Georg Cantor - Contributions to the Founding of the Theory of Transfinite Numbers - Section 15

Welcome to this video. Video number 15 in my series of videos on George Cantor's 1895 and 1897 articles on the theory of aggregates and transfinite numbers. Section 15 is titled The Numbers of the Second Number Class. On page 160, theorem A shows that omega, being the least number of the second number class, is also the smallest transfinite ordinal. Recall that the first number class is the aggregate of all finite ordinal numbers. The representation of omega as the limit of the fundamental sequence of finite ordinal numbers, which is given at the bottom of page 160, will often prove to be quite a useful representation. Theorem B shows that by adding 1 to an ordinal number, we obtain the next ordinal number. That is, as we might expect, there are no ordinal numbers between alpha and alpha plus 1. This method of obtaining new ordinal numbers by adding 1 may be called the first principle of generation. The proof of theorem B is not difficult, but I'll just make a few remarks. At point 7, that the ordinal number of the aggregate FG is equal to alpha plus 1 follows from point 1 of section 8. And for point 8, that alpha plus 1 is greater than alpha follows from point 2 of section 14. The notation alpha plus 1 with a single bar over it represents the cardinal number of the ordinal number alpha plus 1, and similarly for alpha. The method of adding 1 to an ordinal number to generate new ordinal numbers is not the only way of generating new ordinal numbers. Theorem C shows that we can use fundamental series of ordinal numbers to generate new ordinal numbers. This method of generating new ordinal numbers may be called the second principle of generation. It would seem from theorem C that we're stuck in the second number class, since the first and second principles of generation do not seem to allow us to go beyond the numbers of the second number class. This will be taken up in more detail in the next section. Interestingly, it's noted on page 205 that Cantor unconsciously uses the axiom of infinite selection, or the axiom of choice, in theorem C. This is indicated by Cantor choosing any fundamental series of numbers of the first or second number class. There is no indication about how the numbers of this fundamental series are to be chosen. A lot of the details of this proof run along the same lines as the arguments presented on pages 156-158 in section 14, so I won't reiterate all of that here. The proof that the cardinal number of the aggregate G is aleph-naught, and thus the ordinal number of G, and hence the limit of the fundamental series, is a number of the second number class, used as point 8 from section 6, that aleph-naught times aleph-naught is equal to aleph-naught. This is applicable here, since each of the numbers beta-nu has cardinal number which is less than or equal to aleph-naught. And of course there are aleph-naught terms in the sum, beta-1 plus beta-2 plus, and so on. However, since G is a transfinite aggregate, then the cardinal number of G cannot be less than aleph-naught, since that would imply that G was a finite aggregate. Recall that aleph-naught is the smallest transfinite aggregate, and so, any ordinal number less than aleph-naught must be finite. The definition of coherent fundamental series of numbers of the first or second number class is not new. It's merely a special case of the already existing notion of coherent ascending fundamental series given in section 10, but, in the present section, with the less than symbol, replacing the more general simple order symbol used in section 10. We also refer to limiting numbers of fundamental series of ordinal numbers, rather than the more general limiting element. Theorem D is arguably an immediate consequence of theorems C and D of section 10. I gave a proof of theorem C of section 10 for coherent ascending fundamental series, which would apply equally to the proof of the first part of theorem D of the current section. However, Cantor's proof of the first part of theorem D of the present section uses a contradiction argument to achieve the same result. The claim in the proof of the first part of theorem D, that of two unequal ordinal numbers one must be smaller, is substantiated by theorem A of section 14. For the proof of the second part, the conclusion that from a certain lambda onwards we have alpha lambda primed is greater than alpha nu, is directly from the definition of limits of an ascending fundamental series. For if M0 is the limit of an ascending fundamental series in an aggregate M, then for any M which precedes M0 in M, we have elements of the fundamental series between M and M0. Since the fundamental series alpha nu primed and alpha nu have a common limiting element gamma, then alpha nu is less than gamma for every nu, and so there must be a lambda 0 for which alpha lambda primed is greater than