[FOM] On Myhill on Gödel on paradoxes

[Frode Bjørdal]

>From chapter 8, in particular its section 5, in Hao Wang's book *A Logical
Journey* we learn that Gödel preoccupied himeslf with what he called a
"Concept Theory".  For example in the numbered paragraph 8.5.5 on p. 270
Wang (ealrlier also in from Mathemartics to Philosophy) summarized Gödels
view to his approval as follows:

"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."

Chapter 8 section 6 discusses Gödel's quest for a Concept Theory, and is
recommended

I feel that the term "concept theory" is much more appropriate than the term
"property theory". However, I have an analogous preference for my term "sort
theory". The term "concept" is quite prevalently also used for many objects
which cannot possibly be accounted for by theories which attempt to account
for paradoxes in as pure a manneras possible presupposing just a formal
language, and formal or super-formal (semi-formal) means.  Wang's book
contains examples where also Gödel invokes such extra-formal concepts. One
telling example is on p. 119 near the top of the page where Gödel comments
on his ontological proof (from CW 3, 433-437):

"1.  The proof must be grounded on the concept of *value* and on the axioms
for value....."

Perhaps Gödel envisioned a Concept Theory which indeed included all types of
concepts such as 'value', 'cause', positive', God', 'love' etc. ?

Perhaps all properties should be thoought of as concepts? But is the
property of *being in pain* a concept? I understand that such a question
belongs to the philosophical seminar, but this does not exclude that it may
also be of interest to many philosophically inclined mathematicians.

Wang is certainly an authority when it comes to Gödel's way of thinking. Was
John Myhill's remark, and its following influence, based on a
mis-remembering?
-- 


Frode Bjørdal
Professor i filosofi
IFIKK, Universitetet i Oslo

www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html
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