creating an infinite model

From the syntactic compactness theorem for first order logic, we get this nice (and useful) result:

Let T be a theory of first-order logic. If T has finite models of unboundedly large sizes, then T also has an infiniteMathworldPlanetmath model.


Define the propositionsPlanetmathPlanetmathPlanetmath


(Φn says “there exist (at least) n different elements in the world”). Note that


Define a new theory


For any finite subset 𝐓𝐓, we claim that 𝐓 is consistent: Indeed, 𝐓 contains axioms of T, along with finitely many of {Φn}n1. Let Φm correspond to the largest index appearing in 𝐓. If m𝐓 is a model of T with at least m elements (and by hypothesisMathworldPlanetmathPlanetmath, such as model exists), then m𝐓{Φm}𝐓.

So every finite subset of 𝐓 is consistent; by the compactness theorem for first-order logic, 𝐓 is consistent, and by Gödel’s completeness theorem for first-order logic it has a model . Then 𝐓𝐓, so is a model of T with infinitely many elements (Φn for any n, so has at least n elements for all n). ∎

Title creating an infinite model
Canonical name CreatingAnInfiniteModel
Date of creation 2013-03-22 12:44:29
Last modified on 2013-03-22 12:44:29
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Example
Classification msc 03B10
Classification msc 03C07
Related topic CompactnessTheoremForFirstOrderLogic
Related topic GettingModelsIModelsConstructedFromConstants