alpha nu for all lambda greater than or equal to lambda 0. And similarly for alpha mu greater than alpha nu primed for all mu greater than or equal to mu 0. The details of the highlighted section of page 164 are shown on the screen. For the second proof of theorem F, the two fundamental series nu 0 nu and nu, and with limiting numbers nu 0 omega and omega respectively, are clearly coherent, since for any value nu 0 nu, there is always a value mu 0, such that mu is greater than nu 0 nu for all mu greater than or equal to mu 0. Similarly, for any value of nu, there is always a value lambda 0, such that nu 0 lambda is greater than nu for all lambda greater than lambda 0. Therefore, an application of theorem D of this section gives the desired result. The details of the highlighted section of page 165 are shown on the screen. In the proof of theorem H, that F double bar is equal to alph0 implies that F has parts that are finite and also, by theorem A of section 6, has parts with the cardinal number alph0. Therefore, the cardinal number of A primed is either a finite number or is alph0. By section 13, every element of a well-ordered aggregate, except for the smallest element, determines a segment of the aggregate and also, every segment of a well-ordered aggregate is determined by an element of the aggregate. The aggregate alph0 is the aggregate of all ordinal numbers which are less than alph0. An imaging of F' onto the aggregate alph0 implies that these two aggregates are similar and therefore have the same ordinal type. The correspondence between F' and the aggregate alph0 is effectively a correspondence between the segments of F and the aggregate alph0 since the elements of F' are those elements of F which determine segments of F. Since F is equal to F1 F' then F bar is equal to F1 F' bar. Therefore alph0 equals 1 plus beta by 1 of section 14 where beta is the ordinal type of F' therefore beta equals alph0 minus 1 by section 10 of section 14 and the rest of the proof is straightforward. Theorem I follows immediately. A small typographical error on page 167 is shown on the screen. We've already seen earlier in this section, theorems B and C, how new ordinal numbers of the second number class can be generated using the first and second principles of generation. Theorem K shows that all numbers of the second number class arise through these two principles of generation, that is, the first and second principles of generations account for every number of the second number class. The notation alpha sub minus 1 is used here and on the pages which follow to denote the immediate predecessor of a number alpha, wherever alpha is the immediate successor of a number. That is, alpha sub minus 1 plus 1 is equal to alpha. However, alpha sub minus 1 must not be understood as meaning the same as or being equal to the number denoted alpha minus 1. Note that by 10 of section 14 we have in general that 1 plus alpha minus 1 in brackets is equal to alpha, but it is not the case in general that 1 plus alpha sub minus 1 is equal to alpha. That is, alpha sub minus 1 does not coincide with the unique solution of the equation 1 plus gamma equals alpha, which is denoted alpha minus 1. Indeed, it may be that alpha minus 1 is equal to alpha, but it is never the case that alpha sub minus 1 is equal to alpha, whenever both exist. Some details relevant to page 167 are shown on the screen. Alpha sub minus 1 represents the ordinal type of A, which is a segment of F and is, therefore, a well-ordered aggregate. That the aggregate alpha primed can be brought into the form of a simply infinite series, that is, of the type omega, follows since the cardinal number of both alpha primed and omega is aleph-naught, and therefore, there is a one-to-one correspondence between the two aggregates. Thus, we set alpha-nu primed to be lower in rank than alpha-sigma primed, whenever nu is less than sigma. Note that alpha-nu primed lower in rank than alpha-sigma primed does not imply that alpha-nu primed is less than alpha-sigma primed. That is, the elements of the aggregate alpha-nu primed will not necessarily be, and in fact almost certainly won't be, in order of magnitude. A typographical error on page 168 is shown on the screen. The point of this is that a fundamental series alpha-rho-nu primed, such that alpha-rho-nu primed is less than alpha, for every index rho-nu, and for every number alpha primed less than alpha, we have that alpha primed is less than alpha-rho-nu primed for rho-nu greater than some suitable index. Note that the alpha-nu primed account for all numbers less than alpha, so it really is the case that, for all numbers less than alpha, there are elements of the fundamental series which exceed that value. Thus, alpha is the limit element of the fundamental series of ordinal numbers formed. Another typographical error on this page is shown on the screen. 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.