Or, formal logic not a model theory?(adsbygoogle = window.adsbygoogle || []).push({});

(Sorry about the puns! Well, not really... )

For a while I've suspected what I consider to be a flaw in second-order logic (at least its interpretation), and it's recently been confirmed.

From Mathematical Logic by H.-D. Ebbinghaus, J. Flum, and W. Thomas, we have part of the definition of "satisfaction" for a second-order assignment [itex]\gamma[/itex] with [itex]\mathcal{J} = (\mathcal{U}, \gamma)[/itex]:

Forn-aryX:

[itex]\mathcal{J} \models \exists X \varphi[/itex] iff there is a [itex]C \subset A^n[/itex] such that [itex]\mathcal{J} \frac{C}{X} \models \varphi[/itex]

In other words, assuming I follow it correctly, the statement [itex]\exists X: \varphi[/itex] (whereXis a variable denoting ann-ary relation onA) is satisfied by a second-order assignment if and only if we can mapXto some subset of [itex]A^n[/itex] such that the result will be a model of [itex]\varphi[/itex].

This is a nice and dandy definition. I just think it's got the wrong idea.

The important problem, I think, is that the existential quantifer is interpreted as ranging over allset-theoreticrelations. In essence, I get the feeling that it's slipping ZFC into the theory under consideration through the "back-door", and suspect that all of the not-nice things about second-order logic (such as failing the compactness theorem) arise from this.

But let's step aside and consider one of my favorite examples: the theory of real closed fields.

There are several equivalent characterizations of a real closed field. IMHO, the simplest is that R is a real closed field iff it is an ordered field, and R(= [itex]R[x]/(x^2+1)[/itex]) is algebraically closed. Examples are the real algebraic numbers, the real numbers themselves, and IIRC, the set of real formal Puiseaux series. (Formal Laurent series in a k-th root of x. I think we consider x to be infinitessimal)

As we all know, the real numbersRform the onlycompleteordered field. But in some sense,allreal closed fields are complete ordered fields. Consider the following theorem:

Theorem: LetRbe a real closed field. Let [itex]\phi[/itex] and [itex]\psi[/itex] be unary relations onRin the (first-order) language of real closed fields such that neither is identically false, and [itex]\phi(x) \wedge \psi(y) \implies x \leq y[/itex]. Then, [itex]\exists y: \forall x: (\phi(x) \implies x \leq y) \wedge (\psi(x) \implies x \geq y)[/itex].

If we let [itex]\phi[/itex] and [itex]\psi[/itex] range over all subsets ofR, then this theorem would simply be stating thatRis Dedekind complete, and would be false in general. But, by restricting ourselves only to the subsets ofRthat can be defined in the theory of real closed fields, weareable to prove Dedekind completeness!

By restricting ourselves to "internal" subsets like this -- that is, subsets that can be defined as relations in the language of the theory -- we can sometimes to useful things. For example, for any real closed fieldR, [itex]R^n[/itex] has a beautiful geometric structure... as long as you pretend that only semi-algebraic sets exist! (sets defined by equations and inequalities)

Now back to my criticism of second-order logic: in light of the above, it seems that the theorem I quoted above about the "Dedekind completeness" of any real closed field ought to be a theorem in the second-order theory of real closed fields, because it's a statement aboutallfirst-order relations.

But, due to the way second-order logic is done, the universal quantifier is allowed to run over all set-theoretic relations (i.e. all subsets), and not just the internal relations that can be defined in the first-order theory (i.e. semi-algebraic subsets)

So this prompts a question: is there a currently developed theory that remedies this "problem"? I imagine that would merely require that an interpretation of a second-order theory to explicitly specify the domain of any relation variables, rather than assuming the domain to be the collection of all set-theoretic relations. I would like to not have to rediscover it, if it exists already.

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# Dissatisfaction with second-order logic

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