[FOM] SOL vs. ZFC

[Joe Shipman]

It's not a pointless semantic debate if we care about which statements of arithmetic are "true". SOL will not only guarantee the meaningfulness of statements of arithmetic, it will recognize a distinction between Con(PA) which is a consequence of logical validities and can be thought of as unqualifiedly "true", and Con(Z) which is a consequence of set existence axioms that are not part of "logic". 

The question of "justifying the axioms" then goes away for most math because logic comes "for free". Although SOL makes no ontological commitments, that is not necessary when you can characterize a mathematical object up to isomorphism by a single sentence.

-- JS

Sent from my iPhone

On Jun 4, 2013, at 5:07 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

Joe Shipman wrote:

> 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".
[...]
> but Logicism gets pretty far and I wonder why it seems to be unpopular.

A large part of the answer is historical.  The original efforts to carry out the logicist program (e.g., by Dedekind and Frege) ran into difficulties because of the paradoxes.  Hilbert also didn't like logicism. It doesn't take much to make a philosophy unpopular.

One could ask why more people haven't tried to revive logicism.  I think part of the answer is that many people see it as a somewhat pointless semantic debate, over what you consider to be "logic."

I imagine that only a minority of FOM readers take Alain Badiou seriously, but for those who do, I'll remark that the only way I've been able to make sense of Badiou's philosophy is to hypothesize that he regards much of set theory as "logic" and that since logic applies universally, we should expect set-theoretic concepts such as forcing extensions to show up in the real world, and not just in mathematics papers.  So I personally think of Badiou as a kind of logicist, though I doubt he would describe himself in those terms.

Tim
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