This video is number seventeen in my series of videos covering the contents of the translations into English of Georg Cantor's famous 1895 and 1897 articles on the theory of aggregates and tranfinite numbers.
This video covers the material from section 17 of the 1897 article titled The Numbers of the Form 𝜔^𝜇 𝜈_0+𝜔^(𝜇−1) 𝜈_1+…+𝜈_𝜇. This section, therefore, looks at the properties of those numbers of the second number-class which can be written as a finite sum of (finite) multiples of finite powers of omega. As will be seen later in section 19, this type of number by no means accounts for all of the numbers of the second number-class.
This video goes into considerably more detail than some of the previous videos. I have tried to fill in as many of the missing details from this section as I thought was necessary. Sections 16 to 20, as presented by Cantor, are more outline presentations - many of the details are omitted and those details which are omitted are not at all obvious in some cases.
For various reasons, the final three videos of this series will be delayed for as long as it takes for me to get some things sorted out. I will be making the videos whenever I get round to making them, though, I'm confident that no-one will notice anyway.
Welcome to this video, video number 17 in my series of videos on George Cantor's 1895 and
1897 articles on the theory of aggregates and transfinite numbers. This section really starts
to focus in on the numbers of the second number class, rather than parts or segments of it. We
will see some of the basic arithmetical properties of a particular form of number from this class.
The numbers dealt with in this section do not account for all of the numbers of the second
number class. The remaining chapters of this article, however, gradually build a more complete
picture of the numbers of this number class. I'll be going into quite considerable detail
with this section and the sections which follow, and which will be covered in subsequent videos.
Point 3 of this section follows from point 8 of section 14. The details of some of the results
of page 175 will be shown on the screen. For the final case that's considered, we have as shown on
the screen. The three cases just considered show that only the case where phi and psi are identical
expressions will lead to the equality of phi and psi. This implies that the representation of phi
in the form given in point 1 of this section is unique. For A of page 175, we see that since mu
is less than lambda, then all terms of phi in the sum phi plus psi are followed on the right by at
least one term of psi, which is of greater degree than them. And so by 2, they may be omitted, giving
phi plus psi equals psi. For B and C, we have the following. Point 5 is obtained by repeated
application of B of page 175. The details of some of the results of page 176 will be shown on the
power of lambda times kappa plus 1 is equal to omega to the power of lambda plus 1, all multiplied by
kappa, follows from 5 of this section. Point 9 follows immediately from this result. Notice that
in the resolution of phi of point 9, kappa tau is leftmost and the subscript descends as we move to
the right until we reach kappa naught at the rightmost end. This is the reverse order in which
the kappas appear in the form of phi in point 8, where the leftmost is kappa naught, and the
subscript increases as we move to the right of the expression until we have kappa tau at the right
hand end of the expression. Of course, due to issues with commutativity of multiplication of
ordinal numbers, then we cannot simply collect all of the kappa at one end of the expression in 9.
In the form for phi given in 8, then we see that if mu tau is equal to 0 and kappa tau is greater
than 0, then phi can be obtained from some other number by the first principle of generation,
that is, by repeatedly adding 1 until phi is reached, and thus phi will be of the first kind
in this case. In the case where mu tau is not equal to 0, then phi is not obtainable by the
first principle of generation, but is obtainable by the second principle of generation. Thus,
in this latter case, phi is a number of the second kind. Thank you for watching. If you
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