Georg Cantor - Contributions to the Founding of the Theory of Transfinite Numbers - Section 13

Welcome to this video, the thirteenth video in my series of videos on George Cantor's 1895 and 1897 articles on the theory of aggregates and transfinite numbers. Section 13 is titled the segments of well-ordered aggregates. Segments of well-ordered aggregates are defined on page 141. The bulk of the current section is made up of a number of theorems regarding properties of segments of well-ordered aggregates. In this section we see some of the structure and some of the most important properties of well-ordered aggregates which are essential for the considerations that will come up in later sections. This is one of the longest sections in the two combined articles but it's also one of the driest sections to read. There's not much to get excited about but the results are nonetheless extremely important going forward. Despite being one of the longer sections of the two combined articles there's not a great deal for me to say about this section other than a few miscellaneous comments which possibly clarify a couple of minor points here and there. Most of the theorems and proofs present little difficulty. Remainders of well-ordered aggregates are defined on page 142 and Cantor presents a few simple examples to illustrate the concepts of segment and remainder. A typograph clearer on page 143 is shown on the screen. The notation less than applies only to segments of a well-ordered aggregate F and F itself and not general parts of a well-ordered aggregate. Recall that F is not a part of F nor a segment of F. Note that A' less than A a segment does not imply that A' is lower in rank than A as partial aggregates since there will be some overlap of the segments A' and A and thus it's not the case that all of the elements of A' precede all elements of A in F. The proof of theorem B uses transitivity of similarity of aggregates which was mentioned after point 3 of section 7. The proof of theorem B is straightforward. The approach that we see in the proof of theorem B is something that we'll see quite often later in the work namely arriving at a contradiction by taking an infinite decreasing sequence of elements from a well-ordered aggregate. On page 145, Cantor points out that even though a well-ordered aggregate can never be similar to one of its segments as just shown in theorem B, the well-ordered aggregate can be similar to a remainder and he gives an example to illustrate this. Theorem C would seem to be obvious in light of what's been proved in theorem B. The proof of theorem C follows a very similar idea to the proof of theorem B. The theorems and proofs of page 146 are very straightforward. The statements of theorem H on page 147 is quite poorly worded though. We're to understand the segments B and B' both segments of the well-ordered aggregate G, as being similar respectively to the segments A and A' of the well-ordered aggregate F. I find the wording given in the book to be a little bit confusing. Theorem H seems very similar to theorem G but there is a distinction. Theorem G merely says that for A and B similar segments of well-ordered aggregates F and G respectively, that for every A' less than A, then there is a B' less than B which corresponds to A'. However, there is the possibility that there is a B'' of G which corresponds to A' for which B is less than B'. However, theorem F rules out the possibility of A' being similar to more than one segment of G and thus rules out the possibility of there being such a B''. As indicated for the proof, theorem H follows immediately from theorems F and G. There is a small typographical error in the statement of theorem I. The D after B' is greater than B should be G. Theorem I does not rule out the possibility that B itself is similar to F. It simply says that any segment of G which is greater than B or G itself is not similar to F. On page 148 if F' is lower in rank than F, then A' is less than A follows by the definition of less than as applied to segments of well-ordered aggregates on page 143 and similarly for B' less than B. Theorem L fills a bit of a gap that was left open by theorem I. I commented regarding theorem I that using the notation of theorem I the possibility of B itself being similar to F is not excluded. Here we see that it certainly is possible for the B of theorem I to be similar to F. On page 149, since A1 is the least segment in F for which there is no corresponding segment in G, therefore any segment of A1 which is also a segment of F, must have a corresponding part in G. Theorem N will be of particular significance for theorem A of section 14 once ordinal numbers have been introduced. Theorem N will effectively say that any two ordinal numbers have one of the given relations alpha is less than beta, alpha equals beta or alpha is greater than beta and that these relationships are mutually exclusive. The proof of theorem N is straightforward. 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.