[FOM] Questions on Cantor
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.
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