[FOM] Remarks on the Concept Calculus

Frode Bjørdal frode.bjordal at ifikk.uio.no
Sun Aug 11 11:48:32 EDT 2013

In the very interesting Fr(11) Harvey Friedman isolates a concept calculus
with notions < for «better than» and << for «much better than». He shows
that the theory MBT for «much better than» interprets ZF. MBT is B+DE+SUI
(see Fr(11) p. 13).

In interpreting < and << informally Friedmann invokes mereological notions
so that x<y suggests «x is a part of y» and x<<y suggests «x is an
infinitesimal part of y».

It occurs to me that we in interpreting MBT should take inspiration from
Friedman's important article Fr(73), where he shows in Theorem 1 that a
system S which is ZF with collection minus extensionality and a weak power
set principle interprets ZF. The failure of extensionality of the
interpreting theory may in the context of Concept Calculus be taken as
reflected in MBP by DIVERSE EXACTNESS (DE). The domain of discourse of MBT
may be taken as "ordinals" which are well founded, transitive and only
contain transitive sets. a<b accordingly signifies that a is an element of
b. << is only partially implicitly defined by Fr(11), but perhaps one could
simply define a<<b to hold iff a<b and there is a c s.t. a<c<b and for all
c s.t. a<c<b there is a d s.t. c<d<b. STRONG UNLIMITED IMPROVEMENT (SUI),
Fr(11) p. 13, then corresponds to a reflection principle.

It seems to me that an interpretation of Concept Calculus along such lines
as here better connects the Concept Calculus with ordinal hierarchies.

Fr(73): Harvey Friedman, The Consistency of Classical Set Theory Relative
to a Set Theory with Intuitionistic Logic in The Journal of Symbolic Logic,
Vol. 38, No. 2 (Jun., 1973), pp. 315-319

Fr(11): Harvey Friedman, Concept Calculus: Much Better Than, in: Infinity,
New Research Frontiers, ed. Michael Heller, W. Hugh Woodin, Cambridge
University Press, 2011, p. 130 - 164. In references here I use the
pagination of the preprint

Professor Dr. Frode Bjørdal
Universitetet i Oslo Universidade Federal do Rio Grande do Norte
quicumque vult hinc potest accedere ad paginam virtualem
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