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• W2334156613 abstract "sort I propose to help alleviate the problem. Adherents of such views will have to find some other program for epistemological studies. Leaving this possibility aside, what ideas are there in the classical positions which might help us carry out the program I propose? The chief virtue of formalism (aside being precise enough to refute!) seems to be that it uncovered an interesting dass of self verifying propositions, that is propositions which if true can be known to be true (namely the r.e. formulas of arithmetic). Unfortunately, this does not seem to carry us much further. It would not be surprising, however, if this perspective could lead to a proof of the consistency of 2) with U an r.e. formula. Nevertheless, although there are various schemas of recursive ordinals, it is not clear that these can be turned to the purposes at hand with insights from below. Platonistic and even conceptualistic perspectives do suggest other possibilities. At the very least, they suggest working in a richer framework for carrying out the program. For example, it may be that there is a more general theory of humanly intelligible concepts: perhaps at least one for each ordinal. (Compare to G6del's suggestion in Davis (1946, p. 85)) that the ordinal definable sets correspond to the humanly definable concepts. It is by no means obvious that this would be incompatible with the Post-Turing thesis!) From the Platonistic perspective, there is the possibility of tying concepts and proofs to truth and more generally to the real world. While it is not clear exactly how this would go, we do know that what is definable depends on the structure of the real world; it is plausible to think that the concepts which are available depend on what can be defined, and that the proofs that are available depend in turn on the available concepts. It is in any case very plausible that the notion of provability should be dependent on the notion of truth (compare the traditional true belief and the more recent Gettier inspired hereditarily justified true belief). A suggestion the Platonistic tradition (specifically, G6del in Davis (1946, p. 85) which has never been followed up is that axioms might be the instances of certain r.e. schemas which in addition happen to be true. Again, it is by no means obvious that this would be incompatible with the Post-Turing thesis. Another possibility is suggested by George Bealer's approach to the correspondence between thought and reality (1982). Briefly, there are ways of combining concepts (thoughts), perhaps analogous to syntactic operations, but one starts with concepts which are part of the actual structure of reality (qualities and connections). The latter satisfy (at least according to one theory) a weak extensionality This content downloaded 157.55.39.35 on Thu, 01 Sep 2016 04:57:00 UTC All use subject to http://about.jstor.org/terms INCOMPLETENESS THEOREMS 339 condition: they are the same if they necessarily have the same extension. Again, it is not apparent that anything in this would violate the Post-Turing thesis. (It is true that if the logical necessities satisfy S5, i.e. possible implies necessarily possible, then concepts having necessarily the same extension turn out not r.e. However, the qualities and connections are a restricted class of concepts.) I have emphasized that the results pose problems for epistemology generally and not merely for formalist or nominalist epistemology. I do not wish to leave the impression that the nominalist has no special difficulties. The nominalist can of course reject the framework into which I have cast the epistemological issues (for example by denying the meaningfulness of 'it is provable that'). This seems rather unnatural and unintuitive, however, as would rejection of its meaningful applicability to arithmetic sentences, or its iterability in this context. Moreover, it does not seem to be necessary in order to preserve the mechanist thesis. (Some such move would be required if one wishes to present any particular formal system as an analysis of B -namely, then one must at least reject natural axioms for B. But such an analysis is surely very implausible.) If we do cast the epistemological issues into the framework I advocate, then the nominalist does have special problems. The reason for this is that for the nominalist the natural theoretical framework is arithmetic (or something close) and not some esoteric theory involving truth, predication, or other metaphysical notions. The basic problem is not specifically epistemological. It is this. Since the nominalist theory of concepts is the syntactic theory (I take it this excludes the concept of truth), the nominalist is stuck with the fixed point theorems in a special way. This is best explained by contrast with a theory which allows propositions. In such a theory [p] will denote the proposition expressed by p (if there is one) instead of the sentence expressing it. But now one can distinguish between the case where p expresses a proposition, and the case where it does not. There is the possibility of natural restrictions blocking or disarming the fixed point theorems. To be more specific, let us consider an almost-nominalistic theory of propositions. The theory is framed in terms of Kripke's theory of partial truth (Kripke, 1975). An elegant version of this theory which will do for our purposes is due to Feferman. Consider a classical first order language L extending ordinary arithmetic, with predicates T for true and F for false; T is supposed to include the true atomic sentence of arithmetic, F to include the false ones. The special axioms for T and F are the Kripke-Kleene rules. For example a This content downloaded 157.55.39.35 on Thu, 01 Sep 2016 04:57:00 UTC All use subject to http://about.jstor.org/terms" @default.
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• W2334156613 title "Absolute Versions of Incompleteness Theorems" @default.
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