The Sorites paradox and non-standard models of arithmeticDecember 16, 2008
A standard Sorites paradox might run as follows:
- 1 is small.
- For every n, if n is small then n+1 is small.
- There are non-small numbers.
On the face of it, these three principles are inconsistent, since the first two premisses entail that every number is small by the principle of induction. As far as I know, there is no theory of vagueness that gives us that these three sentences are true (and none of them false.) Nonetheless, it would be desirable if these sentences could be satisfied.
The principle of induction seems to do fine when we are dealing with precise concepts. Thus the induction schema for PA is fine, since it only says that it holds for properties definable in arithmetical vocabulary – all of which is precise. However, if we read the induction schema as open ended, that is, to hold even if we were to extend the language with new vocabulary, it is false. For it fails when we introduce into the language vague predicates.
The induction schema is usually proved by appealing to the fact that the naturals are well-ordered: every subset of the naturals has a least element. If the induction schema is going to fail if we allow vague sets, so should the well ordering principle. And that seems right: the set of large numbers doesn’t appear to have a least element – there is no first large number. So we have:
- The set of large numbers has no smallest member.
Again no theory I know of delivers this verdict. The best we get is with non classical logics, where it is at best vague whether there exists a least element of the set of large numbers.
Finally, I think we should also hold the following:
- For any particular number, n, you cannot assert that n is large.
That is, to assert of a given number, n, that it is large is to invite the Sorites paradox. You may assert that there exist large numbers, its just you can’t say exactly which they are. To assert that n is large, is to commit yourself to an inconsistency by standard Sorites reasoning, from n-1 true conditionals and the fact that 0 is not large.
The proposal I want to consider verifies all three of the bulletted points above. As it turns out, given a background of PA, the initial trio isn’t inconsistent after all. It’s merely -inconsistent (given we’re not assuming open ended induction.) But this doesn’t strike me as a bad thing in the context of vagueness, since after all, you can go through each of the natural numbers and convince me its not large by Sorites reasoning, but that shouldn’t shake my belief that there are large numbers.
-inconsistent theories are formally consistent with the PA axioms, and thus have models by Gödel’s completeness theorem. These are called non-standard models of arithmetic. They basically have all the sets of naturals the ordinary natural numbers have, except they admit more subsets of the naturals – they admit vague sets of natural numbers as well as the old precise sets. Intuitively this is right – when we only had precise sets we got into all sorts of trouble. We couldn’t even talk about the set of large numbers because it didn’t exist; it was a vague set.
What is interesting is that some of these new sets of natural numbers don’t have smallest members. In fact, the set of all non-standard elements is one of these sets, but there are many others. So my suggestion here is that the set of large numbers is one of these non-standard sets of naturals.
Finally, we don’t want to be able to assert that n is large, for any given n, since that would lead us to true contradiction (via a long series of conditionals.) The idea is we may assert that there are large numbers out there, but we just cannot say which ones. On first glance this might seem incoherent, however, it is just another case of -inconsistency. a numeral is formally consistent. For example, this is satisfied in any non-standard model of PA with L interpreted as the set of non-standard elements.
How to make sense of all this? Well, the first thing to bear in mind is that the non-standard models of arithmetic are not to be taken too seriously. They show that the view in question is consistent, and are also a good guide to seeing what sentences are in fact true. For example in a non-standard model the second order universally quantified induction axiom is false, since the second order quantifiers range over vague sets, however the induction schema is true, provided it only allows instances of properties definable in the language of arithmetic (this is how the schema is usually stated) since those instances define only precise sets. We should not think of the non-standard models as accurate guides to reality, however, since they are constructed from purely precise sets, of the kind ZFC deals with. For example, the set of non-standard elements is a precise set being used to model a vague set. Furthermore, the non-standard models are described as having an initial segment which are the “real” natural numbers, and then a block of non-standard naturals coming after them. The intended model of our theory shouldn’t have these extra elements, it should have the same numbers, just with more sets of numbers, vague and precise ones.
Another question is, which non-standard model makes the right (second order) sentences true? Since there are only countably many naturals, we can add a second order sentence stating this to our theory (we’d have to check it still means the same thing once the quantifiers range over vague sets as well.) This would force the model to be countable. Call the first order sentences true in the standard model plus the second order sentence saying the universe is countable, plus the statements: (i) 0 is small, (ii) for every n, if in is small, n+1 is small and (iii) there are non small numbers, T. T is still consistent (by the Lowenheim-Skolem theorem), and I think this will uniquely pick out our model as by a result from Skolem (I can’t quite remember the result right now, but maybe someone can correct me if its wrong.) This only gives us the interpretation for the second order quantifiers and the arithmetic vocabulary, obviously it won’t tell us how to interpret the vague vocabulary.