The Critical Rationalist Vol. 03 No. 01 ISSN: 1393-3809 [DRAFT: 11-Feb-1998]
(21) Yet the idea that infinite content implies the existence of infinitely many independent nuggets of information can be pressed a little further. Using a corollary stated (but not explicitly proved) by Popper (1966, p. 349), it is possible to establish the somewhat unexpected proposition that, although t is not equivalent to any infinite independent subset, under the conditions already propounded, it does include, within its content, such a set. This shows that there is one sense (though one that I shall claim to be unimportant) in which we can correctly assert that t does have infinitely many different things to say. Those who want to concentrate on the main problem of the paper, and those who can't be bothered with proofs, are once again permitted to move on, either to 2.10 or, if they are desperate, directly to section 3.
(22) The result of Popper's that we need is this: if
the set of all true statements is not
axiomatizable, and t is axiomatizable and false, then
its truth content
is not axiomatizable.
Phrased more generally, this asserts that if
is an unaxiomatizable maximal theory that does not
imply t, then
is an
unaxiomatizable subtheory of t. It may be shown that
if t has infinite content then there must exist such
an unaxiomatizable
that does not imply t. It
follows that
is not
axiomatizable. Now by a theorem of Tarski (proved
informally on p. 362 of
1935), every
theory is logically equivalent to an independent set,
and hence
must be equivalent to
an infinite independent set (for otherwise, as already
noted, it would be finitely axiomatizable).
(23) To complete the proof we need to establish the
result attributed to Popper, and to establish also that
if t has infinite content then at least one
unaxiomatizable maximal theory does not imply
it.
(24) First suppose that does not imply t.
Since it is maximal, it implies
. Now if
were axiomatizable, so would be its
conjunction with
. But
meaning that too would be axiomatizable. This proves
Popper's corollary.
(25) If t has infinite content then, as noted in
2.3, has an infinite number
of maximal extensions. That these cannot all be
axiomatizable is part of Theorem 8 of (Mostowski
1937; Tarski
1936,
p. 370). Here is the simple proof. If a theory
(whether axiomatizable or not) has infinitely many
axiomatizable maximal extensions
, then it is consistent with each finite
subset of
, and
hence (by the principle of finitude)
is consistent.
By Lindenbaum's theorem,
has a maximal extension
, which cannot be identical with any element of
(for maximal
theories are pairwise incompatible). Thus
has an unaxiomatizable extension
, which cannot
also be an extension of t.
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