The Critical Rationalist                       Vol. 03  No. 01
ISSN: 1393-3809                                    [DRAFT: 11-Feb-1998]


next 2.9 An Example (Kepler's Laws)
previous 2.7 Logical Independence
contents

2.8 A Result in the Other Direction

(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 tex2html_wrap_inline1281 of all true statements is not axiomatizable, and t is axiomatizable and false, then its truth content tex2html_wrap_inline1285 is not axiomatizable. Phrased more generally, this asserts that if tex2html_wrap_inline1287 is an unaxiomatizable maximal theory that does not imply t, then tex2html_wrap_inline1291 is an unaxiomatizable subtheory of t. It may be shown that if t has infinite content then there must exist such an unaxiomatizable tex2html_wrap_inline1287 that does not imply t. It follows that tex2html_wrap_inline1291 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 tex2html_wrap_inline1291 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 tex2html_wrap_inline1287 does not imply it.

(24) First suppose that tex2html_wrap_inline1287 does not imply t. Since it is maximal, it implies tex2html_wrap_inline1191 . Now if tex2html_wrap_inline1291 were axiomatizable, so would be its conjunction with tex2html_wrap_inline1191 . But

displaymath1275

meaning that tex2html_wrap_inline1287 too would be axiomatizable. This proves Popper's corollary.

(25) If t has infinite content then, as noted in 2.3, tex2html_wrap_inline1191 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 tex2html_wrap_inline1069 (whether axiomatizable or not) has infinitely many axiomatizable maximal extensions tex2html_wrap_inline1337 , then it is consistent with each finite subset of tex2html_wrap_inline1339 , and hence (by the principle of finitude) tex2html_wrap_inline1341 is consistent. By Lindenbaum's theorem, tex2html_wrap_inline1069 has a maximal extension tex2html_wrap_inline1345 , which cannot be identical with any element of tex2html_wrap_inline1337 (for maximal theories are pairwise incompatible). Thus tex2html_wrap_inline1191 has an unaxiomatizable extension tex2html_wrap_inline1345 , which cannot also be an extension of t.



next 2.9 An Example (Kepler's Laws)
previous 2.7 Logical Independence
contents

The Critical Rationalist                       Vol. 03  No. 01
ISSN: 1393-3809                                    [DRAFT: 11-Feb-1998]


Copyright © 1998 All Rights Reserved.
TCR Issue Timestamp: Fri Mar 27 14:21:33 GMT 1998

tcr-editors@www.eeng.dcu.ie