[FOM] Questions on Cantor

Vaughan Pratt pratt at cs.stanford.edu
Sun Jan 27 04:12:50 EST 2013

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

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 

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>
> Questions on C
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