# [FOM] Peirce and Cantor on continuity

Vaughan Pratt pratt at cs.stanford.edu
Wed Jan 21 17:21:16 EST 2009

Matthew Moore (Philosophy) wrote:
> 	As examples of a concatenated system not perfect, Cantor gives
> 	the rational and also the irrational numbers in any interval.

Another example would be the set D of decimal rationals, understood as
the formal language D = {0,1,...,9}^* of finite strings on that
alphabet.  D is concatenated and countable.

> 	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.

Call that example B = {0,9}^\omega, the infinite strings on {0,9}.  (I
thought Peirce used {0,1}, but no matter.)  B is perfect (according to
one of Peirce or Cantor: does Cantor give that example or did it come
from Peirce?), not concatenated, uncountable, yet still of measure zero
like D (not that the notion of measure was available to Cantor formally
but he might well have surmised an informal intuition about it.

> (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.)

It's in The Law of Mind, 1892 (see e.g. The Essential Peirce p.320).

> (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.)

This is the Cantor set stated for the decimal number system.  One would
only call that "botched" if one regarded the decimal number system as a
botched attempt at the ternary number system.

> 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
> appreciated.

D is evidently concatenated.  Whatever Cantor had in mind by perfect, if
his example B is intended to illustrate it (I was unable to connect the
example to the definition as stated by Peirce myself) then we can
suppose that any superset of B is also perfect, in particular DB, which
is also concatenated since D is.  But DB lacks 1/3, 1/7, etc. and so
cannot be the continuum.

Moreover since B is (at least intuitively in those days) of measure
zero, and a countable union of sets of measure zero has measure zero
(also surmisable informally by Cantor perhaps though the reasoning
is much trickier in more general cases), DB is an uncountable
perfect concatenated subset of [0,1] of measure zero.

Among Peirce's objections to Cantor's definition of a continuum is that
it is based on metrical considerations.  Cantor could have satisfied
Peirce here by forgetting the metric on Q and hence forgetting measure,
instead taking the rationals to be a linearly order set.  But DB even as
a metric counterexample still refutes Cantor because as a linear order
(definable lexicographically) it is not continuous by Dedekind's
cut-based criterion for continuity (modern terminology: completeness)
because the order would have the kind of cuts that would allow 1/3 to be
inserted, even though we could no longer say which particular cut was
1/3.  Topology (not yet invented) is not needed here, Dedekind cuts are
all that's needed to make that refutation work and these had been
available since 1872, twenty years before Peirce's The Law of Mind.

On the other hand, that the completeness property for the Dedekind cuts
in Q wasn't immediately recognized by Cantor and Peirce (if I've
interpreted the scenario correctly) as the route to defining the
continuum abstractly says something about the logical difficulties
people were having in defining the continuum back then.

Vaughan Pratt