[FOM] Questions on Cantor
Arik Hinkis
arikhinkis at gmail.com
Tue Jan 29 14:56:16 EST 2013
A set that contains itself was discussed in correspondence between Dedekind
and Weber in 1888. See Dugac 1976 p 273.
Isn't any well-order relation also a well-founded relation? In his 1883
Grundlagen Cantor characterized the order relation between his transfinite
numbers (later ordinals) in terms of its well-founded properties.
Well-order he characterized by a different property and he did not show
there that the properties are equivalent.
On Mon, Jan 28, 2013 at 9:46 AM, Vaughan Pratt <pratt at cs.stanford.edu>wrote:
> 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 hindsight.
>
> 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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