In order for mathematical fictionalists to be entitled to use the compactness theorem, they will have to show that the nominalistic formulation (Compact#) follows from the platonistic one (Compact).
Recall that the motivation for Field to use the compactness theorem was to reformulate conservativeness without having to assume abstract entities (namely, those required by the semantic and the proof-theoretic accounts of consistency).
In fact Robinson used the compactness theorem to build a structure R′ that is a model of exactly the same sentences of L as R, but which has infinitesimals.
Then, by using the compactness criterion and Schauder's fixed point theorem, we present an existence theorem for an asymptotically almost periodic solution to the addressed Volterra-type difference equation.
The proof of Theorem 1.5 From Lemma 2.1, Lemma 2.2, Lemma 3.1 and Lemma 3.2, using the compactness theory (cf. [17]), there is a sequence (still denoted as { u k } ) of { u k } such that when k → ∞, we have (4.1).
We can employ the compactness theorem to construct non-standard general models of the Peano postulates containing infinitely large numbers.
The fictionalist cannot rely on the conservativeness of mathematics to justify the use of a mathematical result (the compactness theorem) that is required for the formulation of the notion of conservativeness itself.
To derive the second statement from the first, note that 'T ⊨ φ' is true if and only if there is no model of the theory T ∪.) Anatolii Mal'tsev first gave the compactness theorem in 1938 (for first-order logic of any signature), and used it in 1940/1 to prove several theorems about groups; this seems to have been the first application of model theory within classical mathematics.
On the other hand, the failure of the compactness theorem to generalize to L ω1,ω) in any useful fashion is a severe drawback as far as applications are concerned.
This may be shown by appealing to the Compactness Theorem again.
As we have seen, the compactness theorem in its usual form fails for all infinitary languages.
