[FOM] Peirce and Cantor on continuity

Matthew Moore (Philosophy) MatthewM at brooklyn.cuny.edu
Sun Jan 18 12:56:28 EST 2009

I need some historical and technical assistance with a paper I am
writing on C.S. Peirce's reading of Cantor's works. In his definition
of 'continuity' for the Century Dictionary, Peirce clearly relies on
Cantor's definition of a continuous point set from the Grundlagen of
1883. Having (more or less accurately) explained Cantor's definitions
of perfect and connected set, Peirce says:
	As examples of a concatenated system not perfect, Cantor gives 
	the rational and also the irrational numbers in any interval.
	As an example of a perfect system not concatenated, he gives
	all the numbers whose expression in decimals, however far
	carried out, would contain no figures except 0 and 9.
('Concatenated' is Peirce's translation of 'zusammenhaengenden'; he
was working from the French translation in Acta Mathematica, which has

My questions have to do with the three examples he gives in this

(1) Are the first two examples (the rational and irrational numbers in
    any interval) to be found anywhere in Cantor's writings? (I
    haven't been able to find them.)

(2) The last example looks to me like a botched description of
    Cantor's ternary set; does anyone know of anything else close to
    this in Cantor? (Again, I haven't been able to find it.)

Also, I've been having a hell of a time figuring out whether the last
example is perfect or connected, so any help with that would be much


Matthew Moore

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