Properties of infinite integers

If there are difficulties in conceiving of immeasurably large, self-contained, integers whose magnitudes are comparable to each other, and also to finite numbers, where their relationships to each other and to finite numbers are governed by fixed rules, then these difficulties are connected with a perception that while the new numbers may certainly have the properties of the traditional ones in many respects, in several other respects they have a very strange nature. This perception may be due to the fact that different properties can occur together in a single new number, but which never appear together in a traditional number; they only appear separately in the traditional numbers.

In a passage cited in the previous section, one can find the idea that an infinite integer, if it existed, would have to be both even and odd, and since these two properties cannot appear in combination, which leads to a conclusion that such numbers cannot exist.

There is a tacit assumption that properties which are separate in the traditional numbers must also have this relationship to each other in the new numbers, and from this one concludes that the infinite numbers are impossible.

Who doesn’t see the fallacy here?

Isn’t every generalization, or extension of terms, associated with giving up specific features, and is even unthinkable without such occurring? Only recently the idea of complex numbers, so important for the development of analysis and leading to the greatest advances, was introduced in spite of the apparent obstacle in the fact that they can be called neither positive nor negative? I am daring only to take a similar step here.

Perhaps the common perception will become even more likely to follow me than it was possible to progress from the real numbers to the complex ones; for the new integers, even if they are distinguished by a deeper and more substantial determinateness than the traditional ones, nevertheless, as ordinal-numbers, they have just like the same kind of reality as them, whereas the introduction of the complex numbers was beset by difficulties until they were given a geometrical representation as points or lines in a plane.

In order to come back briefly to that consideration of the even and the odd, let us again consider the number ω, in order to show how those features which are incompatible in finite numbers occur together here without any contradiction.

In Section 3 the general definitions for addition and multiplication are given, and I have emphasized that in these operations the commutative rule is generally not valid; in this I see an essential difference between infinite and finite numbers.

Note that in a product βα the multiplicator is β and the multiplicand is α.

It then follows that for ω the following two forms result: ω = ω · 2 and ω = 1 + ω · 2. Accordingly, ω can be understood as both an even and an odd number.

But from another point of view, namely, if 2 is taken as the multiplier, it could also be said that ω is neither an even nor an odd number, because, as one can easily prove, it cannot be either in the form of 2 · α nor in the form of 2 · α + 1. The number ω has a very peculiar nature in comparison to the traditional numbers, since all these characteristics and properties are consolidated in it. The other numbers in the second number-class are even more peculiar, as I will later demonstrate.