Peirce's version of the proof for Cantor's theorem can be mapped in a quite
straightforward way to the structure of the New List of 1867. At the same
time
the proof of Cantor's theorem can be extended by continued diagonalization
(which latter, by the way, Peirce discovered not later than 1867 and under
a
different name and in a much more general form than afterwards discovered
and
used by Georg Cantor, Kurt Goedel and Alan Turing) to a derivation of the
system
of Existential Graphs, which can thus be seen, as Peirce himself said, to
be
"expressive of the properties of the continuum" and fulfills the criteria
Peirce
gave for "true continuity", namely "Kanticity" and "Aristotelicity".
I could probably show in strict terms what the above means, but this does
not
seem to me to make any sense in an email forum, since it involves "a lot
of"
logic and mathematics and is by no means impossible, but difficult to
express in
words. Anyway, I've written it down and so maybe one day... . One of the
main
difficulties is perhaps generally, that it is impossible to understand
Peirce
from a "set theoretical" point of view (even if this be "only used as a
language" and however implicitly) and it seems to me equally and definitely
impossible to understand Peirce's continuum in terms of any form of
"nonstandard
analysis".
This sounds perhaps complicated, but it is in fact simple and only
difficult to
understand, as it seems. Anyway, this is the end of the road for me, since
I
surprisingly found what I have been looking for over long years and Peirce,
according to my understanding, is not so much about a "body of knowledge",
but
what he found out is meant to be used and that's the only meaning it has.
So I
leave it at this point and shall now do something completely different.
Let me finish with two concluding remarks: What regards a "fourth
category",
this means for me to simply go into the wrong direction. A reduction to two
categories might be "progress", but Kant already tried that, as is well
known,
and he failed.
Secondly, Douglas Adams once described how "flying" works: "You throw
yourself
at the ground, and miss it completely". This seems to me to apply
beautifully to
induction in particular and signs in general, too;-)
Bye,
Thomas.
P.S. I might be completely wrong of course.
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