Georg Cantor - Contributions to the Founding of the Theory of Transfinite Numbers - Section 4

Welcome to this video, the fourth video in my series of videos on Cantor's 1895 and 1897 articles on the theory of aggregates and transfinite numbers. Section 4 is titled, The Exponentiation of Powers. I've got quite a bit to say about the contents of this section. In section 4 we have the notion of a covering of the aggregate n with elements of the aggregate m, or a covering of n with m, which is nothing more than a function, not necessarily one-to-one, such that for each element n of the aggregate n, there is associated an element m in the aggregate m with Fn equals m. Cantor says when we should consider two coverings of n with m to be the same, and he also gives some examples of interesting cases of coverings. Firstly, the constant covering, and secondly, a law which is constant but for one exceptional value. The totality, or aggregate, of coverings of n with m is called the covering aggregate of n with m. We're assured on page 95 that the coverings of n with m form a definite aggregate, and it's worth noting that no consideration is given here as to whether the aggregates m and or n are finite or not, although, admittedly, we don't have a notion of finite and transfinite cardinal numbers or aggregates yet. This assertion is a very bold assertion and shows that Cantor certainly had no objection to the idea of having completed infinite totalities of potentially all sizes, something which several mathematicians of the 20th century objected to very fiercely regarding completed infinite totalities as nonsense. A to the power of b is thus defined to be the cardinal number of the covering aggregate of n with m. We see that the cardinal number of the covering aggregate depends only on the cardinal numbers of the aggregates m and n, and does not depend on the specific nature of the elements of those aggregates. Points 5, 6 and 7 indicate what would need to be shown in order to prove the properties 8, 9 and 10 of cardinal numbers. These properties can easily be recognised as bearing a strong similarity to the familiar laws of indices. It's important to note that points 8, 9 and 10 place no restriction on the cardinal numbers. That is, the cardinal numbers are not required to be finite. Again, we don't really have a notion of finite or infinite at this stage, and as such, the symbol a to the power of b cannot be taken as literally meaning a multiplied by itself b times, which is often our intuitive way of thinking about exponentiation, particularly of integers. The symbol a to the power of b is merely a symbol used to represent the cardinal number of the covering aggregate of n with m. Clearly, the notation is very suggestive and closely resembles the symbolism used to represent exponentiation using finite positive integers. Indeed, for a and b, both finite positive integers, the symbol a to the power of b exactly coincides with the usual notion of exponentiation. But in general, the symbol a to the power of b cannot, and possibly should not, be interpreted in an analogous way. Let's just have a look at how the equivalences 5, 6 and 7 might be shown. In my solutions to these, I've slightly abused the notation so that things don't get excessively fussy, and the ideas remain as clear as possible. I'll leave some of the more fussy, yet fairly obvious details for you to fill in yourself. An outline of the details of possible solutions to 5, 6 and 7 will be shown on the screen. Page 96 gives an application of the properties 8, 9 and 10. I'll go through pages 96 and 97 in detail, but I'll just point out in advance that some concepts that are used on these two pages have not yet been properly introduced. Nonetheless, I'll indicate wherever necessary the relevant sections of the work to refer to. A typographical error at the top of page 96 is that x is greater than or equal to should be x is greater than or equal to 0. And a little bit further down the page, 2 to the power of aleph should be 2 to the power of aleph-naughts. As I mentioned in section 1, Cantor seems to be very sure that every aggregate has an unambiguous cardinal number. Thus it stands to reason that the linear continuum X has a cardinal number which Cantor denotes using the symbol factor O. We're to prove that factor O is equal to 2 to the power of aleph-naughts. The symbol aleph-naughts will be defined in section 6, but it suffices to say that aleph-naughts here has the meaning usually associated with it and the cardinal number 2 will be defined in section 5. The numbers which are elements of X are to be understood as existing in the usual sense. The formula 12 is indicating that all real numbers of X can be represented by a not necessarily unique sequence of binary digits. For example, the number 3 quarters is equal to half plus 1 over 2 squared. And so we can think of 3 