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Topic: [HM] Axiom of Infinity
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Posts: 2
Registered: 4/25/06
Re: [HM] Axiom of Infinity
Posted: Apr 22, 2006 9:24 AM
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--On Tuesday, April 18, 2006 9:05 AM +0300 HISTORIA MATEMATICA
<> 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.
148-150 of "Dedekind's Analysis of Number: systems and axioms", Synthese
147 (2005), pp, 121-170. 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

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,


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. 37-8 and 54ff., and Ferreirós 1999, pp. 243-6; 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 412-268-8565 (office)
Carnegie Mellon 412-268-1440 (fax)
Pittsburgh, PA 15213 412-531-3608 (home)

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