FOM: Proper Names and the Diagonal Proof

[friedman@math.ohio-state.edu]

Buckner writes:
 
> Cantor's proof has it otherwise.  It says that we cannot "match" a set of
> ordinary names (numerals) to a set of names formed by infinite decimal
> expansion.  Each row is "complete": it is a proper name formed out of an
> "infinite" collection of digits.  It assumes not just an infinite collection
> of names, but an infinite collection of names with infinite number of parts.
> But it's not clear which object is named by any name in the the sequence,
> since it is not clear what the name actually is.  Even if we suppose that
> "the square root of two" names something, what does "1.23245467 ..."
> name,when the dots signify not a specific method of carrying on the
> sequence, but no specific method at all.
..
> I'm not arguing against Cantor's proof, I'm saying I don't understand what
> it's meant to prove.

Cantor's diagonal theorem (in mathematics) does not involve names. One standard 
formulation is that in any infinite sequence of sets of natural numbers 
(indexed by the natural numbers), some set of natural numbers is missing. 

The progress of mathematics and science has necessary lead to the consideration 
of concepts that do not coincide with any that are commonly used in ordinary 
informal discourse. For instance, physicists and chemists do not convey their 
discoveries in terms of "fire" and "hot" and "wet" and "dry". It was only 
through great progress that "water" was, to a substantial extent, preserved.  

It is not clear whether mathematics (ultimately formulated in set theoretic 
terms) has an interpretation in ordinary informal discourse. I have conjectured 
that even set theory with large cardinals has a convincing interpretation in 
ordinary informal discourse, sufficient to prove its consistency (freedom from 
contradiction) within ordinary informal commonsense reasoning. However, it 
appears that we are some distance from establishing that conjecture.