[FOM] On_Myhill_on_Goedel_on_paradoxes?

[Frode Bjørdal]

The posting of mlink at bu.edu.math very much bolsters the view that
Gödel was questing for a *theory of concepts*, and, indirectly, that
Myhill had it wrong when he ascribed to Gödel the view that there
still were *property theoretic* paradoxes. The terms are close, and so
such a subtle mis-remembrance can very much be understood.

The term 'concept' is philosophically very loaded and has a strong
lore, see e.g. http://plato.stanford.edu/entries/concepts/ and
http://en.wikipedia.org/wiki/Concept. The same can be said for the
term 'property'. Both terms also have everyday usages interconnected
with the philosophical ones which makes it very awkward, almost
repellent, to think of theories that attempt to approximate naive
abstraction type-freely as either concept-theories or
property-theories. In everyday language we talk about the concept of
'blue' and the property of 'being in pain', and it is for such reasons
to my mind very misleading to offer theories built upon pure formal
languages as either property theories or concept theories.

Alternatively, in these days where we also have on offer alternative
set theories in which both wellfoundedness and extensionality fail,
one might think that type-free accounts approximating naive
abstraction might be considered precisely that: alternative set
theories. I believe such a choice of terminology would be disingenious
for the librationist theory I offered a small introduction to in a
posting on f.o.m. on June 13 this year:

http://www.cs.nyu.edu/pipermail/fom/2011-June/015571.html

A third *caveat lector*, which slipped my mind and which unfortunately
did not make it through the proof-reading, is that the objects
isolated in the librationist framework are to be considered *sorts*.
Those sorts that are well-behaved in not being paradoxical are called
*kinds*. Kinds that are also hereditary kinds are called *goods*. Even
for goods, it turns out that extensionality fails. If we e.g. isolate
the least Jensen closure of von Neumann's omega, JCO, (which exists
according to librationism and is a good), the goods in JCO may for
contextual purposes be considered *sets* as extensionality relative to
JCO holds. In certain settings extensionality may be simulated by
bisimulation. New terminology is called for in order to make useful
and appropriate distinctions.

-- 
Frode Bjørdal
Professor i filosofi
IFIKK, Universitetet i Oslo
www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html