[FOM] On_Myhill_on_Goedel_on_paradoxes?

[mlink at math.bu.edu]

_____________

"In relation to logic as opposed to mathematics, I believe that the unsolved
difficulties are mainly in connection with the intensional paradoxes (such
as the concept of not applying to itself) rather than with either the
extensional or the semantic paradoxes. In terms of the contrast between
bankruptcy and misunderstanding [MP:190-193], my view is that the paradoxes
in mathematics, which I dentify with set theory, are due to
misunderstanding, while logic, as far as its true principles are concerned,
is bankrupt on account of the intensional paradoxes. This observation by no
means intends to deny the fact that *some* of the principles of logic have
been *formulated* quite satisfactorily, in particular all those which are
used in the application of logic to the sciences including mathematics as it
has just been defined."
_____________


It seems that on your topic of properties, concepts, and theory
in connection to philosophy, in addition to what has already been
noted, there is also Goedel's Gibbs lecture (1951 in CW III,
particularly beginning on p. 311, which is p. 16 in the
original).  As you've mentioned, Goedel is sharply distinguishing
between extensional and intensional paradoxes.  There is the
strong statement of the paradoxes of set theory not being a
problem in his (1964, pp.\ 262-263).

He had been quite critical of logical intuitions in his (1944,
p. 131).  But he states the situation with respect to the
intensional paradoxes differently in other places.  Charles
Parsons says (forthcoming) that the "impression one gets from
[fragment Q] is that [the intensional] paradoxes are the main
obstacle to a satisfactory formal theory of concepts''.
According to Parsons, Goedel saw solving these paradoxes and
developing a theory of concepts as two aspects of the same issue,
but Goedel never developed a theory of concepts.  Although Goedel
does not specify what these intensional paradoxes might be,
Parsons suggests an intensional version of Russell's paradox, and
directs the teader to LJ, remarks 8.6.24-8.6.26.  Fragment Q
(partly in Wang (1977a) and LJ) has a lot on the theory of
concepts.

Also informative is his shorthand draft (1961? in CW III) in
which he asks: "In what manner, however, is it possible to extend
our knowledge of these abstract concepts, i.e., to make these
concepts themselves precise and to gain comprehensive and secure
insight into the fundamental relations that subsist among them,
i.e., [into] the axioms that hold for them?" (p. 383; 7 in the
original pages). He says that explicit definitions alone will not
work.

You may also have mentioned reflection principles, for which see
Wang (1977a) and fragment Q.  One difference, following Parsons
(forthcoming), is that Wang's account features sets and classes
whereas in fragment Q classes are derived from concepts and the
focus is on concepts.  W.N. Reinhardt (in a note on p. 189 of
"Remarks on reflection principles, large cardinals, and
elementary embeddings" of 1974) says that Goedel did not buy the
way Reinhardt justified extendible cardinals (see also Peter
Koellner ("On reflection principles", (2009), in particular
section 7)).

In regard to your question: "Perhaps all properties should be
thought of as concepts?", it bears noting that in at least one
place an analogy Goedel makes is between properties of concepts
and properties of matter (1953/9 in CW III, p. 9 in original).
The properties of concepts are as objective as the properties of
matter because, he says, each is composed of primitive
consituents and their properties.  Goedel seems to be focussed on
the objective reality of concepts, for which see Dagfinn
Follesdal's introduction to 1961 (CW III, particulary p. 369),
and the 3 February 1959 letter quoted there.  Goedel writes
(1944, p. 142): "Logic and mathematics (just as physics) are
built up on axioms with a real content which cannot be 'explained
away'".

I am no expert on Goedel.  I apologize for errors in the
preceding.

--ml