Reals by Abstractiont

ions like the Direction equivalence and Hume’s principle and that it is reasonable to regard it as one. We might bring EM into line with the characterisation of abstraction principles with which I began by first defining an equivalence relation on ordered pairs of quantities: E[(a,b), (c,d)] ↔ ∀m,n (ma nb ↔ mc ⇔ nd), and then setting: Ratio(a,b) = Ratio(c,d) ↔ E[(a,b), (c,d)]. Alternatively, if it were felt desirable to avoid reliance on the notion of an ordered pair, we could introduce an extension of the notion of an equivalence relation so as to allow relations of arity greater than 2 to qualify as equivalence relations. Later we shall meet another abstraction principle which does not, as it stands, conform to the usual characterisation, but which may readily be brought into line in one or other of these ways. 15 Although I am not identifying quantities, as such, with numbers of any kind, it should be fairly clear that a full domain, and likewise the domain of ratios on it, is dense, and that we can develop an ‘arithmetic’ of ratios structurally analogous to that of the positive rationals. Reals by Abstraction 183 to any, much less all, (positive) irrational numbers.16 If ratio-abstraction is to yield all the positive reals, we require a complete domain. Indulging—for convenience, but avoidably—in set-theoretic language, we say that a subset S of quantities belonging to a q-domain Q is bounded above by b iff for every quantity a in S, a ≤ b. A quantity b ∈ Q is a least upper bound of S ⊆ Q iff b bounds S above & ∀c(c bounds S above → b ≤ c), and finally that a q-domain Q is complete iff Q is full and every bounded above non-empty S ⊆ Q has a least upper bound.