This re-instated what passes for faith in humanity

[wakim]

It doesn't make sense in general to add cardinal numbers.
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I wish I'd known that before I had the hubris to add 2 + 2 in expectation of meaningful result.

With more seriousness, I doubt I can master the particular field of abstract mathematics that you speak of fast enough to respond to your statements in a meaningful manner; and when I speak I find that I lose you in what appears to be "philosophical pyrotechnics." Allow me to try to explain clearly:

My difficulty is that it would seem that you claim a class of numbers neither finite nor infinite in its properties: not infinite in that it is bounded, yet not finite in that it has no upper limit. This strike me as contradictory, as what is bounded is bounded by limits, and what is unbounded is unlimited:

If some set of items is infinite, a property I would assume one should rightly attribute to the natural numbers, then how can its members be numbered, or bounded, or limited? (How many natural numbers are there? Infinite. What is the largest natural number? There isn't one.) If some other set of items is also infinite, a property that I would assume one should rightly attribute to the divisibility of any continuous interval, then how can its members be numbered, or bounded, or limited? (How many divisions are there in a continuous interval? Infinite. What is the smallest division? There isn't one.)

It would seem that to compare one of these infinite sets to another one introduces language that denotes difference between magnitudes of infinity; yet if infinity is unbounded, unlimited, and un-numbered, then how can one assign magnitude to it? But perhaps it isn't magnitude, but correspondence that is introduced? Correspondence between what? Correspondence between the magnitude of the quantity. What is the magnitude of the quantity? Infinite. Magnitude cannot be assigned to infinity, but perhaps it isn't magnitude, but correspondence that is introduced? Correspondence between what? ...