Linnebo's Trick

Last time, at the end, we talked about class models:

A set model looks like this %\[\langle D,\{\langle P,\{x| x\text{ is a person}\} \rangle ,\dotsc \}\rangle\]%

A class model looks like this %\[\langle \delta (x),\{\langle P, \ulcorner x\text{ is a person}\urcorner \rangle ,\dotsc \}\rangle\]%

Now, the background set theory is a model: It is %\[\langle x=x,\{\langle \in ,\ulcorner x\in y\urcorner \rangle\}\rangle.\]%

In addition, every set model is a class model. For example, the set model above is, re-expressed as a class model %\[\langle \ulcorner x\in D\urcorner,\{\langle P,\ulcorner x\in\{x| x\text{ is a person}\}\urcorner \rangle ,\dotsc \}\rangle\]%

I have defined class models of a theory of sets in the metalanguage of that theory of sets. My assertions aren't in the set theory, they are about the set theory. For example, in %\[\langle \in ,\ulcorner x\in y\urcorner \rangle ,\]% the definition of %$\in$%, the left-hand %$\in$% is in the object language, but the right-hand one is in the metalanguage.

Linnebo has noticed that putting the set-models into the object structure doesn't yield semantic closure. (Not surprising since even metalinguistically they didn't give the relevant semantics.) He therefore puts class-models into the object structure. That looks more promising initially because of cardinality considerations: there are no more properties of sets (that is, formulas with sets as parameters) than there are sets, and so, unlike building one set theory over another, we do not have a blow-up of size.

We take properties not to be sets. So, in addition to sets, we have nonsets, which are called urelements. We have in our language, besides %$\in$%, two new unary predicates: %$S$% and %$P$% for "is a set" and "is a property." Also, we have a new binary relation %$\eta$% that, intuitively, stands for "holds of." For example, if %$M$% is the membership property, then %$\langle a, b\rangle\eta M$% will be true if and only if %$a\in b$%. In the example that is like class models, that would be %$\langle a, b\rangle\eta \ulcorner x\in y\urcorner$%. In that case, we might read %$\eta$% as "is true of."

Here is an obvious, but false, principle concerning properties: %\[\exists x(P(x)\land\forall y(y\eta x\leftrightarrow \phi (x)),\]% where %$\phi$% is any formula. I don't know how to type a buzzer sound or one of those bamboo canes, but that is what is required by the formula %$\phi$% when it is taken to be %$\lnot x\eta x$%.

What to do? Well, Russell had a solution: types. He also had a kind of motivation for it: predicativity. We can't, it seems, do that here, since all the properties are already in the domain. However, there is an intuitive sense in which even if a property is already in the domain and hence another property is applied to it, it wasn't used in the definition. There is a rough and ready mathematical notion which is that S is the base of a function on V if defining the function of S determines the values on the rest of V. Linnebo's idea is that the definition of a property Q does not depend on the properties in a set D if there is a definition of Q from a basis that includes no members of D. Say that two objects are _indiscernibles with respect to a language L if exactly the same things are true of them in L. Say that a property Q is definable on a basis set S if

• Q is definable in our background language plus constant symbols for members of S. (Call that language %$L_S$%
• Every object that is not in S is indiscernible with some member of S in the language %$L_S$%.

How do we do this? Take the initial language %$L_0$% to be %$L_\varnothing$%. That doesn't quite work, because of %$\eta$%. So, I lied. We need to take %$L_S$% to be the language that only has constant symbols for members of %$S$% and that only includes %$\eta$% restricted to members of %$S$%. Now, %$L_1$% is the language %$L_{S_1}$% where %$S_1$% is the union of %$\varnothing$% with the properties defined in %$L_0$%. And so on.

Each stage of this language has a semantics that can be defined at the next stage, and the domain never changes. The central problem of the type view was that we had no way to say things across types. Can we say things about all properties? Sure, they are all in at the first level. Can we give a semantics for %$\eta$%? Well, we can give for each stage %$\alpha$% a semantics for %$\eta$% restricted to stage %$\alpha$% at stage %$\alpha + 1$%, and we can, Linnebo claims, quantify over the %$\alpha$%s. Thus, we can say things about %$\bigcup _{\alpha}\eta _\alpha$% which is %$\eta$%.

-- ShaughanLavine - 26 Feb 2009