[FOM] Peirce and Cantor on continuity

[Matthew Moore (Philosophy)]

I just received the FOM digest with Vaughn Pratt's very helpful response
to my query, for
which I thank him. A couple of people have been kind enough to email me
directly to set me
straight on the ternary set. Professor Pratt is quite right: far from
being botched, Peirce's
description shows that he understands Cantor's construction well enough
to put it into a form
better suited to the readers of the *Century Dictionary*. I am sorry to
have bothered the list
with such an elementary question.

Just to clarify: I was trying to find the first two examples (rationals
or irrationals in any
interval) not in Peirce's writings, but in Cantor's. I believe that I
may have found the source
in Cantor's 1872 paper, just after his definition of a limit point. Here
is the passage (sorry,
I don't have Zermelo's edition in front of me, and so can't give the
page number):
	Besteht beispielsweise die Menge P aus allen Punkten der
Geraden,
	denen rationale Abszissen zwischen 0 und 1, die Grenzen ein-
oder
	ausgeschlossen, zukommen, so besteht die abgeleitete Menge P'
aus
	allen Punkten des Intervalles (0,1), die Grenzen 0 und 1 mit
	eingeschlossen.
This doesn't exactly answer to Peirce's characterization, but a
mathematician as astute as
Peirce could easily have derived those examples from this passage; and
there is independent
reason to think that he had seen this before he wrote the definition
that I quoted. (That
was his definition of 'continuity' for the *Century Dictionary*, by the
way, written around 1884.)

So I think I might now have the answers to all of my questions. I
certainly have been saved
from a real gaffe about the ternary set, and I've got at least a
reasonable hypothesis as to
the provenance of Peirce's examples. All that said, if anyone has any
pointers to other possible
sources in Cantor, I'd be most grateful.

Matthew Moore