[FOM] Questions on Cantor

Vaughan Pratt pratt at cs.stanford.edu
Mon Jan 28 02:46:10 EST 2013

On 1/27/2013 5:22 PM, Frode Bjørdal wrote:
> 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.)

There are three notions:

A.  That of a well-founded relation, meaning a pair (X, R) where X is a 
set and R is a binary relation on X.

B.  That of a well-founded set X.

C.  That of a well-ordered set (X, <) where < linearly orders X.

In hindsight, definitionally A precedes B and C because a well-founded 
set X can be defined as one for which the membership relation on the 
transitive closure of X is well-founded, while a well-ordered set (X, <) 
can be defined as a linearly (totally) ordered set whose ordering 
relation < (understood as strict) is a well-founded relation.

That B and C have simple definitions in terms of A as a common root 
makes it easy to see the connections between A, B, and C, at least in 

Chronologically the reverse is true: C (understood by Cantor) predates B 
by decades assuming the validity of Mirimanoff's 1917 priority claim, 
which I see no reason to doubt.  Bernays came after Mirimanoff, both in 
age (he was 27 years younger) and in considering well-founded sets, 
which he could not have done without Mirimanoff's invention of the 
concept.  (Bernays certainly didn't invent it himself.)

I don't know where A fits in chronologically.  Certainly no later than 
Mirimanoff 1917.  Anyone know of an earlier reference to well-founded 
relations?  But even if A predated B, B further requires the notion of 
transitive closure of a set, which even implicitly does not predate 
Mirimanoff as far as I know.  Russell's weaker "ensemble de premiere 
sorte" certainly does not require that notion (which as an aside could 
be taken as a benefit of Quine's NF suitably debugged).

I'm sympathetic to the notion that the above might not be immediately 
obvious from a literal reading of Mirimanoff's text.  Both the 
historical context (which Mirimanoff does quite a good job of 
explaining, at least up to 1917) and our modern understanding of the 
notions (which he certainly could not have anticipated) are needed to 
appreciate precisely what he contributed.

Vaughan Pratt

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