At a conference in 1965 there were some interesting comments made by Kreisel and Mostowski asserting that Gödel later changed his mind regarding his1938 note on his set theory results (see Problems in the Philosophy of Mathematics, Imre Lakatos, North Holland, 1972; in these and other quotes I enclose my comments in square brackets).

(Kreisel comment, p. 97): It is true that in Gödel's original note in the Proc. National Acad., 1938, he remarks that V=L added as a new axiom seems to give a natural completion of the axioms of set theory, in so far as it determines the vague notion of an arbitrary infinite set in a definite way. This remark is certainly not in the spirit of the article on the continuum problem in the American Math. Monthly, 1947, and even less in the spirit of the addendum to this article in the anthology by Benacerraf-Putnam, 1964. It is a pity that Gödel himself does not draw explicit attention to the change in his point of view.

(Mostowski comment, p.108): The view that the notion of set is vague and that one can make it more precise was expressed by Gödel in his [1938] paper…But Kreisel is right in stating that Gödel did not maintain this view in his (later) paper on the Continuum Hypothesis. I personally found the view expressed in the earlier paper of Gödel extremely illuminating and I regret that I did not at once quote the source.

Similar observations appeared on p. 235 of the Martin Davis paper, What did Gödel Believe and When did He Believe It:

This passage is clearly at odds with some of Gödel's later utterances. Here he suggests that there is something “vague” about the “notion of an arbitrary infinite set”.

Davis also remarks here that …”In an article nine years later [Cantor 1947] instead of suggesting that A [V=L] might be accepted as an axiom Gödel strongly suggests that the axiom is false.” So Davis (and apparently also Kreisel) believed that Gödel accepted V=L in 1938 as a new axiom.

The key phrases of concern in the above are “vague concept of set” and “natural completion of axioms of set theory.” If Kreisel is asserting that Gödel would never say such things later, Gödel does not seem to be cooperating with Kreisel on that.

For a few years after the Kreisel, Mostowski comments, in Wang’s 1974 book From Mathematics to Philosophy, Gödel appears in a section on perception of concepts to indicate that “vague concept of set” is completely consistent with what he says in his 1964 Cantor paper (pp. 84-85): If we begin with a vague intuitive concept, how can we find a sharp concept to correspond to it faithfully? The answer Gödel gives is that the sharp concept is there all along, only we do not perceive it clearly at first. This is similar to our perception of an animal first far away and then nearby. ... There are more similarities than differences between sense perceptions and the perception of concepts. Gödel conjectures that some physical organ is necessary to make the handling of abstract impressions (as opposed to sense impressions) possible [Is this Gödel’s response to what is now known as Benacerraf’s access problem that Platonism requires an unfortunately non-existent sensory organ to gain access to concepts and abstract objects?; see also pp. 211-212 of Wang’s 1996 Logical Journey book where Gödel says that “the perception of concepts may either be done by some internal organ or …no special organ. I conjecture that some physical organ is necessary for this. … I believe there is some causal connection in the perception of concepts.”]

To see that Gödel is clarifying his epistemology given in Cantor 1964, here is the corresponding section in Cantor 1964 which the above paragraph in Wang 1974 is apparently clarifying (p. 268 in the Collected Works of Gödel, Vol. II):

It should be noted that mathematical intuition need not be conceived of as a faculty giving an immediate knowledge of the objects concerned. Rather it seems that, as in the case of physical experience, we form our ideas also of those objects on the basis of something else which is immediately given. Only this something else here is not, or not primarily, the sensations. That something besides the sensations actually is immediately given follows (independently of mathematics) from the fact that even our ideas referring to physical objects contain constituents qualitatively different from sensations or mere combinations of sensations, e.g., the idea of object itself, whereas, on the other hand, by our thinking we cannot create any qualitatively new elements, but only reproduce and combine those that are given [Gödel is not saying here that thinking is trivial but that our thinking and sensory apparatus require each other because thinking without an input facility (sensory apparatus for physical objects and abstract intuition for concepts and abstract objects) is empty and an input process without thinking is blind (Gödel's version of Kant's epistemological dualism?). In fact, Gödel on p. 256 in Wang’s 1996 Logical Journey book asserts that abstract or mathematical intuition is obtained when our thinking idealizes our strict native intuition, which only provides access to small sets, so that infinite sets can be accessed. Idealization is one way of extending the range of intuition with the help of thought, Gödel says.] Evidently the “given” underlying mathematics is closely related to the abstract elements contained in our empirical ideas. It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted. Rather they, too, may represent an aspect of objective reality, but, as opposed to the other sensations, their presence in us may be due to another kind of relationship between ourselves and reality.

