Our intuitions about the existence and nature of a continuum arise from perceptions of space and time . But the existence of a fundamental Planck length suggests that spacetime may not be a continuum. In that case, our intuitions originate from something (an idealization) that is not actu ally realized in Nature.I would go further and say that probability does not exist in the physical world.
Quantum mechanics is formulated using continuous structures such as Hilbert space and a smoothly varying wavefunction, incorporating complex numbers of arbitrary precision. However beautiful these structures may be, it is possible that they are idealizations that do not exist in the physical world.
It may come as a surprise to physicists that infinity and the continuum are even today the subject of debate in mathematics and the philosophy of mathematics. Some mathematicians, called finitists, accept only finite mathematical objects and procedures . The fact that physics does not require infinity or a continuum is an important empirical input to the debate over finitism.Yes, but it is hard to prove much unless you assume mathematical infinities.
There was a concerted effort beginning in the 20th century to place infinity and the continuum on a rigorous foundation using logic and set theory. However, these efforts have not been successful. For example, the standard axioms of Zermelo-Fraenkel (ZFC) set theory applied to infinite sets lead to many counterintuitive results such as the Banach-Tarski Paradox: given any two solid objects, the cut pieces of either one can be reassembled into the other .No, this is wrong. ZFC is a perfectly foundation for mathematics, and is widely accepted. Those Banach-Tarski subsets are not measurable, and do not undermine ZFC.
Post-Godel there is no general agreement as to what is meant by "rigorous foundations"...No, this is a common misconception. Mathematics was on shaky foundations in the 1800s. Basic concepts like real numbers and sets had not been rigorously defined. Goedel helped show that first order logic had the properties that mathematicians needed, and helped prove that axiomatizations of set theory could be used for foundations. Soon mathematician settled on ZFC as a suitable foundation for all of mathematics.