quarters as being represented by the binary sequence 1, 1. We may consider any sequence of binary digits as infinite by placing an infinite sequence of zeros after the final 1, if a final 1 exists of course. Thus 1, 1 can be thought of as 1, 1, 0, 0, 0 and so on. Here we see that f of 1 is equal to 1, f of 2 is equal to 1 and f of nu is equal to 0 for all nu greater than or equal to 3. As a further example, the sequence 1, 0, 0, 1, 1, 0, 0, 0 and so on represents the number 19 over 32. That every real number in X has such a binary representation is not proved here. The remark relating to point 12 which starts 2 to the power of aleph-naughts is the power of all representations and so on means that the cardinality of the totality of all binary sequences has cardinal number 2 to the power of aleph-naughts which follows immediately from point 4 when we think of m as the aggregate with elements 0 and 1 and n as the aggregate 1, 2, 3 and so on of finite positive integers. m has cardinal number 2 and n has cardinal number aleph-naughts. The totality that we're concerned with is then the covering aggregate of n with m which of course has cardinal number 2 to the power of aleph-naughts. Cantor points out that some numbers have more than one binary representation. For example, a half can be represented by the sequence 1 with an infinite number of 0s after it or 0 with an infinite number of 1s after it. This can be compared to the fact that 0.9 recurring is numerically equal to 1. The totality of numbers of X which do not have unique representations is denoted s-nu in braces. Cantor describes this as an innumerable totality which means that its cardinal number is aleph-naughts. The symbol aleph-naughts has not yet been defined in this work but is, as we might expect, the cardinal number of the set or aggregate of natural numbers. Note that the aggregate s-nu is the totality of numbers of the form indicated in the text where nu and mu both range over the natural numbers and so the cardinal number of the aggregate s-nu is equal to aleph-naught times aleph-naught which is equal to aleph-naught using 0.8 from section 6. Each number in the aggregate s-nu has exactly two binary sequence representations and we see that the totality of all binary sequences has the same cardinal number as the union aggregate of X along with all of the elements of the aggregate s-nu in order to account for the repetitions. X accounts for one of the binary representations for each of the numbers s-nu of the aggregate s-nu and the aggregate s-nu itself accounts for the second binary representation of each aforementioned number s-nu. It seems evident then that each number in the aggregate s-nu is also a number in X but that we are to consider these as distinct instances that is, we are not to consider them as common elements. Therefore, the equality 2 to the power of aleph-naught is equal to the cardinal number of the union aggregate of s-nu and X is justified. Recall that we have already established that the totality of binary sequences has cardinal number equal to 2 to the power of aleph-naught. A fairly straightforward argument shows that X is equivalent to the union aggregate of the aggregates s-nu and X. The argument uses that T-nu can be considered as the union aggregate of T2-nu-1 and T2-nu and that the aggregates T2-nu-1, T2-nu T-nu and s-nu are all innumerable aggregates that is, that they all have cardinal number aleph-naught. Recall also that the cardinal number of a union aggregate depends only on the cardinal numbers of the individual aggregates. And so by section 1, the cardinal number of X is equal to the cardinal number of the union aggregate of the aggregates s-nu and X and therefore 2 to the power of aleph-naught is equal to the cardinal number of the union aggregate of s-nu and X which in turn is equal to the cardinal number of X which is equal to factor O. The results immediately following this depend on the results from section 6. However, that 2 to the power of aleph-naught times 2 to the power of aleph-naught is equal to 2 to the power of aleph-naught plus aleph-naught follows from point 8 of section 4 and that aleph-naught plus aleph-naught is equal to aleph-naught follows from point 6 of section 6 which of course we haven't got to yet. The correction of a typographical error on this page is shown on the screen. The ambiguity that's mentioned next to the word power is because power may, depending on the context refer to either exponent or cardinality. Here, power is meant in the sense of exponent. The 1878 work that's referred to, I can't make any further comments on at this stage. I have a copy of this article but since it's in German, I'm not yet able to understand it. Thank you for watching. If you found this video useful, then please hit the like button and subscribe to my channel. 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