Pretty straight forward questions.
I assume that ZFC would be considered a logicist program if its founding axioms were logical and analytic in nature, yet ZFC is often referred to as "extra-logical". Is this the same as saying that it is synthetic a priori and depends on human intuition?
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The notion of analytic appearing in Kant is equivocal. This has led to two different accounts and analyses attempting to declare the distinction useless.
You have heard of analycity with respect to the law contradiction. Kant formulated the notion of analycity in these terms in an effort to accommodate Aristotle's notion of essence without requiring completed infinities. Aristotle had suggested that essence could be obtained through a regressive descent. Aquinas used the idea to account for God's ability to know souls individually. Leibniz adapted Aquinas' idea to his relational view of space. In the attempt to accommodate the incompatible views of Leibnizians and Newtonians, Kant distinguished logic from mathematics by retaining the principle of the identity of indiscernibles for logic and asserted mathematical reasoning to be gronded upon spatial perception of numerical difference. This left him with the problem of explaining essence without infinite descent.
The classic example,
"A bachelor is an unmarried man"
would be false if denied. But what makes it false is the meaning of its words. So, one speaks of an analytic conception of truth based upon the meaning of words. This conception of truth is associated with the work of Carnap. His introduction of meaning postulates still plays a role in some theories of intensional semantics.
It is simplistic to think that logicism is motivated by the analytic-synthetic distinction. Logicism is associated with certain trends in mathematics leading to Cantor and Dedekind. But this trend took a decidedly different character with the comprehensionalist account of natural numbers introduced by Frege and popularized in the philosophical community by Russell. According to advocates for Lawvere's conception of set, Zermelo had to make a choice between Cantorian and Fregean views. He chose Frege's view.
The perception that the modern set theory is extra-logical would probably lie with the difference between a set as a collection taken as an object and the comprehensionalist account of Frege and Russell. Set theory speaks of types based upon indiscernibility with respect to formulas. Hamkins' paper on pointwise definable models makes this rather clear. Using ranks and Skolem functions, one member from these types can be chosen to obtain a HOD model for any given ground model. Comprehensions do not individuate elements as had been assumed in the Fregean construction.
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You say in the7 comments that you are familiar with Kant's "Critique of Pure Reason".
Kant's association of mathematics with sensibility has been misunderstood historically. Kant speaks of different logics. The logic which would correspond with the modern development of first-order logic would be his pure general logic (see A54/B78).. The logic he associates with the phenomena of sensibility is transcendental analytic (see A63/B88). In places where he speaks of whether or not sensibility is an object, he is quite clear. It is not. What he says (in agreement with Leibnizian principles) about what is given through sensibility is that it can only provide relations. Consequently, no logic driving extensionality to individuals can be an implementation of transcendental analytic.
The notion of synthetic thus corresponds to what is given by empirical experience. People who have read Kant and are familiar with his schematism know that Hilbert's metamathematical numerals and Markov's iteration of strokes mark a return to Kantian explanations. When a constructive mathematician in the manner of Markov draws a mark on a page, that mathematician is purposely generating a spatial phenomenon.
Kant does warn about the possibility of misapplying transcendental aesthetic. One might compare this with the objections against completed infinities or objections against the extension of finitary logical principles to infinite domains of discourse.
Chapter 19 of Russell's "Principles of Mathematics" is very instructive. He considers accounts of quantitative identity different from logicist conceptions. The relative view includes axioms which do not assert the necessary truth of reflexive equality, although he then states that this detail may be avoided by subsumption under a type.
Naturally, axioms of equality without subsumption under a type reflect constructive methods in the manner of Markov.
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The analytic-synthetic distinction concerns whether a proposition is true by definition (analytic), or false by definition (also analytic), or doesn't have its truth value determined by definition alone (synthetic). We owe this terminology to Immanuel Kant. But the "by definition" concept depends on what you think definitions are, and since his idea we've drawn a distinction between explicit and implicit definitions.
In the 1700s, Kant argued some mathematical truths are synthetic, but of course he didn't have set theory in mind. He would argue, for example, that if you look up definitions of $7$, $5$, addition and equality, you won't encounter $12$ as a concept, and therefore $7+5=12$ is synthetic. You might not find that view persuasive in 2020. If you define $5$ as $1+1+1+1+1$ etc., then all you need to make the inference is the associativity of $+$, so it comes down to whether you can prove that from $+$'s definition. Let's discuss whether you can.
Modern axiomatic systems are regarded as implicit definitions. The Peano axioms implicitly define non-negative integers and their arithmetic, and ensure $+$ associates. What is a set in ZFC? It's the kind of thing its axioms truthfully describe, just as a group element is the kind of thing the axioms of group theory truthfully describe. On that view, ZFC axioms are no more or less synthetic than $7+5=12$. I'll leave it to you to decide whether the "by definition" in the distinction's definition should refer to explicit definitions or allow implicit ones.