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In their paper, "On Second-Order Characterizability", Hyttinen, Kangas, and V$\ddot a$$\ddot a$n$\ddot a$nen define that notion as follows:

Let us call a structure $\mathfrak U$ second order characterizable if there is a second order sentence $\varphi$ in the vocabulary of $\mathfrak U$ such that

$\mathfrak B$$\vDash$$\varphi$ $\Leftrightarrow$$\mathfrak U$$\cong$$\mathfrak B$ [where, I believe, "$\cong$" means "isomorphic to" though their paper does not explicitly say that--my comment] for all structures $\mathfrak B$.

The theory $T$, containing such second-order sentences $\varphi$ as axioms (along with first-order axioms) from the language $\mathscr L$ formalizing the vocabulary of $\mathfrak U$ would therefore be categorical.

As is well-known, such a second-order theory $T$ has two types of semantics; full models (where the second-order variables range over all subsets and relations of the domain), and Henkin models (which make $T$ a many-sorted first-order theory). The authors, in section 4 of their paper, define the distinction as follows (using monadic second-order logic as a concrete example):

Monadic second order logic over a structure ($A$, $R_0$,...,$R_n$) can be thought of as first order logic over the enhanced structure

(1) ($A$$\cup$$P_1$, $A$, $P_1$, $\in$, $R_0$,...,$R_n$)

where $P_1$=$\mathcal P$($A$) [ which might be the power set of some ambient set theory--my comment] and $\in$ is restricted to $A$$\times$$\mathcal P$($A$). Respectively, monadic third order logic can be reduced to first order logic of

(2) ($A$$\cup$$P_1$$\cup$$P_2$, $A$, $P_1$, $P_2$, $\in$, $R_0$,...,$R_n$)

where $P_2$=$\mathcal P$($P_1$). For non-monadic higher-order logics similar translations exist. One can reduce the entire type theory to first order logic in this way. The price one pays is that structures are limited to to the very special form of (1) and (2). Henkin...took the natural step of considering the following more general structures than (1):

(3) ($A$$\cup$$P_1$, $A$, $P_1$, $E$, $R_0$,...,$R_n$), nary predicate $\subseteq$ $A$$\times$$P_1$ satisfying the Extensio where $E$ is just a binary predicate $\subseteq$ $A$$\times$$P_1$ satisfying the Extensionality Axiom. In addition, the Comprehension Axioms from [Hilbert and Ackermann's Grundz$\ddot u$ge der theoretischen Logik--my comment] are assumed. The Comprehension Axioms say that any definable relation on $A$ is canonically represented by an element of $P_1$. Henkin [in his paper, "Completeness in the theory of types"--my comment] proved that such models yield a Completeness Theorem for second (or higher) order logic with respect to the obvious rules of inference that were introduced in [the Grundz$\ddot u$ge--my comment]. The original model (1), called the full model, is of course a special case of (3) and satisfies the Comprehension Axioms. Thus results about the models (1) can be considered generalizations of results about the models (3). When we prove existence results this generality means that our results are weaker than corresponding results about full models. However, our results use respectively weaker assumptions.

It is in this context that one can make sense of the following remark, made in Section 1 regarding Section 4:

In Section 4 we study second order characterizability in the more general framework of normal models, that is, general models in the sense of Henkin [in his paper, "Completeness in the theory of types"] satisfying the Comprehension axioms of Hilbert and Ackermann [in the Grundz$\ddot u$ge]. The more general framework permits us to get results without cardinal arithmetic assumptions. The lesson then is, that non-categoricity of second order theories is a common phenomenon but if we want to manifest non-categoricity by means of full models (models in which the second order variables range over all subsets and relations of the domain), we have to make cardinal arithmetic assumptions or use forcing.

My purely philosophical concern is as follows....

Consider Hyttinen, Kangas, and V$\ddot a$$\ddot a$n$\ddot a$nen's description of a Henkin model:

($A$$\cup$$P_1$, $A$, $P_1$, $E$, $R_0$,...,$R_n$), where $E$ is a binary predicate $\subseteq$$A$$\times$$P_1$ satisfying the Extensionality Axiom, and $P_1$=$\mathcal P$($A$), the power set of $A$. Via analogy (as I said, this is a philosophical argument), one can define the generalized model for monadic third-order logic as follows:

