Georg Cantor - Contributions to the Founding of the Theory of Transfinite Numbers - Section 18

Welcome to this video, the 18th video in my series of videos on Cantor's 1895 and 1897 articles on the theory of aggregates and transfinite numbers. Section 18 is titled The Power Gamma to the Power of Alpha in the Domain of the Second Number Class. The focus of this section is to make sense of the operation of exponentiation as applied to ordinal numbers. This operation has been conspicuously absent so far. The only type of exponentiation, at least in relation to ordinal numbers that we've seen so far, really is in relation to finite powers of omega, which entirely agree with our intuitive notion of exponentiation. That is, omega squared equals omega times omega, omega cubed equals omega times omega times omega, and so on. We've not yet dealt with exponentiation by a transfinite number. By the end of this section, we'll have a notion of exponentiation by those transfinite numbers of the second number class. There's a lot to get through in this section, and I'll be going into quite considerable detail. Page 178 gives the statement of a theorem to be proved, which is the key theorem of this section. Before we get started with the proof, I'll just give a bit of an idea about what the aim is and how the proof will be executed. We want to show that there is a function defined on all numbers of the first and second number classes, which satisfies the conditions A to D of the theorem. This is done by actually constructing a function f from scratch with the desired properties. Moreover, we also need to show that the function constructed is unique. That is, any second function g constructed, also satisfying the conditions of the theorem, must in fact be the same function as f. The construction of the function which satisfies the conditions of the theorem proceeds in three fairly distinct parts. The first part is to construct a function satisfying the properties of the theorem, but restricted to the numbers of the first number class, that is, the finite numbers. It's then shown that this function can be extended to apply to all numbers of the second number class in two stages. Firstly, that the function can be extended using the first principle of generation, and secondly, by using the second principle of generation. This covers the existence of a suitable function. Finally, there is the uniqueness of the function. Note that in the proof of the existence, it's not necessary to check all of the conditions A to D at each stage. I'll indicate this as we progress through the proof. I'll go through the proof of theorem A quite carefully. Let's start with the construction of the function on the numbers of the first number class. There are no fundamental series which have a limit which is a finite number, and so D is not applicable at this stage. Therefore, D may be considered as trivially holding for finite numbers, or even simply not applicable. By choosing F0 to be equal to delta, then A is satisfied, and by choosing F nu plus 1 equals F nu times gamma for each nu, then C is satisfied in the case of finite numbers. Note that since gamma is greater than 1, and F nu plus 1 is equal to F nu times gamma by C, therefore by theorem H of section 14, we necessarily have that F nu times gamma is greater than F of nu. Therefore, F nu plus 1 is greater than F of nu, and the ascending sequence F1 less than F2 less than F3 and so on is justified. Therefore, B is satisfied in the case of finite numbers. So we see that there is a function which can be constructed and which satisfies the criteria of theorem A, at least those that are applicable on finite numbers. Now let's move on to numbers of the second number class. We start with the assumption that there is a function satisfying the conditions of theorem A up to a number alpha sub minus 1. That is, A, B, C and D hold for all numbers up to and including alpha sub minus 1. Note that alpha sub minus 1 may be a number of the first or second kind. It makes no difference. There may also be numbers of the first kind and numbers of the second kind that are less than alpha sub minus 1. The number alpha is taken to be equal to alpha sub minus 1 and so alpha is a number of the first kind since it is obtained by the first principle of generation. Since alpha is a number of the first kind, we do not need to check condition D, since a number of the first kind cannot be a limit of a fundamental series, and so D again trivially holds up to and including alpha. Also, we don't need to check A, since this only applies to 0, and so A again trivially holds for all numbers up to and including alpha. Since alpha is a number of the first kind, then it has an immediately preceding number alpha sub minus 1, such that alpha sub minus 1 plus 1 is equal to alpha. So by setting f of alpha equal to f of alpha sub minus 1 times gamma, then we have C satisfied for alpha and moreover, by theorem H of section 14, f of alpha sub minus 1 times gamma is greater than f of alpha sub minus 1, and so f of alpha is greater than f of alpha sub minus 1. It follows that since B holds by assumption for all numbers up to and including alpha sub minus 1, that f alpha is greater than f alpha primed for all alpha primed less than alpha, and so B is satisfied for alpha. Therefore, conditions B and C hold up to and including the number alpha. Since a function satisfying the conditions of theorem A was assumed to exist for all values up to alpha sub minus 1, we've shown that this implies that the function can be extended so that all conditions of theorem A are satisfied for all numbers up to and including alpha. That is, the function can be extended by the first principle of generation. Now we assume that there is a function satisfying the conditions of theorem A for all numbers less than alpha, where alpha is a number of the second kind. Since alpha is of the second kind, then it has no immediate predecessor, but also, since it is a number of the second kind, it is by definition the limit of a fundamental series of numbers from the second number class. Moreover, the function satisfies the conditions of theorem A for all values alpha new of the fundamental series, since alpha new is less than alpha for all new. In particular, condition B is satisfied for all numbers less than alpha, and so the series f alpha new is a fundamental series, and so has a limit which is the number which is next in order of magnitude to all of the f alpha new. If we set f alpha equal to the limit of f alpha new, then condition D is satisfied for all numbers up to and including alpha. It's easy to show using B that for any fundamental series alpha new primed, which is coherent to alpha new, that the fundamental series f alpha new primed will be coherent