξ is solvable, it is often found that it is satisfied by an infinite number of values of ξ. However, there will always be a smallest one of these different values. We choose the notation β- α for the smallest solution of the equation:

ξ + α = β,

if it is actually solvable at all. In general β- α is different to β - α, which always exists only if α < β.

Consider also the following equation between three integers β, α, γ:

β = γα

(where γ is the multiplier). It is easy to convince oneself that the equation:

β = ξα

has no solution other than ξ = γ, and that one can denote this γ by β⁄α . On the other hand, one finds that the equation:

β = αξ

(where ξ is the multiplicand), if it is actually solvable for ξ, that often there are several solutions and perhaps infinitely many solutions, but one of the values for ξ is always the smallest, and this is designated by β⁄α .

The numbers α of the second number-class are of two kinds:

those α which have a preceding member in the sequence, which is then α-1 and I call these numbers of the first kind, and those α which do not have a preceding member in the sequence, for which therefore there is no such α-1 and I call these numbers of the second kind. For example, the numbers ω, 2ω, ωv + ω, ωω are of the second kind, whereas ω + 1, ω2 + ω + 2, ωω + 3 are of the first kind.

Correspondingly, the prime numbers of the second number-class, which I defined in general terms in § 3, split into prime numbers of the second kind and of the first kind.

Prime numbers of the second kind are, in the order of their appearance in the second number-class (II), as follows:

ω, ωω, ωω2, ωω3, …

so that among all the numbers of the form:

φ = v0 ω μ + v1 ω μ - 1 + … + vμ - 1 ω + vμ

there is only the one prime number ω of the second kind. But one can not conclude from this comparatively sparse distribution of the prime numbers of the second kind that the set of all of them has a cardinal-number that is smaller than that of the number-class (II) itself; it will be found that this set has the same cardinal-number as (II).

The prime numbers of the first kind are initially:

ω + 1, ω2 + 1, …, ωμ + 1

These are the only prime numbers of the first kind that occur among the numbers previously designated as φ. The set of all prime numbers of the first kind in (II) also has the cardinal-number of (II).

The primes of the second kind have a property which gives them a very distinctive character. If η is such a prime number (of the second kind), then ηα = η whenever α is any number smaller than η. From this it follows that if α and β are any two numbers, both of which are smaller than η, then the product αβ is always smaller than η.

If we restrict ourselves here to the numbers of the second number-class, which have the form φ, the following addition and multiplication rules can be found for these. Let

φ = v0 ω μ + v1 ω μ - 1 + … + vμ

ψ = ρ0 ω λ + ρ1 ω λ - 1 + … + ρλ

where we assume that v0 and ρ0 are different from zero.

Addition

If μ < λ, then one has: φ + ψ = ψ If μ > λ one has: φ + ψ = v0ωμ + … + vμ - λ - 1 ωλ  + 1 + (vμ - λ + ρ0) + ρ1ωλ  - 1 + ρ2ωλ - 2 + … + ρλ

For μ = λ φ + ψ = (v0 + ρ0) ωλ + ρ1ωλ   - 1 + … + ρλ

Multiplication

If vμ  is different from zero, then one has: φψ = v0ωμ + λ + v1ωμ  +  λ  -  1 + … + vμ-1ωλ   +  1 + vμ ρ0ωλ + ρ1ωλ   -  1 + … + ρλ If λ = 0, the last term on the right is vμ ρ0.

If vμ = 0, then one has: φψ = v0ωμ  +  λ + v1ωμ  +   λ  -   1 + … + vμ-1ωλ + 1 = φωλ

The decomposition of a number φ into its prime factors is as follows. Given:

φ = c0ωμ + c1ωμ1 + c2ωμ2 + … + cδωμδ

where

μ > μ1 > μ2 > · · · > μδ

and

c0, c1, … cδ

are positive finite numbers other than zero, then:

φ = c0 (ωμ - μ1 + 1) c1 (ωμ1 - μ2 + 1) c2 … cδ-1 (ωμδ - 1 - μδ  + 1) cδ ωμδ

If one thinks of c0, c1, … cδ - 1 cδ according to the rules of the first number-class, one then has the prime factorization of φ, since, as noted above, the factors ωx + 1 and ω are themselves prime numbers. This decomposition of numbers of the form φ is uniquely determined, also with regard to the order of the factors, if one abstracts from the commutability of the prime factors within the individual c and if it is determined that the last factor has the cardinal-number ω or equal to 1 and that ω may only be a factor in the last position. I will write about the generalization of this decomposition into prime factors for arbitrary numbers α of the second number-class (II) at another occasion.