Your question is reminiscent of Skolem's paradox:
Skolem's Paradox involves a seeming conflict between two theorems from classical logic. The Löwenheim-Skolem theorem says that if a first-order theory has infinite models, then it has models whose domains are only countable. Cantor's theorem says that some sets are uncountable. Skolem's Paradox arises when we notice that the basic principles of Cantorian set theory—i.e., the very principles used to prove Cantor's theorem on the existence of uncountable sets—can themselves be formulated as a collection of first-order sentences. How can the very principles which prove the existence of uncountable sets be satisfied by a model which is itself only countable? How can a countable model satisfy the first-order sentence which says that there are uncountably many mathematical objects—e.g., uncountably many real numbers?
A full answer to your question would require perspicuous definitions of understanding and word meaning that for now, perhaps, much escape us. We do have the ability to think over infinitary logics along with other matters of higher set theory, to think through sequences of greater and greater cardinality. Arguably, plural quantification even does much of the needed work: we can have the expression "∃rr" in play, indeterminately ranging over the (indeterminate) manifold of the alephs (say).
But then again, what is it for a symbol to mean something, and how many symbols are allowable? In principle, we could use a continuous stream of percepts symbolically, could have continuously many symbols, but repeating our use of percepts in this way would not comprise a very stable language.
So there's an aspect of the problem: not mere meaningfulness "just like that" but the stability thereof, the communicative recurrence of the icons we intend to use to convey information to ourselves and each other. At any given time, we will have ever written down only finitely many notations of individual alephs, even as we skip up the cumulative hierarchy to talk about having k-many of this or that large cardinal. Our ability to target absolute infinity in our sights, and the appearance of openness that this "thing" has to its name (it is open under all other predicates besides "is open under all other predicates"), with absolute infinity being yet seemingly "one thing," gives us the capacity to think over all the alephs at once, we wish we could unhesitatingly contend. However:
At the core of the problem lies the assumption that the set-theoretic paradoxes cast doubt upon the existence of a comprehensive domain of all objects. What they reveal, according to Dummett (1991, 1993), is the existence of indefinitely extensible concepts like set, ordinal, and object. For Dummett, the indefinite extensibility of set is incompatible with the existence of a comprehensive domain of all sets, since no matter what putative domain of all sets we isolate, we find that we can employ Russell’s reasoning to characterize further sets that lie beyond the putative domain of all sets with which we began. The set of all non-self-membered sets in the initial domain cannot, on pain of contradiction, be in that domain, which means that it must lie in a more comprehensive domain of all sets. If there is no domain of all sets, there is, the thought continues, no hope for a domain of all objects. One may respond to this line of argument by contesting Dummett’s diagnosis of the set-theoretic antinomies. Boolos (1993) suggests that we take Russell’s paradox, for example, to establish that not every condition determines a set. The moral of Russell’s paradox is that there is no set of all non-self-membered sets, not that it lies beyond the initial domain of quantification.
The Boolos just mentioned is the progenitor of plural quantification theory by the by, incidentally. For more on issues of indefinite/indeterminate extendibility, see Storer, esp. §§3.2, 3.3, 5, and 6.5.