Cantor s two views of ordinal numbers

Size: px

Start display at page:

Download "Cantor s two views of ordinal numbers"

Vincent Fox
6 years ago
Views:

1 University of Barcelona July 2008

2 The precursor s of Cantor s ordinal numbers are the symbols of infinity. They arose as indexes in infinite iterations of the derivative operation P P, where P R is bounded and P is the set of its accumulation points.

3 In 1872 Cantor considered finite iterations of the derivative operation: P (1) P (2) P (3) P (n), where P (1) = P, and P (k+1) = (P (k) )

4 In 1880, Cantor extended the iteration beyond the finite. He started with P ( ) = P (n) n By example, he showed how to get iterates of the general form: P (n 0 m +n 1 m 1 + +n m) He concluded that with the suggested procedure one successively reaches the further concepts : P ( ), P ( ), P ( ),......

5 Cantor did not iterate along the symbols of infinity. They had no independent status: They were mere indices of the iteration. They indicated how each iterate was built from previous ones. Cantor gave no definition of them, and he was elusive as to how far the iteration can proceed (he only said that the process goes always farther). In particular: Can one iterate past the indices,,,...,...?

6 1882 Theorem. If some P (α) is empty, then P, and hence P is countable. Cantor s account of the iteration was enough for the proof of the theorem. But it was not precise enough to establish the following conjecture: Conjecture. If P is countable, then P (α) is empty, for some α. Work on the conjecture led him to the discovery of the transfinite ordinal numbers.

7 First Cantor reached the countable ordinals, which he viewed as the old symbols of infinity. He saw that they form an uncountable set. This was crucial for the proof of the conjecture. The countable ordinals were the first natural extension of the finite number sequence. Cantor then conceived of the ultimate extension: one provably admitting no further extensions.

8 Cantor s generating principles. Grundlagen (1883) 1. If α is a generated number, so is its immediate successor α Whenever a definite succession of generated numbers is given, having no largest term, a new number is created which we conceive as the limit of those numbers, i.e., it is defined as the number immediately greater than all of them. ( A definite succession of numbers is a set of numbers.)

9 Numbers are generated from 0 by means of the generating principles. It is immediate that there is no definite succession (no set) containing all numbers. This, however, does not imply that the extended number sequence is large. That will depend on what sets of numbers there are (can be given). Cantor intended it to be inconceivably large: The extended number sequence is a symbol of the absolute.

10 A crucial feature of the absolute according to Cantor is that it cannot be known, not even approximately. In a positive formulation: If a succession of numbers can be characterized or defined, then it is a definite succession, i.e. a set. This leads to conditions on sethood, from which every number less than the first weakly inaccessible is secured.

11 Conditions on sethood 1. The totality of finite numbers is a set. 2. The totality of numbers equivalent to a given number is a set. 3. If α is a number, any succession {α ξ : ξ < α} of numbers indexed by α is a set.

12 In Grundlagen numbers were not defined as ordinals, but they were applied to play the role of ordinals: The Anzahl of a well-ordered set is its order-type: A and B have same Anzahl iff A = B. Every well-ordered set A has the same Anzahl as the set of predecessors of some number α. Cantor set the Anzahl of A equal to the number α.

13 Soon after Grundlagen Cantor abandoned the generating account of numbers on the grounds that it lacked mathematical rigor.

14 I limited myself to treat the matter in a purely mathematical way [1883]. I have already fulfilled to a certain extent your desire for a treatment of the matter of my Grundlagen which is purely mathematical and free from metaphysical aspects [1884]. For the rigorous foundation of this matter, discovered in 1882 and exposed in Grundlagen, we make use of the so-called ordinal types [1895].

15 He redefined the numbers in terms of order-types. First, as signs (Zeichen) for order-types of well-ordered sets. The whole numbers are nothing but signs for the various types of well-ordered sets. Then as general concepts obtained by abstracting from the nature of the elements of a well-ordered set, while retaining the order.

16 Two damaging effects of Cantor s turn: 1. He was unable to argue for the absolute infinity of transfinite numbers, because he had no strong principles of set existence. 2. When, later, he was in possession of stronger principles of set existence, he was unable to explain the paradoxes away in a convincing way.

17 After Beiträge (1897), Cantor distinguished between: 1. Completed or Consistent multiplicities, 2. Uncompleted or Inconsistent multiplicities. Completed multiplicities are sets. A multiplicity is a set if and only if their elements exist together, if and only if they all coexist.

18 In Beiträge I defined a set as a collection. But collecting together is only possible when existing together is possible [1897]. All elements of a multiplicity can be collected when their being together, their coexistence, can be thought without contradiction [1898]. A multiplicity is a completed set if all its elements can be thought without contradiction as being together, and hence as a thing in itself [1898].

19 The grounds of the distinction failed to convince even such advocates of set theory as Dedekind or Hilbert.

20 DEDEKIND (1899): Although I have read several times your entire letter, I am still unclear about your division of totalities [Inbegriffe] into consistent and inconsistent. I do not know what you mean when you speak of the being together of the elements of a multiplicity and of its opposite.

21 HILBERT (1904): G. Cantor sensed the contradictions just mentioned and expressed his awareness by differentiating between consistent and inconsistent sets. But, since in my opinion he does not provide a precise criterion for this distinction, I must characterize his conception on this point as one that still leaves latitude for subjective judgment and therefore affords no objective certainty.

22 Cantor s distinction is hard to make sense of under the assumption that all sets and numbers lie in an all encompassing universe. How can they fail to coexist if they inhabit the same such universe? (How, indeed, if, in Cantor s own words, they exist from all eternity as ideas in the divine intellect?)

23 But the distinction is completely natural in a generating setting, at least as it applies to the transfinite numbers. The reason why the multiplicity of all numbers is not a set is simply that the generating principles have no closure. This is what it means that the numbers do not all coexist.

24 As Cantor said, number generation is not purely mathematical. It cannot be appealed to in a mathematical proof. However, it may have a place in the justification of the set-theoretical axioms.

Cantor s Letter to Dedekind

Cantor s Letter to Dedekind (Halle, 28 July 1899) Áron Dombrovszki October 24, 2017 Introducion in the first part of his letter, Cantor deals with the same contradiction as Burali-Forti about the multiplicity