[FOM] Diagonalization and Self-Reference

[Sandy Hodges]

I am still unclear whether it is self-reference, or just reference to
left and right-hand sides, that diagonalization does not deliver.

When the {x e N: f(x)} notation for 'the set of x in N such that f(x)'
is introduced by a definition, a formula including this notation is not
itself a formula of the object language, but is a sub-formula (that is,
a noun phrase) of the meta-language, which phrase serves as a name of
the corresponding object-language formula.    For example:

{x e N: f(x)} e z

is the meta-language name of the object language formula:

(E s) [ (\/ x) ( x e s <=> x e N & f(x) ) & s e z ]

Now it is of course possible that the meta-language name of a formula
could be of the form (A <=> B), without the formula itself being of (X
<=> Y) form.     If you had such a name, and referred to the left hand
side of it, and meant to refer to the left hand side of the
object-language formula itself, your reference would of course fail.

Mathematical language, even very precise mathematical language, uses
formulas such as: "{x e N: f(x)} e z."    The claim is made that a
language adequate to mathematics needs only "\/", and "E", and concepts
such as "{x e N: f(x)}" can be defined.  But such definitions are not
satisfactory in any case, as they result in meta-language formulas, and
the meta-language is not so well specified.   When the object language
can refer, by Gödel numbers or otherwise, to its own formulas, the
inadequacy of contextual definitions (such as that of "{x e N: f(x)}")
is even clearer.

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Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.