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sieg
Posts:
2
Registered:
4/25/06


Re: [HM] Axiom of Infinity
Posted:
Apr 22, 2006 9:24 AM


On Tuesday, April 18, 2006 9:05 AM +0300 HISTORIA MATEMATICA <ownerhistoriamatematicadigest@chasque.apc.org> wrote:
>> >>> > Dedekind seems to have been motivated by >>> > Bolzano's "Paradoxien des Unendlichen" (1851) in >>> > giving >>> > his proof, and probably also in realizing the need for it. >>> >>> Jose, I'm not quite sure about this. Remember >>> what Wilfried Sieg (CMU) always stresses, namely, >>> that we have Dedekind's own witness that he was >>> unaware of Bolzano's booklet at that time. Best, >> >> Dear Bernd >> >> well, I am quite sure: Dedekind didn't know Bolzano's work until >> the early 1880s, but the proof of his theorem 66 was not in the >> earlier draft of the book (1870s; as you know, the draft was >> published by Dugac in his book). Dedekind only added it in the >> second draft, 1887, and thus we are entitled to suppose that >> reading Bolzano may have been an inspiration. >> >> Best regards, >> Jose >> >> >>
Dear Jose:
In my paper with Dirk Schlimm I have tried to put all the available evidence for these speculations together; see the "Excursion" on pp. 148150 of "Dedekind's Analysis of Number: systems and axioms", Synthese 147 (2005), pp, 121170. I attach the text of the excursion below.
Apart from those facts, there is at least one other consideration that should weigh in:
Bolzano's considerations do not yield what Dedekind wanted to obtain. Reading Bolzano's essay carefully makes quite clear that he bases his considerations concerning the "objective existence" of the infinite i) implicitly on the existence of the species of integers and ii) explicitly on the existence of the set of sentences and truths in themselves.
Dedekind, in contrast, uses only one universal system, his "Gedankenwelt"; a simply infinite system and the natural numbers are obtained from it.
All the best,
Wilfried
Here is the content of the "Excursion":
Excursion. The need to prove the existence of an infinite system is not even discussed in 1872/78. A proof is given in the manuscript from 1887 that precedes the final writing of 1888, and this seems to be the first appearance in Dedekind's manuscripts or published writings. Through a letter from Cantor to Dedekind dated 7 October 1882 we know that the former sent with his letter also a copy of Bolzano's booklet to Dedekind. These three facts are taken as evidence, for example by Dugac, that Dedekind adapted Bolzano's considerations concerning the objective existence of the infinite. (Cf. Dugac 1976, p. 81 and p. 88, but also Sinaceur 1974, p. 254, Belna 1996 pp. 378 and 54ff., and Ferreirós 1999, pp. 2436; on p. 243 Ferreirós takes it for granted that Dedekind knew Bolzano's proof when giving his own and transformed it to suit his different philosophical ideas and his strict definition of infinity.) We first describe in (almost tedious) detail, where the claim and proof for the existence of an infinite system occur first, namely, in the fourth section of 1887 that is entitled The finite and infinite. It starts out with a definition. 40. Definition. S is called an infinite system, if there is an injective mapping from S, such that the image of S is a proper part of S; in the opposite case S is called a finite system. This is followed by the remark that all hitherto known definitions of the finite and the infinite are completely useless, to be rejected by all means. Next comes a proposition, numbered 41, which states that the union S of the singleton {a} and T is finite, if T is finite. As in other manuscripts of Dedekind's, the pages of 1887 are vertically divided in half. The main text is written on one half, whereas the other half is reserved for later additions. On this particular page a number of important additions have been made. Already its first line indicates that the manuscript is still being reorganized in significant ways: Dedekind refers to remarks on a separate page and writes that the first two propositions of §7 belong here. This is followed by three propositions, numbered 40x, 40xx, and 40xxx, the last of which claims: There are infinite systems. Dedekind adds parenthetically, Remarks on separate page, and mentions there that the following proposition can be added immediately to the fundamental definition 40: Proposition: There are infinite systems; the system S of all those things s ( this word understood in the sense given in the introduction ) that can be objects of my thinking, is infinite (my realm of thoughts). The proposition is established by a proof of roughly the same character as that given in the sources we discussed already. In the preface to the second edition of 1888, Dedekind emphasizes that Cantor and Bolzano had also recognized the property he uses as the definition of an infinite system. However, ? neither of these authors made the attempt to use this property as the definition of the infinite and to establish upon this foundation with rigorous logic the science of numbers. But this is precisely the content of my difficult labor, which in all its essentials I had completed several years before the publication of Cantor's memoir [i.e., Cantor 1878] and at a time when the work of Bolzano was completely unknown to me, even by name. Whether and how Dedekind was influenced by Bolzano's work in formulating his proof of theorem 66 remains a topic of speculation. The known facts, as we recounted them, allow a different interpretation than that given by Dugac and Ferreirós: the manuscript of 1887 is so different from 1872/78 that one might conjecture with good reason that Dedekind had other intermediate manuscripts or, at least, additional notes to bridge this remarkable conceptual and mathematical gap. The issue of providing models for axioms had been pressing already at that time, as we pointed out in section C. Given Dedekind's own remarks concerning the connection with Bolzano's and Cantor's work (indicated in the above quote), we speculate that he must have completed one such intermediate manuscript no later than 1878.  Dedekind's own remarks, quoted above at the end of section C, about these early considerations do not shed decisive light on the issue at hand: in his response to Weber's inquiry concerning the status of Was sind und was sollen die Zahlen? from 1878 he gives a description that fits the available material in 1872/78, and views it as a rough draft; in the Introduction to 1888 the earlier material is said to include also the justification of definition by recursion (that is barely hinted at in the folder of 1872/78); finally, in his remark concerning Cantor and Bolzano in the preface to the second edition of 1888 we just quoted, he claims to have completed the work in all its essentials several years before the appearance of Cantor's 1878 paper. End of Excursion.

Wilfried Sieg
Department of Philosophy 4122688565 (office) Carnegie Mellon 4122681440 (fax) Pittsburgh, PA 15213 4125313608 (home)
http://www.hss.cmu.edu/philosophy/facultysieg.php http://www.phil.cmu.edu/projects/apros/index.php


Date

Subject

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4/22/06


sieg

5/3/06


sieg


