THE INEFFECTIVENESS OF LOEWENHEIM-SKOLEM THEOREM

Loewenheim-Skolem theorem, in its strong form, states that if a theory formulated in first order logic has a non-denumerable model, there is a denumerable model of it which makes true all its sentences; that is, as Quine emphasizes, all objects of the model, except a denumerable quantity of them, are dead wood. In a certain sense, this theorem is obvious, since any communicable theory must be couched in a language with a denumerable number of possible sentences. But this is a limit of our language. It is not sure that in the world there are not a non-denumerable set of objects. For instance mathematical physics assumes that space-time is composed of a non-denumerable set of points. It is clear that this non-denumerability is inessential from a semantical point of view, since possible results of measurements are either finite or denumerably infinite, but the non-denumerability of points is strictly necessary in mathematical physics and it could be an effective characteristic of world.

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4 risposte a “THE INEFFECTIVENESS OF LOEWENHEIM-SKOLEM THEOREM

  1. I am not even sure that in the world there is an infinite number of objects. Clearly, real numbers are useful in thinking about the world; but of course this does not mean that they should be, or correspond to, “objects”. In fact, I doubt that the Löwenheim–Skolem theorem says anything at all about the world.

    By the way, the Löwenheim–Skolem theorem holds for theories with a countable language. You can have theories with an uncountable language (for example, it can be useful to have a name for each real number), and then all you can say is that there is a model whose cardinality is at most the cardinality of the language.

  2. Thank Angelo. Also I am not sure, but till now a realistic interpretation of space-time physics favor the existence of non-denumerable points. Sure LS does not say anything about the world, but it could be used to say that even if in the mathematical machinery of a theory considered at least partially true a non-denumerable infinity of points is presupposed, this non-denumerability is essential to models of that theory.
    I didn’t knew that there are languages with a non-denumerable set of terms. Where can I find some information about the topic?

  3. I would say that any sufficiently modern book on mathematical logic should work with arbitrary languages, as opposed to countable ones. Certainly any book that stresses model theory will do so, for example, the old classic “Model Theory” by Chang and Keisler.

    One case in which it is useful to have uncountable languages is in doing non-standard analysis: you want to have in your language names for all the n-relations on the real numbers, so your language will have a cardinality that is even higher than the continuum.

    I am not sure I understand the sentence “it could be used to say that even if in the mathematical machinery of a theory considered at least partially true a non-denumerable infinity of points is presupposed, this non-denumerability is essential to models of that theory.”

    Anyway, clearly, mathematical analysis, that is built on real numbers, says something about the real world, or experimental physicists wouldn’t be able to use it with such outstanding results. This does not mean, however, that each single real number corresponds to some object; what is relevant is the whole structure. And the Löwenheim–Skolem theorem is a meta-mathematical result, it does not apply within the structure itself.

    I mean the following. Cantor’s diagonal argument, which is, by the way, completely constructive, says that there are uncountably many real numbers. So, in any sufficiently rich theory, the real numbers are uncountable. All the Löwenheim–Skolem theorem says is the following: if you choose countable many functions and relations (for example, you could take all elementary functions), you can find a countable substructure in which all the function and relations in your list have the same properties that they have for the whole set of real numbers. There will be a bijective correspondence between the “new” real numbers and the natural numbers; however, by Cantor’s argument shows that this bijective correspondence can not be defined within the substructure. So, the substructure is “internally” uncountable, even if “externally” it is countable.

    At this point, it does not matter much which of the two structures you use to model reality: the real numbers will be uncountable in any case.

  4. OK, I understood. My moderate realistic approach to scientific theory make me believe that till now space-time is constituted of a non-denumerable quantity of point. I know that there are many nominalist philosophers which would deny for metaphysical reason the reality of a non-denumerable set of objects. I have the feeling that LS theorem could be used by these scholars, but what you say show me that this possibility is precluded.

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