[FOM] From theorems of infinity to axioms of infinity

[Murray Jorgensen]

Indeed the first paragraph of the 1910 edition of Principia Mathematica 
suggests that Whitehead and Russell held these attitudes in 1910:

"THE mathematical treatment of the principles of mathematics, which is
the subject of the present work, has arisen from the conjunction of two
different studies, both in the main very modern. On the one hand we have
the work of analysts and geometers, in the way of formulating and 
systematising their axioms, and the work of Cantor and others on such 
matters as the theory of aggregates. On the other hand we have symbolic 
logic, which, after a necessary period of growth, has now, thanks to 
Peano and his followers, acquired the technical adaptability and the 
logical comprehensiveness that are essential to a mathematical 
instrument for dealing with what have hitherto been the beginnings of 
mathematics.
 From the combination of these two studies two results emerge, namely 
(1) that what were formerly taken, tacitly or explicitly, as axioms, are 
either unnecessary or demonstrable; (2) that the same methods by which 
supposed axioms are demonstrated will give valuable results in regions, 
such as infinite number, which had formerly been regarded
as inaccessible to human knowledge. Hence the scope of mathematics is
enlarged both by the addition of new subjects and by a backward 
extension into provinces hitherto abandoned to philosophy."

Murray

On 12/03/2013 3:20 a.m., Alasdair Urquhart wrote:
> It seems fairly clear that it was the paradoxes that prompted the move
> to postulate the axiom of infinity outright.  You can see this from
> the example of Russell.  In the "Principles of Mathematics" he makes
> the remarkable assertion:
>
>      That there are infinite classes is so evident that it will scarcely be
>      denied.  Since, however, it is capable of formal proof, it may be
>      as well to prove it [PoM p, 357].
>
> He proceeds to give no less than three proofs of the axiom of infinity,
> including versions of the notorious `proofs' of Bolzano and Dedekind.
>
> Even as late as 1904, Russell continued to cling to the view that the
> axiom of infinity was a logical truth, when he engaged in a debate on
> the matter with Cassius J. Keyser.
> In his reply to Keyser, Russell repeats one of his proofs of the axiom
> of infinity from the pages of the "Principles of Mathematics",
> concluding triumphantly: ``Hence, from the abstract principles of logic
> alone, the existence of infinite numbers is rigidly demonstrated.''
>
> In 1905 and 1906, Russell had high hopes for his new substitutional
> theory, and in his reply to Poincaré published in September 1906, proves
> the axiom of infinity in his paper by constructing an infinite series of
> propositions.
>
> In the end, though, the substitutional theory succumbed to the paradoxes,
> and Russell was forced to adopt type theory against his will and logical
> inclinations.  In Principia Mathematica, the Axiom of Infinity is
> not postulated, but is present as an explicit antecedent in a lot
> of arithmetical propositions.
>
> Russell, unlike Zermelo, was committed to a logicist view of mathematics,
> and so clung desperately to the idea that the existence of infinite
> classes was a logical truth.  It was clearly the paradoxes that
> forced him to give up this view.
>
>
>
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-- 
Dr Murray Jorgensen      http://www.stats.waikato.ac.nz/Staff/maj.html
Department of Statistics, University of Waikato, Hamilton, New Zealand
Email: maj at waikato.ac.nz      majmurr at gmail.com         Fax 7 838 4155
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