The next paper in Kurt Gödel, Philosopher-Scientist is by Paola Cantù, on “Peano and Gödel”. The headline claim is that Gödel’s philosophical notebooks indicate that he had read Peano (and in particular Peano’s contributions to the Formulaire des mathématiques /Formulario Mathematico) rather carefully — and that Gödel was alert to the important differences too between Peano and Russell.
Unfortunately, when it comes to trying to spell out carefully what Peano’s views were (e.g. about the nature of functions), how Russell’s differed, and exactly what Gödel’s response comes to, I found Cantù less than ideally clear.
Here’s just one example where Cantù seems muddled. Peano defines the “classe nullo” to be the class of objects which are common to every class (“classe de objecto commune ad omne classe” Formulario, §6). Peano also defines the iota function (Formulario, §7) such that is the singleton class of (the class of such ). And then he adds a [partially defined] inverse iota function which undoes the effect of the iota function. I can’t typographically invert an iota here, so I’ll follow the later Peano and write instead: and then the idea is that, if is a class other than such that any two members of it are equal (i.e. if it is a singleton class), then = iff . Note, by the way, Peano’s , i.e. his inverted iota, is thus significantly different from Russell’s inverted iota!
Peano then shows, inter alia, something of this shape: , where ‘’ is in fact just one obvious way of characterizing the singleton of . The finer details of ‘‘ don’t matter.
So far so good. But Cantù comments on the latter rather trivial result as follows:
The definition of by means of the inverse iota operator allows us to define it as an individual object rather than as a set: nullo is the element associated to a set that contains all the such that , i.e. the elements that, for any property, satisfy that property and its contradictory.
This is surely muddled (even forgetting about Cantù’s symbolic foul-up, though to be honest that doesn’t inspire confidence). For no more defines as an individual object in any sense that contrasts with its being a class than would do the close equivalent = ! is still a class (though of course, it also an element — an element of its own singleton). Peano’s result which Cantù quotes doesn’t make nullo an element associated to a set containing just those things satisfying some contradictory condition (associated how?); rather it still is, as originally defined, a class
To move on, there is a quite separate issue for Peano when he later makes use of the inverted iota operation applied to classes defined by conditions not guaranteed to determine singletons. How are we to understand ‘’ if is not a singleton? Good question. Cantù gives an entry from Max Phil where Gödel (a) holds that Peano commits himself to the idea that there is a contradictory null object (Unding), and “ when has no elements [or] several elements, is this null object”, but (b) Peano can avoid this obscure doctrine. It is not clear, however, that Peano is countenancing a null object in this sense [which is not to be confused with the perfectly good class !]. Gödel’s alternative isn’t clear either, and Cantù’s discussion of it here is not easy to follow. But I am not minded right now to sit down with more of the Formulario to try to work out what’s going on — fun though it is to decode Peano’s simplified Latin! So for the moment, I’ll have to leave things in this unsatisfactory state.