have got no meaning, because we hadn't calculated it yet. But such it doesn't mean that such formulas have no sense, because we still might guess about it. While we wouldn't guess about such formulas as:
[∃*∀2∃*, all, (0)]
indeed, meaningless. And sometimes it appears to be true, but in particularly this case:
This was believed to be true for more than thirty years, and the result was used by other mathematicians to prove other results. But in the mid-1960s, Stål Aanderaa showed that Gödel's proof would not actually work if the formulas contained equality... References to Stål Aanderaa and to the article.
We can think of some formula to have meaning, but to get the answered is to prove the formula. And for that reason a meaningless formula is the formula that has no proof.
But what does it mean for a certain formula to have no proof? It means that there is no values which satisfies it. It's interesting to note that we can say about a sign or a thing that it has sense if it can be applied to something. If a sign or a thing can be used somewhere, even hypothetically, then that sign or thing isn't senseless.
And what about improvable formulas? Do they have any things for applying? - No, they do not. And that's why we can say about a certain improvable formula that because it's meaningless it has no sense. Any improvable formulas are meaningless, therefore all of such are senseless.
Need to note that there are meaningless things beyond proof technique, and for a certain thing to be meaningless is the same to have no sense of being used. So, that's why for any thing it's true that if this thing is meaningless, it is senseless.
This is really good question, because even posting something doesn't mean I am sure in that.
So, here's the difference:
Some complex structure isn't meaningless if for each element in this structure x there is another one element y such that xRy and for each the relation R we can predict to which x which y is corresponded.
Or more briefly:
all provable functions aren't meaningless
And both as things so complex of things can have sense. If a certain thing has sense, then it can be applied somewhere. Like we can use "=", so this sign has sense (in the field we're using it); if we can use "equality" somewhere, then it has sense (in the field we're using it).
And a sentence is senseless if this statement is a proposition function like ϕ(x) and there is no arguments x which satisfies this sentence.
Or more compactly:
Senseless is wide uselessness or potential uselessness is senseless.
Hence, as soon as all the meaningless formulas are useless, they are also senseless.
"And similarly, a single number so big it can't be calculated is it meaningless?"
According to Frege it has got no meaning (for now), but it is not potentially meaningless. If we had got much more powers, we would solve it. And since such a formula is not potentially useless it is not senseless.