I have a question about compactness, non standard models of reals and illusory paradoxes.
Now, we know that because of the compactness theorem in FOL there are, for instance, non standard models of reals. In fact, if we call R the collection of all (and only) the true formulae in the standard model of real numbers, because of compactness there will be a model of R in which there are infinitely large numbers (ie. objects that do not respect the Archimedean property).
Proving that this holds true is extremely simple. Let us extend the signature of real numbers theory with a new constant, ∞. Let's create a new set of formulae and call it T, where T={n<∞|n∈N} and now consider the set of formulae R∪T. Every finite subset of R∪T has a model because the standard model obviously satisfies it (you just keep interpreting ∞ in a large enough real number to validate all the formulae in F). Hence, by the Compactness Theorem, R∪T has a model. This model will make true all the formulae in R, so it is a model of reals, but it is an alternative model of reals because it will make true all the formulae in T. So ∞ must be interpreted in an element of this new model which will be bigger (in the new, non standard, order relation) than any (non standard) natural number. ∞ will indeed be an infinite element that contradicts the Archimedean property for standard real numbers.
The problem is this. The "Archimedean property sentence", ie. the FOL formal sentence that in the standard model expresses the Archimedean property, being true in the standard model, is obviously one of the formulae in R, so it must also be true in the non-standard model.
There seems to be a contradiction between this sentence being true in the non-standard model, and the existence of a real number with the properties of ∞. I sort of understand that there really isn't a contradiction because, being a non standard model, there will be something in the interpretation of fundamental symbols that is different from the standard one in a way that dissolves the contradiction, but I am unable to point out exactly and with clarity why there isn't a contradiction. Can anyone help me? Thanks a lot.