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One of the quickest ways to demonstrate that there exist Lebesgue measurable subsets of the real line that are not Borel measurable is to compute the cardinality of the Lebesgue $\sigma$-algebra and the Borel $\sigma$-algebra. The former has cardinality $2^{2^{\aleph_0}}$ (it contains the power set of the Cantor set), whereas the latter has cardinality $2^{\aleph_0}$ (by the transfinite induction construction of the Borel $\sigma$-algebra).

EDIT: Another potential place for larger cardinality sets to appear (though one which is still currently somewhat rare) is in nonstandard analysis. The usual construction of nonstandard models requires only countable ultraproducts, which do not increase cardinality that much. On the other hand, as a consequence, the models that one gets are only countably saturated. One can ask for more saturation by taking larger ultraproducts. In fact, if one iterates this process out to an inaccessible cardinal, one eventually obtains a monstrously large model which has saturation at all cardinalities smaller than that of the model. Such models have occasionally been used in analysis (e.g. in a recent paper of Hrushovski to attack the "noncommutative Freiman theorem" conjecture) but one can take the position that these tools are largely a convenience, and that one could work with a much less saturated model and still get the same applications at the end of the day (but perhaps with a lengthier argument).

I gather that something analogous happens in arithmetic geometry, in which it is convenient to work with Grothendieck universes which are again the size of inaccessible cardinals in order to obtain saturation-like properties, but that this is not absolutely necessary. (Though, I believe that the only extant proofs of Fermat's last theorem, for instance, still ultimately use Grothendieck universes, though perhaps not in a particularly essential fashion.)

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EDIT: Another potential place for larger cardinality sets to appear (though one which is still currently somewhat rare) is in nonstandard analysis. The usual construction of nonstandard models requires only countable ultraproducts, which do not increase cardinality that much. On the other hand, as a consequence, the models that one gets are only countably saturated. One can ask for more saturation by taking larger ultraproducts. In fact, if one iterates this process out to an inaccessible cardinal, one eventually obtains a monstrously large model which has saturation at all cardinalities smaller than that of the model. Such models have occasionally been used in analysis (e.g. in a recent paper of Hrushovski to attack the "noncommutative Freiman theorem" conjecture) but one can take the position that these tools are largely a convenience, and that one could work with a much less saturated model and still get the same applications at the end of the day (but perhaps with a lengthier argument).

I gather that something analogous happens in arithmetic geometry, in which it is convenient to work with Grothendieck universes which are again the size of inaccessible cardinals in order to obtain saturation-like properties, but that this is not absolutely necessary. (Though, I believe that the only extant proofs of Fermat's last theorem, for instance, still ultimately use Grothendieck universes, though perhaps not in a particularly essential fashion.)

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One of the quickest ways to demonstrate that there exist Lebesgue measurable subsets of the real line that are not Borel measurable is to compute the cardinality of the Lebesgue $\sigma$-algebra and the Borel $\sigma$-algebra. The former has cardinality $2^{2^{\aleph_0}}$ (it contains the power set of the Cantor set), whereas the latter has cardinality $2^{\aleph_0}$ (by the transfinite induction construction of the Borel $\sigma$-algebra).