Georg Cantor - Contributions to the Founding of the Theory of Transfinite Numbers - Section 16

Welcome to this video. Video number 16 in my series of videos on George Cantor's 1895 and 1897 articles on the theory of aggregates and transfinite numbers. Section 16 is titled, The power of the second number class is equal to the second greatest cardinal number, Aleph 1. Given the statement of theorem A on page 170, it would follow then that the aggregate or totality of the numbers of the first and second number classes, in their order of magnitude, has an ordinal number, since this totality is a well-ordered aggregate. The aggregate A sub alpha does not include the numbers of the first number class, that is, the finite ordinal numbers. Some additional details of the proof of theorem A are shown on the screen now. A small typographical error on this page is shown on the screen now. Thus alpha-naught is a number greater than any number in the aggregate J. A sub alpha-naught plus 1 is the aggregate of all of the numbers of the second number class, which are smaller than alpha-naught plus 1. Since alpha-naught is less than alpha-naught plus 1, then alpha-naught must be an element of A sub alpha-naught plus 1. Therefore, A alpha-naught plus 1 has at least one element, which is greater than all of the elements of J. That there is a number alpha-primed in A alpha-naught plus 1, which follows immediately from all of the numbers of J, follows from criterion 2 of page 137 in section 12. For theorem D, that the aggregate alpha can be brought into the form of a simply infinite series, that is, of type omega, on the assumption that the cardinal number of the aggregate alpha is equal to LF-naught, is for the same reason that the M of the theorem in section 9 can be brought into the form of type omega, and which I explained in the video for that section, and so I won't repeat it here. The numbers gamma sub rho-2, gamma sub rho-3, and so on, must all exist, otherwise it would indicate that the aggregate alpha had the largest number, which is not true. Most of the details of this proof of theorem D are fairly straightforward and follows a contradiction argument, and doesn't really require any additional comments. For theorem E, the aggregate beta is not necessarily a segment of the aggregate alpha, but merely a part of it. The part beta of alpha may be similar to alpha without being equal to it. This does not, therefore, contradict the definition of partial aggregate, given in section 1, and which suggests that a partial aggregate of an aggregate cannot be equal to the whole aggregate of which it is considered a part. The symbols alpha-naught minus omega with a double bar on this page should, in my opinion, be alpha-naught minus omega with a single bar, since alpha-naught minus omega is an ordinal type, and the cardinal number of an ordinal type is indicated by a single bar, as in point 5 of section 7. I wanted to say that these were just typographical errors in the English translation, but they're written with two bars in the original German articles also. For the proof of theorem F, it's worth bearing in mind the result of theorem D of this section, which says that the power of the totality of all of the numbers of the second number class is not equal to alph-naught. Thus we see from theorem F, that since the power of the second number class is not equal to alph-naught, and that there are no cardinal numbers between alph-naught and the power of the second number class, then the power of the second number class is the next greatest transfinite cardinal after alph-naught, which is denoted alph-1. It would seem that, from Cantor's perspective, the totality of numbers of the second number class already exists. It's shown in section 16 that the totality of the numbers of the second number class form a well-ordered aggregate, and therefore, this totality has an ordinal number. However, it's also shown that the totality of the numbers of the second number class has cardinal number greater than alph-naught, and therefore, the ordinal number of the second number class must be in a higher number class. Therefore, despite the results of theorem A and B of the previous section, the first and second principles of generation do not necessarily restrict us to the second number class only. The principle of generation is not to be confused with creation of new numbers. New numbers can be generated using the principles of generation, but it would seem that Cantor sees these numbers as already existing. If these numbers didn't already exist, then it wouldn't be possible to say that a number was the limit of a fundamental series of numbers, since the limit has to be a number in the aggregate in which the fundamental series is a part. Thank you for watching. If you found this video useful, then please hit the like button and subscribe to my channel. You can see more videos on various topics on my channel, and if you have any suggestions for topics for future videos, then please feel free to let me know, and I'll try my best to put something together. Don't forget to hit that subscribe button, and keep an eye out for new videos being uploaded to my channel.