Problems with Logicism
Although the pursuit of the logicist program gave rise to a great many insights into the nature of mathematical concepts, not long after its inception it began encountering substantial difficulties.
For Frege, the most calamitous came early on in a letter from Russell, in which Russell pointed out that one of Frege’s crucial axioms for arithmetic was actually inconsistent.
His intuitively quite plausible “Basic Law V” (sometimes called “the unrestricted Comprehension Axiom”) had committed him to the existence of a set for every predicate.
But what, asked Russell, of the predicate “x is not a member of itself”?
If there were a set for that predicate, that set itself would be a member of itself if and only if it wasn’t; consequently, there could be no such set.
Frege’s Basic Law V couldn’t be true
(but see Frege’s Logic, Theorem, and Foundations for Arithmetic and recent discussion of Frege’s program in §5 below).
What was especially upsetting about “Russell’s paradox” was that there seemed to be no intuitively satisfactory way to repair set theory in a way that could lay claim to being as obvious and merely a matter of logic or meaning in the way that Positivists had hoped to show it to be.
Various proposals were made, but all of them were tailored precisely to avoid the paradox, and seemed to have little independent appeal.
Certainly none of them appeared to be analytic.
As Quine (1956/76, §V) observed, in the actual practice of choosing axioms for set theory, we are left “making deliberate choices and setting them forth unaccompanied by any attempt at justification other than in terms of their elegance and convenience,” appeals to the meanings of terms be hanged (although see Boolos 1971).