($A$$\cup$$P_1$$\cup$$P_2$, $A$, $P_1$, $P_2$, $E$, $R_0$,...,$R_n$), where $E$ is a binary predicate $\subseteq$ $A$$\times$$P_1$ and $E$$\subseteq$$P_1$$\times$$P_2$ satisfying the extensionality axiom, and where $P_2$=$\mathcal P$($P_1$) (where $\mathcal P$ is the power set operator),

and so on until the whole of type theory is reduced to first order logic (and a rudimentary cumulative hierarchy is produced from $A$$\cup$$P_1$$\cup$$P_2$$\cup$..., which I will denote as the proper class ($V$, $E$), so as to avoid the Burali-Forti Paradox). It should be noted that ($V^{'}$, $\in$), which the authors (presumably) would denote as the cumulative hierarchy of the full models (as defined in their paper), is a special case of ($V$, $E$) (to simplify matters, I will let $A$ be the class of urelements satisfying $R_0$,...,$R_n$), and the cumulative hierarchy ($V^{'}$, $\in$) will contain every set since naively, a set is just a class which can be an element of another class). Now by Mostowski's Collapsing Theorem (assuming $E$ is wellfounded, which, by my extended definition, is certainly reasonable), there is a transitive class $M$ and an isomorphism $\pi$ between ($V$, $E$) and ($M$, $\in$) and the transitive class $M$ and the isomorphism $\pi$ are unique. From this it is clear that $V$$\subseteq$$V^{'}$ (since $E$$\subseteq$$\in$). But, (as regards forcing) one seems to be in the position Carl Mummert states allows for the contradiction user8996 found (see the mathoverflow question, "Generic filter over $V$"):

...the contradiction you mention only happens if you also assume our $V$ contains every set [my ($V^{'}$, $\in$), formed from the cumulative hierarchy of the first-order characterization of the full model of type theory]. In the sort of systems where you can consistently force over $V$ (e.g. the Gitman/Hamkins multiverse) that will not be the case [i.e. the universes of the Gitman/Hamkins multiverse are the cumulative hierarchies ($V$, $E$) of type theory, i.e., the Henkin models of type theory].

Question: Regarding forcing, is this the correct assessment (conceptually) of the situation (that it is seems to be confirmed by Herbert Enderton in his SEP entry "Second-order and Higher-order Logic" when he refers to the full models as defined in "On Second-order Characterizability" as "Absolute")? But if this is the case, then how can one properly define forcing on such full models? Andreas Blass seems to provide the correct path in his answer to user8996's question by reference to Boolean-valued models but then how would one properly define the notion of Second-order Characterizability in terms of Boolean-valued models so as to allow forcing on full models of a first-order reduction of a second-order theory as described in Section 4 of "On Second-Order Characterizability"?

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  • $\begingroup$ You seem to be asking two questions - one about developing forcing over models of set theory defined via some second-order conditions (which I'm honestly not too clear on - I think it amounts to the "true" universe $V$, so the problem (as usual) is that by definition the "true" universe has no extensions, forcing or otherwise, since it already contains every set), and another about second-order characterizability and forcing (which I don't actually see - but I don't see the role of second-order characterizability in your main question). Can you clarify what exactly you are asking? $\endgroup$ Mar 25, 2017 at 17:04
  • $\begingroup$ @NoahSchweber: In his paper, "Second-Order Logic and Foundations of Mathematics", V$\ddot a$$\ddot a$n$\ddot a$nen defines Second-Order $ZFC$ (denoted $ZFC_2$ in that paper ) as follows: "Suppose we want to do set theory using second-order logic. Subset existence is built into the language, so we do not need the usual set existence axioms. We still need axioms for itering set construction, i.e. forming sets of sets, sets of set of sets, etc. It is natural to proceed as in first-order set theory with individual variables ranging over sets and second-order variables ranging over subsets of $\endgroup$ Mar 31, 2017 at 10:17
  • $\begingroup$ (cont.) the universe.". Furthermore (in the same paper), he makes the following claim: "If we add to $ZFC_2$ the following axiom 'There are no inaccessible cardinals', the resulting system is categorical. I will use this example to (hopefully) clarify my question. Since $ZFC_2$ + "There are no inacessible cardinals" is categorical, all of its models are isomorphic. However, $ZFC_2$ + "There are no inaccessible cardinals" has both Full models and Henkin models-- the Full models contain every set (so have, as you point out, no forcing extensions), yet the Henkin models (not $\endgroup$ Mar 31, 2017 at 10:42
  • $\begingroup$ (cont.) containing every set) can have forcing extensions (this is what Mummert, in his comment/answer, seems to suggest). If one follows Hyttinen, Kangas, and V$\ddot a$$\ddot a$n$\ddot a$nen in reducing second-order theories to first-order theories with enhanced structures (as quoted in my question from their paper--$ZFC_2$ + "There are no inaccessible cardinals" could so also be reduced), the 'Full' enhanced structures (since the authors claim they contain every set) still should be isomorphic, while the Henkin enhanced structures (where $E$ $\subset$$\in$) need not be, (and Mummert's $\endgroup$ Mar 31, 2017 at 11:00
  • $\begingroup$ (cont.) comment still holds). Yet Hytinnen. Kangas, and V$\ddot a$$\ddot a$n$\ddot a$nen, claim that one can force over the enhanced structures to produce non-categorical models--how can that be? Also, Blass states that what one is actually doing is surreptitiously using boolean-valued models, but the question I asked in my last paragraph comes into play. I hope I have not further muddied the waters.... $\endgroup$ Mar 31, 2017 at 11:10

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