to f alpha new, and so the limit of f alpha new is equal to the limit of f alpha new primed, which equals f alpha. Thus the value of f alpha is independent of the choice of fundamental series with limit alpha. We also see then that f alpha is greater than f alpha new for all values of new, and this is true for any fundamental series with limits alpha, and therefore f alpha is greater than f alpha primed for all values alpha primed less than alpha. Therefore B holds for all values up to and including alpha. Condition C does not need to be checked for alpha since it only applies to finite numbers and numbers of the first kind, and so C trivially holds up to and including alpha. And finally condition A trivially holds up to and including alpha as well. Therefore the function which was assumed to hold for all values less than alpha is shown to hold for all values up to and including alpha. Therefore with all of these cases accounted for, we can deduce that there is a function which satisfies the four conditions of theorem A, and which is valid for all numbers of the first and second number classes. Cantor shows that there cannot be any exceptional values for which the conditions of the theorem doesn't hold. This part doesn't require any further explanation, but we still have uniqueness to prove. The uniqueness of the function will be proved on the screen. For theorem B, we need to check that it's possible to construct the function gamma to the power of xi, which has the properties described in A to D of theorem B. Once this is done, then it can be seen that by replacing gamma to the power of xi, gamma to the power of xi primed etc, with f of xi, f of xi primed etc respectively, throughout the statements of theorem B, and replacing 1 with delta, we see that theorem B is essentially nothing more than a restatement of theorem A. Therefore, the function of theorem B must be unique by the result of theorem A. The construction of the function of theorem B proceeds in a practically identical way to the construction of the function of theorem A, with f of xi etc replaced with the corresponding expressions of the form gamma to the power of xi, and with delta of theorem A replaced with 1. Due to the similarity of the constructions, and to avoid unnecessary repetition, I'll omit the construction of the function of theorem B. It may seem that a construction of the function of theorem B is not necessary. This is not the case. We have to be aware that the function of theorem B, although it appears to have a familiar form, is initially without meaning until a specific value is assigned to it for each of its arguments. Of course, we seem to have an intuitive understanding of how we feel the function of theorem B should behave, based on our understanding of how such a function behaves of finite numbers, or at least finite indices. And indeed, the properties that the function must satisfy, outlined in the statement of theorem B, will assign a meaning to gamma to the power of xi, which I believe closely reflects our intuitive expectations. However, the expression gamma to the power of xi acquires its meaning through the construction that we give to it, and therefore, how this function behaves is not to be assumed or taken for granted. Theorem C shows that the unique function constructed in theorem A has the form of the function given in theorem C. Therefore, it's a simple case of showing that the function on the right hand side of f of xi equals delta gamma to the power of xi satisfies all of the conditions of theorem A. It's necessary in places to use the result of theorem B to do this. To check that delta gamma to the power of xi satisfies all of the conditions of theorem A is not difficult. The only real yet minor difficulty arises when we consider fundamental series in relation to condition D, in the case of a number of the second kind. But as is suggested by Cantor, we make use of point 24 from section 14. The steps to show that the function of theorem C satisfies D is shown on the screen. For theorem D, we show that the function phi of xi satisfies the conditions of theorem D, which are effectively the conditions of theorem A with delta equal to gamma to the power of alpha. This will go as shown on the screen. Since the function phi of xi has been shown to satisfy the conditions stated on page 181, then as stated previously, these are effectively the conditions of theorem A with delta equal to gamma to the power of alpha. Now because the function of theorem D satisfies theorem A, then it's unique. But we also know that any function which satisfies the conditions of theorem A can be represented in the form of theorem C. Note that the function of theorem D has delta equals gamma to the power of alpha. Then theorem C tells us that phi of xi equals gamma to the power of alpha times gamma to the power of xi. But we also know that phi of xi equals gamma to the power of alpha plus xi. Therefore by uniqueness we must have that gamma to the power of alpha plus xi is equal to gamma to the power of alpha times gamma to the power of xi. Replacing xi with beta gives the result. Of course theorem D represents a kind of generalisation of the usual laws of indices for positive integers. We see that formally we have the same law which applies to all numbers of the first and second number classes. Theorem E is very similar to theorem D. We can recognise that the function of theorem E is a kind of stretched version of the function from theorem B. It's required to show that the function of theorem E satisfies the conditions A to D stated in the proof of the theorem. This will go as shown on the screen. Therefore the function of theorem E satisfies all of the conditions stated on page 182. These conditions are effectively those conditions of theorem A with delta equals 1 and with the gamma of theorem A replaced with gamma to the power of alpha. Therefore the function psi of xi can be represented in the form of theorem C with delta equals 1 and with the gamma of theorem C replaced with gamma to the power of alpha giving gamma to the power of alpha to the power of xi equals gamma to the power of the product of alpha and xi. Finally replacing xi with beta gives the result. Again we see that theorem E is a kind of generalisation of the usual laws of indices as applied to positive integers. Formally we have the same law which applies to transfinite numbers of the second number class. When it comes to theorem F there isn't very much for me to say. There's a few details that I want to add to what's in the proof already of theorem F regarding some of the inequalities in the case where alpha is a number of the first kind. These are 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. 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