[FOM] Questions on Cantor

[Frode Bjørdal]

Thank you for usefully hinting to the fact that Cantor held well ordering
to be a fundamental principle, Vaughan.

But it is a result of Bernays that Foundation, or the Axiom of Regularity,
is independent of the other axioms of ZFC. So it is unclear to me what you
mean by "passage", or what connection it is that that you think is easy to
see in hindsight. (Is there something in Mirimanoff's text that resolves
this? Not that I noticed from reading it.)

Also, it seems to me that Russell's and Poincare's full vicious circle
principles which founded the predicativist program should be mentioned as
more full fledged forerunners to foundation.

By the way, it took many years for the EPR paradox to be dismantled in
physics.

Frode


2013/1/27 Vaughan Pratt <pratt at cs.stanford.edu>

> As far as I'm aware the passage from well-ordered sets to well-founded
> sets was first made by Dmitry Mirimanoff in 1917, see
> http://retro.seals.ch/digbib/**view?rid=ensmat-001:1917:19::**
> 170&id=hitlist&id2=&id3=<http://retro.seals.ch/digbib/view?rid=ensmat-001:1917:19::170&id=hitlist&id2=&id3=>
>
> In hindsight this is an easy connection to see, but before Mirimanoff it
> seems highly unlikely that anyone saw it.  In 1904 Koenig presented an
> argument that some sets could not be well-ordered, which within a day or so
> was shown to be unsound (the same fate that befell the
> Einstein-Podolsky-Rosen paradox a third of a century later).  Koenig
> continued to write about well-ordered sets in his 1914 book, which
> Mirimanoff acknowledges and then says "Je donnerai dans un autre travail
> les raisons qui m'ont determine a ne pas rattacher cette etude a la theorie
> de J. Konig."
>
> By this sentence I understand that Mirimanoff is claiming priority for the
> adaptation of well-ordering in general to well-founded sets in particular.
>
> Since this is decades after Cantor had contributed anything further of
> significance to set theory, it seems safe to infer from Mirimanoff's
> priority claim that Cantor had never given any thought whatsoever to the
> concept of a well-founded set, since otherwise people would have noticed
> long before Mirimanoff.
>
> The closest thing before well-founded sets would seem to be Russell's idea
> of a set that does not belong to itself, "un ensemble de premiere sorte" in
> Mirimanoff's preamble.
>
> Vaughan Pratt
>
> On 1/26/2013 3:32 AM, Frode Bjørdal wrote:
>
>> I have not studied Cantor's texts, but from what I recall I have heard
>> and seen conflicting accounts as to how and whether he implicitly
>> presupposed a well-founded notion of sets. Could some please
>>
>> (1) give textual evidence for him assuming i) well-foundedness, ii)
>> non-wellfoundedness and (perhaps) iii) full naivety,
>>
>> and
>>
>> (2) confirm textually that he presupposed extensionality?
>>
>>
>> Best regards from
>>
>> Frode
>>
>> ***********************************************************
>>
>>
>> Frode Bjørdal
>> Professor i filosofi
>> IFIKK, Universitetet i Oslo
>> www.hf.uio.no/ifikk/personer/**vit/fbjordal/index.html<http://www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html> <
>> http://www.hf.uio.no/ifikk/**personer/vit/fbjordal/index.**html<http://www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html>
>> >
>>
>> Questions on C
>>
>>
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-- 


Frode Bjørdal
Professor i filosofi
IFIKK, Universitetet i Oslowww.hf.uio.no/ifikk/personer/vit/fbjordal/index.html
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