[FOM] A FOL truth behind the Cantor lemma

[David Auerbach]

Hi Dick,
 Yep. But that article of Thomson’s that I keep touting was previous. I always assign it in my intro puzzles, paradoxes, etc. class. I’m not sure if there is an earlier explicit rendition of the logical truth as such. But surely…  (Some years ago a student of mine got a tattoo of the Fregeschrift version.)
The Thomson piece is in a volume called Analytical Philosophy ed. by Butler; Cartwright’s “Propositions”, a early piece of Putnam’s, etc. are in there too. Here’s a snippet from near the beginning:


Let S be a set and R a binary relation defined on S or on
some set of which S is a subset. (' R is defined on S ' means that
for any x and y in S, not necessarily distinct, there is a clear sense
true or false in saying that x has R to y). Then to each x in S we
can associate a subset f(x) of S; f(x) is the set of all and only the
elements y in S such that Rxy holds. Let S' be that subset of S
which contains all and only the elements x in S which are not
elements of their associated subsets. Thus x is in S' if and only if
x is not in f(x). Then it is easy to see that S' is not the associated
subset of any element z in S, i.e., S' is not f(z) for any z in S. For
suppose it were: for each x in S, x is in f(z) if and only if x is not
in f(x), so, supposing z to be in S, it follows that z both is and is
not in f(z).
We can now state a small theorem :
(1) Let S be any set and R any relation defined at least on S.
Then no element of S has R to all and only those S-elements
which do not have R to themselves.
Although this result provides a foundation on which many
paradoxes are built, it is not itself in any way ' paradoxical ', but
a plain and simple logical truth. And this is more easily seen if we
notice just how much it says. In particular it does not deny that
there may be something which has R to all and only the S-elements
which do not have R to themselves; it says only that if there is
such a thing, it is not in S.
 
> On Aug 14, 2015, at 4:21 PM, Richard Grandy <rgrandy at rice.edu> wrote:
> 
>  There is a first order logical truth that seems to me to be an even more general and fundamental principle behind Cantor's lemma.
> 
> I don't remember where I first saw this noted, I vaguely think it may have been in something by Richard Cartwright.  It makes a nice exam question in either semantic or derivational version.
> 
> 
> Richard Grandy
> Philosophy & Cognitive Sciences
> Rice University
> Houston TX (still in the USA so far)
> 
> 
> 
> 
> <First order Cantor formula.pdf>
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David Auerbach                                                      auerbach at ncsu.edu
Department of Philosophy and Religious Studies
NCSU
Raleigh, NC 27695-8103

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