Carnap's Higher Order Numbers
In §27 of Meaning and Necessity, Carnap announces that all mathematical concepts can be defined without the use of any class expressions. The basic idea is to use Frege's system, but to replace all occurrences of class variables with higher order variables. In particular, the cardinal number of a property F is defined as the second order property of being equinumerous to F (definition 27-4). "Thus, for example, '2' is a predicator of second level" (p.117).
It would be interesting to know if this announcement can be carried out.
Identity statements between number expressions then have to be interpreted
as expressing an equivalence between the corresponding second order
properties: A=B abbreviates 
F (Ax(Fx) 
Bx(Fx)). "Predecessor" would also express a third order relation:
Predecessor(A,B) abbreviates something like 
F
G (Ax(Fx) 
Bx(Gx) y (Gy z(Fz
Gz 
(z=y)))) (compare Grundlagen §76).
I'm not sure how to express Hume's Principle in this framework.
Provided that this, and the rest of the programme, can be carried out, has Carnap rescued logicism? If not, why not? Only because Carnap ignores the surface structure of arithmetical sentences, which appear to be about individuals? But to a large extent this is also true of structuralist accounts, where simple equations are interpreted as complex quantifications. Or is the use of higher order logic cheating, because the standard semantics of higher order logic involves set theory? But so does the standard semantics of first order logic.
In support of his view -- support by appeal to authority --, Carnap cites a footnote on p.80 (§68) of Frege's Grundlagen, where Frege himself seems to say that instead of identifying the number 2 with the class of all concepts having exactly two instances, one might as well identify 2 with the concept of being one of these concepts: "I believe that instead of 'extension of the concept' one might say simply 'concept'". Very puzzling. He adds that the seeming conflict between this and his account of numbers as objects can be avoided, but he doesn't explain how.
In "Ãber den Begriff der Zahl 2: Auseinandersetzung mit Kerry", an early draft of "Ãber Begriff und Gegenstand", Frege says that in that footnote he didn't really suggest an identification of numbers with concepts, because replacing "extension of the concept" with "concept" in the relevant definition on p.80 of Grundlagen results in
The number belonging to the concept F is the concept "equinumerous to the concept F"
and here "the concept 'equinumerous...'" is a singular term, and hence designates an object, not a concept. In fact, Frege says, it designates the extension of the concept "equinumerous...". However, I doubt that Frege really had this in mind when he wrote the passage in Grundlagen.