[FOM] Peirce and Cantor on continuity

Matthew Moore (Philosophy) MatthewM at brooklyn.cuny.edu
Thu Jan 22 20:59:04 EST 2009

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
	denen rationale Abszissen zwischen 0 und 1, die Grenzen ein-
	ausgeschlossen, zukommen, so besteht die abgeleitete Menge P'
	allen Punkten des Intervalles (0,1), die Grenzen 0 und 1 mit
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

More information about the FOM mailing list