Correction
for
BUNGE, Br J Philos Sci XI (42) 153-156.
The British Journal for the Philosophy of Science 1960 XI(43):251; doi:10.1093/bjps/XI.43.251-s
© 1960 by British Society for the Philosophy of Science
Correction to RFM
Correction to RFM. On page 153 (9.1) of
RFMI I asserted that
Wittgenstein's remarks on the ambiguity of the notion of an
enumeration of (number theoretic) functions were unfounded.
This assertion was correct because he was explicitly concerned
with Cantor's set theory where a function
f is identified with
the set of pairs <
n,f(n) >, and the notion of enumeration
is unambiguous. However, his remarks can be given a little more
sense if an intensional notion of function (
rule of calculation)
is considered. If we take the familiar notion of a rule of calculation
expressed by a set of (recursion) equations, at least three
meanings of enumeration suggest themselves: (1) an enumeration
of the symbolic expressions of the rules, that is, of the equations,
m which distinct equations get distinct numbers; (2) one in
which distinct rules which define functions with the same course
of values, get the same numbers, others get different numbers;
(3) one which associates numbers not with the symbolic expressions
of the rules, but with (suitable finite) sets of values of the
functions defined by means of such rules. Now, (1) is evidently
possible, (3) is trivially impossible since the number associated
with a function depends only on a finite set of its values,
(2) is seen to be impossible by means of a not altogether trivial
argument,
1 if we require the enumeration to be given by a rule
of calculation too. And even though non-enumerability is established
both m senses (2) and (3), these senses are certainly different.
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