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Is it too pedantic to ask, why in the definition of a structure in model theory sets are assigned to the relational symbols $P, R, ...$ of a language and not to corresponding formulas $Px, Rxy, ...$ (modulo choice of variables)? It would seem to me more consistent with the interpretation of arbitrary open formulas inside model theory and compared to set theory where by the comprehension axiom sets are assigned to open formulas, not to symbols.

Is it just a notational abbreviation - to name a set by $P$ instead of $Px$ - or is there something deeper behind it? If it's an abbreviation: Why is this so seldom (if ever) made explicit?

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How would you then define the meaning of P(f(x))? – Levon Haykazyan Jun 27 '11 at 9:24
What does this have to do with my question? – Hans Stricker Jun 27 '11 at 9:39
You suggested to change somehow the definition of a structure claiming that it will make matters more consistent. I tried to imagine how will this affect other basic definitions. I think that in defining the semantics of formulas (and by that I mean the relation ${\cal A} \models \phi$), you'll either need to define some complicated operations on sets or, at least implicitly, state that relations are assigned to relation symbols (not formulas). I don't think this is more consistent and wondered if you have a better definition. – Levon Haykazyan Jun 27 '11 at 11:43
@Levon: I didn't want to change the definition of a structure but asked for an alternative reading of the standard definition: wether the assignment of sets to symbols couldn't be understood as the assignment of sets to (equivalence classes of) formulas (just represented by symbols). – Hans Stricker Jun 27 '11 at 12:38
@Hans: There is a natural one-to-one correspondence between relation symbols and their corresponding formulas (modulo choice of variables). So from formalist point of view, there is no difference whether you assign something to the former or to the later. It is like talking about functions and their graphs. I interpreted your question as an aesthetic question of preferring one over the other. If this is not the case, I am not sure I understand your question, at least the first part. – Levon Haykazyan Jun 27 '11 at 13:25

I don't see why you think this would be more consistent, and I think the analogy to the comprehension axiom (schema) doesn't fit.

First, I wouldn't say that the axiom schema of comprehension assigns sets to formulas, at least not in the same sense as sets are assigned to symbols in the definition of a structure; rather, it generates one closed formula as an axiom for each open formula with at least two free variables, and this formula happens to assert the existence of a set with a certain property -- if anything can be said to be assigned here, I'd say a closed formula is assigned to each open formula with at least two free variables.

Second, the axiom schema of comprehension does this for every open formula with at least two free variables; in your case, you want to assign sets only to very specific formulas, namely the atomic formulas corresponding to each of the relational symbols. In fact, to be precise you'd need to assign sets to equivalence classes of these atomic formulas so you don't distinguish between free variables with different names. This problem doesn't occur with respect to the axiom schema of comprehension since there's no harm done in asserting the existence of a set multiple times with different variables names, which shows again that asserting the existence of sets and assigning sets are two quite different things. If you do form the equivalence classes, there's no longer any real difference between assigning a set to each relational symbol and assigning a set to each equivalence class of atomic formulas with relational symbols, since they're in one-to-one correspondence, so you'd just be complicating the matter without changing the content. The same is not true for the axiom schema of comprehension, which is fundamentally about formulas and can't be equivalently formulated with respect to (equivalence classes of) symbols.

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I agree with your explanation that the relational symbol $P$ is a perfect name for the equivalence class of formulas $Px$, $Py$, $Pz$, ... and I will take this for the answer. (I still see a difference between the symbol and the equivalence class of formulas which it names, and dare to suggest to take this difference serious.) – Hans Stricker Jun 27 '11 at 9:38
@Hans: That's not how I'd summarize my explanation :-) By saying $P$ is a perfect name for the equivalence class, you still seem to be implying that somehow it's more natural to think of the set being assigned to the equivalence class than to the symbol, and the symbol is just standing in for the equivalence class. I don't see it that way; it seems most natural to me to assign a set to each symbol, without any reference to the equivalence class. – joriki Jun 27 '11 at 9:43
The question then is: Is there a "measurable" difference between these two point of views? – Hans Stricker Jun 27 '11 at 9:47
@Hans: Well, there's no "measurable" difference inside the theory; that was one of my points; but there's a measurable difference in complexity between the two prescriptions "assign a set to each relational symbol" and "assign a set to each equivalence class of relational atomic formulas where two formulas are equivalent iff they differ only by renaming variables", so it seems there would have to be some reason to induce us to prefer the second over the first. In a sense, the first is preferable precisely because the two are the same :-) – joriki Jun 27 '11 at 9:58
And I find the second conceptually clearer and simpler, while then sets can always be seen as extensions of predicates (= formulas), restricted to the domain of discourse (which brings the comprehension axiom schema into play again). But maybe we should stop this discussion, it goes round in circles. – Hans Stricker Jun 27 '11 at 10:04

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