So, just like in his Cantor 1964 paper, Gödel in Wang 1974 views our abstract intuition facility as analogous to our sensory facility where, instead of sensory data, abstract intuition provides data of the second kind, which Gödel refers to in Wang 1974 as abstract impressions of concepts. Our vague intuitive notion of set is such an abstract impression (which is itself a concept, here a vague concept), and it can, but hasn’t yet, converge on the sharp concept of set. The examples of vague intuitive concepts converging on the corresponding sharp concept are given in this section only for some cases where the sharp concept is known, such as formal system and velocity. Although Kreisel and Mostowski (or even concept of arbitrary infinite set specifically, just vague notions of sharp concepts in general) aren’t mentioned in this section, it aooears to have been motivated by the Mostowski Kreisel comments. Gödel also does not address the question of his opinion change directly, but what he says implies that vague intuitive notion of arbitrary infinite set being clarified is part of his epistemology given in Cantor 1964, and is thus not inconsistent with it. At any rate Gödel clearly wanted to straighten us out on “vague intuitive concepts” in this section, but he didn’t directly affirm or deny the Kreisel change of view observation, even though Kreisel asked him to in that observation.

As for the "natural completion of set theory axiom systems" phrase, Gödel appears to paraphrase it in a conversation with Wang recorded on p. 261 in Wang's 1996 Logical Journey book in a way completely consistent with Gödel's Cantor 1947 paper in that both the axiom of constructability and an axiom that refutes GCH are considered (and rejected) as new axioms. The proposition that all sets are constructable is a natural completion of subjective set theory for human beings. Ronald Jensen has shown that it leads to unnatural consequences such as Souslin's hypothesis...The "axiom" of determinacy is another example which in its general form contradicts the axiom of choice.

What does Gödel mean by "subjective set theory for human beings"? We claim that he means axiomatic set theories. For to what kind of systems are the axiom of constructability and axiom of determinacy added? To axiomatic set theories, not to no mathematical theories of seven element sets. Also Gödel defined in his 1951 Gibbs lecture (see pp. 305, 309 in Collected Works of Gödel, Vol. III) subjective mathematics (equivalently subjective set theory) as being the statements that can be proven in axiomatic set theories obtained by human mathematicians through a series of extensions.

Before the above paraphrase from A Logical Journey appears the following comments about the subjective concept of set (p. 260):

Subjectively a set is something which we can overview in one thought … for every set there is some mind which can overview it in the strictest sense…to say that the universe of all sets is an unfinishable totality does not mean objective indeterminedness, but merely a subjective inability to finish it…

(p. 256): By our native intuition we only see clear propositions about physically given sets and then merely simple examples of them…if you give up idealization, then mathematics disappears… according to George Miller (1956) we are psychologically capable of taking in with one glimpse a collection of about seven items…in order to overview an infinite set, it is necessary to resort to an extension of intuition in the Kantian sense-to show some sort of infinite intuition [what Gödel calls idealized intuition or idealized subjectivity]. See also p.182 in Wang 1974 for a discussion of idealized intuition, and p.189 of that book for Gödel suggesting that such idealized intuition existing for some sets provides one of five principles used to judge new axioms.

So, Gödel felt, using our idealized intuition we obtain an axiomatic set theory that provides a concept of arbitrary infinite set. How powerful such a system is Gödel may have thought depends on how powerful our intuition is in terms of recognizing individual sets (Gödel says given any set there is some mind that can view it strictly, where humans can only view small sets strictly).

So Gödel did indeed use the phrases “vague intuitive concept” and “seems to be natural completion of axiomatic set theories (or at least of subjective set theories for humans)“ much later than 1938 (namely in the 1970s). The vague intuitive concept is the perception of a sharp concept not the sharp concept in itself (which would be an oxymoron). V=L seems to be a completion of axiomatic set theories means that possibility is suggested by the nature of those theories, not necessarily that it’s true. That’s what Gödel meant by those words in the 1970s but it would be nice to see some correspondence that clarifies whether he meant the same things by those words in 1938.

Note that even if Gödel found a nice axiom of infinity that implies the negation of GCH, since Gödel apparently believed what he called the incompletability of subjective mathematics in Gibbs since Gödel’s 1933 Cambridge talk so refuting GCH would only seem to be a completion of subjective mathematics since there is no such completion.

Also please note that Karl Menger, Reminiscences of the Vienna Circle and the Mathematical Colloquium, edited by Golland, McGuinness, and Sklar, reports that Gödel discussed in 1933 and spring 1939 how new axioms are required for set theory but that he never mentioned any specific such axiom, hence didn’t mention axiom of constructability as a potential new axiom (p. 222): “ he undoubtedly meant that no one had given an adequate basic description of that world of sets in which he believed…but I myself never heard from him any indications about where he expected to find such axioms.”

Indeed even if Gödel believed in 1938 that GCH was false and another axiom was required to prove that, it would be a distraction and speculation to mention that here.