The Critical Rationalist Vol. 03 No. 01 ISSN: 1393-3809 [DRAFT: 11-Feb-1998]
(27) Lying behind the view that scientific theories
are infinitely varied in their consequences there is, I
suspect, an atomistic view of content: the idea that
there exist minimal independent morsels of information
from which the contents of all more informative
statements and theories are compounded by finite or
infinite conjunction. The reader is warned not to be
misled by the use of the word `bit' in information
theory to express what sounds like exactly this idea.
It is not the same idea. Bits are not minimal, except
in the logically insignificant sense of being
representable by expressions of minimal length. Indeed,
the atomistic thesis as it stands is untenable. For
the only possible candidates for the role of atomic
contents would be irreducible statements, and no
axiomatizable theory with infinite content can be built
entirely from irreducibles. This is easily shown. For
a set of k irreducibles generates a theory with
distinct consequences; and an axiomatizable
theory that is logically equivalent to the conjunction
of denumerably many irreducible statements is, by
finitude, equivalent to the conjunction of some finite
subset of them, so that we return to the previous case.
A simple example is supplied by the calculus described
in 2.6, elementary logic with identity
as the only relation: for each positive i the
statement
is irreducible, but the
conjunction of all these statements yields a theory
(`the number of objects is not finite', or
`the universe is infinite') that is not finitely
axiomatizable.
(28) The objection may be strengthened by noting that in many calculi there exist no irreducible theories at all. This follows from the result of Mostowski already cited. An example is provided by ordinary classical sentential calculus with denumerably many sentence letters. In such calculi, of course, all non-tautological theories have infinite content.
The Critical Rationalist Vol. 03 No. 01 ISSN: 1393-3809 [DRAFT: 11-Feb-1998]
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TCR Issue Timestamp: Fri Mar 27 14:21:33 GMT 1998