[FOM] SOL vs. ZFC
José Ferreirós Domínguez
josef at us.es
Wed Jun 5 03:26:25 EDT 2013
Dear Joe Shipman,
> I am motivated by the question "how much of mathematics is logic in disguise?". If second order logic in the standard formalism is really "logic", then logic can express almost all of the mathematics anyone cares about, and the question reduces to which axioms can plausibly be described as "logical".
> I'd hesitate to justify results derived from "V_w?3 exists" as purely logical, but Logicism gets pretty far and I wonder why it seems to be unpopular.
Logicism died out in the 1950s and for good reason. Unfortunately many people trained in mathematical logic do not get exposed to the more delicate conceptual issues underlying this, hence it is quite common to find people, like you, who find the idea of logicism attractive.
Second order logic is Ok as a logical system, provided you avoid something that has been traditional. The so-called "full semantics" of SOL has, properly speaking, very little to do with logic. The standard semantics of SOL must be Henkin semantics. The reason for this is simple, namely that the idea of "full" powersets presupposes arbitrary subsets of infinite sets -- and this is a not completely clear notion, that set theory must investigate and clarify.
In fact, calling the set-theoretic semantics the "standard" semantics of SOL is the last remnant of logicism. And we would help students of logic by calling their attention to this. What has logic got to do with arbitrary subsets? Logic has to do with concepts, propositions, conceptual relations, and so on. Would you argue that it makes sense to speak of "arbitrary concepts", which are the intensional correlates of arbitrary sets?
I have written precisely on this topic in a recent paper, On Arbitrary Sets and ZFC. The Bulletin of Symbolic Logic, Volume 17, Issue 3 (2011), 361-393.
The end of logicism in the 1950s is the topic of an old paper of mine, Notes on Types, Sets, and Logicism, 1930-1950 Theoria 12 (1997), 91–124. Also, understanding the emergence of logicism was an important question for me, I have written quite a lot about this in my book Labyrinth of Thought. (Birkhäuser, Basel/Boston, 1999) and in the paper The Road to Modern Logic – An interpretation. The Bulletin of Symbolic Logic 7 (2001), 441–484.
Prof. José Ferreirós
Departamento de Filosofia y Logica
Universidad de Sevilla
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