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Unformatted text preview: Notes on Skolems Paradox and its Philosophical Implications Branden Fitelson 04/18/07 1 Hunter on the Paradox and Its Implications 1.1 Hunter on the Upward LST and Number-Theoretic Concepts I begin with some of the material from pages 205208 of Hunter. Here, Hunter provides some clarification of the technical results surrounding Skolems Paradox, and he offers his own resolution of the para- dox. First, he gives some results pertaining to the upward LST and number-theoretic concepts (two kinds). 49.1 . No first-order theory can have as its only model one whose domain is the set of natural numbers N . Proof. This follows from the key lemma [45.15] of the completeness theorem of QS . Every consistent first-order theory T has a Henkin model M T (in addition to its intended model M T , the domain of which is, say, N ), the domain of which is the set of closed terms of T . Closed terms are symbols, not numbers. So, the domain of M T is not N . [At least, we dont intend to be talking about numbers when we talk about Henkin models! Structuralists (like Quine) need not worry about this weak kind of ambiguity, so long as there is a structure-preserving 11 mapping an isomorphism between the two domains. Our proof of lemma 45.15 did not construct such an isomorphism. But, it seems that one should exist here, since both sets are denumerable, and the models are equivalent . Well return to these issues in depth, below.] 49.2 . Let T be any first-order theory. Fix the meanings of the logical connectives and quantifiers [the logical constants on which, see the Stanford Encyclopedia of Philosophy entry by our own John MacFarlane] in the usual way. And, let the axioms of T fix the meanings of the non-logical symbols (the prdicates, functions, etc .) of T to the extent that they can fix these meanings. Even if T has denumerably many axioms, the axioms of T cannot force us to interpret any predicate symbol in T as meaning is a natural number, and they cannot force us to interpret any expression in T as the name of a natural number. Proof. Again, this follows from 45.15, which says that T will have an unintended model M T M T . As such, any expression of T that is a name of a natural number (or means is a natural number) on the intended interpretation of T will be the name of some closed term of T (or will be a property whose extension is not any subset of N but some subset of the set of closed terms of T ) on M T . So, we are not forced to interpret any expression of T as the name of a natural number (or as meaning is a natural number). The moral here is supposed to be that we cannot unambiguously define numeral nouns using just the narrowly logical structure of any first-order theory. But, again, this weak sense of ambiguity (the intuitive lack of identity isomorphism between M T and M T ) is one that neednt bother a structuralist, since 45.15 is consistent with there being ansince 45....
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- Spring '07