José Ferreirós Domínguez josef at us.es
Mon Jun 10 03:42:14 EDT 2013

Let me try again. You do a disservice to SOL by insisting on the "full" set-theoretic semantics. SOL is an interesting logical system, but one has to make a little effort to interpret it correctly (even if this means going against some ideas that have been traditional in some areas of mathematical logic in the past). 

The principle that, given a set, each and every one of its subsets (whether definable or not, that's immaterial) is likewise given -- this is a strong existential principle. So your version of SOL is not at all innocent. Given N as a set, I can easily see why I should accept also the set of odd numbers and the set of prime numbers, etc. But arbitrary sets of natural numbers? And this, for purely logical reasons? Generation after generation of logicians have found this hard to buy, hence their annoying insistence that there is something fishy about what some people call "standard" SOL. Notice that the "full" semantics is about arbitrary subsets of the domain, so we completely forget about things being conceptually determinable.

Furthermore, that principle has its roots purely and exclusively in mathematics, it has no conceivable background in logic (in the study of conceptual interrelations, to express it somehow). In my paper 'On arbitrary subsets and ZFC' I explain in detail how one can motivate the Powerset principle and arbitrary subsets simply from the real numbers. Not only is this possible: this motivation is the main one behind the postulation of powersets in the standard (arbitrary) sense. If you think about it, it's little wonder that "full" SOL gives the appearance of showing that the real-number structure is categorical. The process is this: first you develop set theoretic principles strong enough to capture your notions about the real numbers (as Cantor, Dedekind etc. did); then you claim that some of those principles are just logic (as Dedekind and Hilbert, but not many others, did); finally you apply this seemingly powerful (though in practice vague) logic in doing mathematics -- and everything is magically fixed.

I insist: there's nothing wrong in SOL, properly interpreted. One does indeed recognize validities in second-order logic, but those do not depend on the "full" semantics. Even logicians who tend to favor the full semantics happen to show that they don't take it seriously. You may find people saying that, if you have qualms about AC, you can always avoid treating it as a logical principle in your system of SOL. This in effect is to acknowledge that the full semantics is not the standard semantics.

Finally, let me add that "being tired of" something is not a very good argument for or against. I know philosophers who are tired of mathematicians and scientists defending geometries other than Euclid's. Also one may be tired of CH remaining aloof from anything known in set theory, of the Continuum problem (which according to "full" SOL should be the most basic and clear question about sets) being intractable. Those who are tired have good reason to avoid set theory. Which reminds me of the following. 
Shelah has written interesting things about the consequences of independence results in set theory, e.g. in a paper entitled ‘The future of set theory’, 1991. Even though he admits to be inclined towards platonism, he states that in his view “the universe of ZFC” is not like “the Sun”, but rather like “a human being” or “a human being of some fixed nationality.” And he wrote the following:

“So a typical universe of set theory is the parallel of Mr. John Smith, the typical American; my typical universe is quite interesting (even pluralistic), it has long intervals where GCH holds, but others in which it is violated badly … (Shelah, 1991)

Best wishes,

Jose Ferreiros

El 09/06/2013, a las 18:04, fom-request at cs.nyu.edu escribió:

> The reason I am sympathetic to Logicism is that it derives most of mathematics without specifically mathematical axioms, and thereby avoids having to justify axioms and allows mathematical propositions to be "true" in the strongest sense. The axioms for second-order logic developed by Hilbert may fairly be called "logical" rather than "mathematical".
> That FOL has some nice properties that SOL lacks doesn't mean SOL isn't logic, it just means that we don't have an algorithm to enumerate logical truths (or "validities" if you don't like using the word truth, but expressing a mathematical proposition as a validity of SOL can reasonably be described as "proving it true"). Since SOL includes FOL, preferring FOL because of its convenient metamathematical properties is like preferring Presburger Arithmetic to Peano Arithmetic because it's a complete theory rather than an undecidable one--fine as a matter of taste, but no reason to discourage people from working in the larger system.
> As a foundation for "ordinary mathematics", SOL works just fine, even if logicians and foundational specialists and set theorists have professional reasons for preferring to found mathematics in ZFC+FOL. I'm willing to accept that propositions like Con(Z) cannot be proven from SOL and that the best SOL can do is to prove "V_(omega+omega) models Z", because I'm not demanding that all mathematics be derived from logic, just as much as possible, with those parts being granted greater epistemic certainty.
> And I am tired of hearing criticism of SOL as merely "sneaking in set theory in disguise". SOL isn't claiming to answer all mathematical questions that can be asked. That fans of ZFC can't settle the Continuum Hypothesis from their axioms doesn't justify the claim that CH is meaningless because the concept of "arbitrary subset of the natural numbers" is vague. The standard semantics of SOL refers to "all" subsets, and the existence of models which don't include all the subsets doesn't mean that MY concept of "all subsets" is incoherent or vague.

Prof. José Ferreirós
Departamento de Filosofia y Logica
Universidad de Sevilla

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