Eckard,
Physics does not require, nor use, transfinite algebra. Every measured result has to be positive or zero. Negative and imaginary numbers are mathematical artifacts.
If one is disposed (I am) to argue for a continuum of mathematical results with physical phenomena, one has to be careful when speciifying domains. No physical domain that is not measure zero, is infinite. To use a simplistic analogy, though, of the uncountable molecules of water that go into making up a river at its source, we recognize finite phase transitions from vapor to moisture to puddle, etc. Extending that process to the origin of the universe is not a leap -- it is continuous.
You wrote: "Galileo Galilei correctly concluded by means of bijection: There are not more natural numbers 1, 2, 3, ... as compared with their squares 1, 4, 9, ... because the comparative relations are not valid for infinite quantities, only for finite ones. G. Cantor claimed having "proved" him wrong by arguing that there must be more irrational numbers than rational ones because something that is neither smaller nor equal to something must be larger." Accordingly Cantor introduced what he first called Maechtigkeit and later renamed cardinality.
Is this "another way of saying that a calculated result must be positive, negative or zero"?"
No. The idea of the cardinality of sets has nothing to do with numbers per se. It describes comparative relations, so it certainly is appropriate for infinite sets. Infinity is not a number.
"Well, explicit finite numerical results are rational numbers and therefore they obey this trichotomy."
We assume so. Intuitionists and some constructivists would disagree, allowing that without an explicit procedure to decide, one cannot know whether a result is positive, negative or zero. Again, though, this has nothing, at least directly, to do with physics.
You wrote, "However, as I tried to explain in my last essay, Cantor's naive transfinite numbers have proven sterile. Already in 1922 Fraenkel admitted: Cantor's definition of sets, including infinite ones, is untenable.
"It is always assumed that a Hilbert space can have no more than a countable infinity of linearly independent state-vectors. This implies that there are no eigenstates of exact position, that the Dirac delta-function is illegitimate." [quoted from Gibbins, p. 90].
Set theory (arithmetic) is useful to physics. Its usefulnes is limited, however, to the counting function. Physics doesn't address infinite sets.
Tom