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Just another anecdote on name changes, Uri Abraham was (and in Hebrew, still is) Uri Avraham. He said that while he was a postdoc in the US, people confused those two names enough for him to give up and change.
My point is that you seem to be asking about some kind of a constructible concept compatible with measurable cardinals, and that already exists. Jech has an introduction to it, as will Kanamori, and at least two chapters in the Handbook of Set Theory come to mind.
Are you asking if measurable cardinal are compatible with the universe being constructible from a set? Yes, they are. $L[D]$, when $D$ is a measure on $\kappa$, is such an example. It is not at all clear what these axioms actually mean, though. Or are you just asking if we can create some kind of a filtration of the universe whose successor steps don't increase in cardinality for infinite indices? In which case, also yes.
@David: The use of $\alpha^+$ as the successor ordinal in some texts which present only the very basic introduction to set theory is somehow understandable (since $x\cup\{x\}$ is, in a good sense, a successor of $x$ as a set). But the notation is unambiguously used in set theory to mean the cardinal successor. Indeed, one of the few cases of notation meaning "pretty much just the one thing" that I can think of in set theory.
