Plenetuindous Platonism tries to resolve worries about access to mathematical objects by saying that there are mathematical objects corresponding to every "coherent theory".
The standard objection to this, based on a point by Boolos, is that if 'coherent' means first-order consistent, then this has to be false because there are first order consistent theories which are jointly inconsistent- but if 'coherent' doesn't mean first-order consistent, the notion is obscure.
I used to think this objection was pretty decisive, but I don't any more.
For, contrast the following two claims:
- TRUE: all consistent first-order theories have models in the universe of sets (completeness theorem)
- FALSE:all consistent first-order theories are true (Boolos point)
Which of these is relevant to the plenetudinous platonist?
What the plenetudinous platonist needs to say is that whichever kind of first-order consistent things we said about math, we would have expressed truths. But remember that quantifier restriction is totally ubiquitous in math and life (if someone says all the beers are in the fridge they don't mean all the beers in the universe, and if some past mathematician says there's no square root of -2, they may be best understood as not quantifying over a domain that includes the complex numbers).
So, what the plenetudinous platonist requires is that that every first order consistent theory comes out true for some suitable restriction of the domain of quantification, and interpretation of the non-logical primitives. And this is something the reductive platonist must agree with, because of the completeness theorem! The only difference is that the reductive platonist thinks there are models of these theories built out of sets, whereas the plenetudinous platonist thinks there's a structure of fundamental mathematical objects corresponding to each such theory.
Thus, plentudinous platonism's ontological commitments can be stated pretty crisply, as in the bold section above. And there's nothing inconsistent or about these commitments, unless normal set theory is inconsistent as well!