I feel like this is too grammatical for the math stack exchange, but I am sorry if it is too mathematical for this stack exchange.
In math there are several different types of infinity, some of which are known as "countable" and others which are known as "uncountable." I have always heard that one should use fewer to denote instances where one can count the objects to compare and less when one cannot count the objects. A beginning math student will often make the error of saying that the number of integers which exist is smaller than the number of rational numbers which exist.
Would it be proper for this student to say "There are less integers than rational numbers," or should he say "There are fewer integers than rational numbers"? Both of these sets are countably infinite, so one could theoretically count them. However this student may make a similar error, saying that there are fewer numbers between one and two than between two and three.
Would this student be correct to say "There are fewer numbers between one and two than between one and three," or should he say "There are less numbers between one and two than between one and three"? Both of these sets are uncountably infinite, so one could not count them.
Do the definitions of countable and uncountable make a difference in this case? Which would be the correct versions of these sentences? This is not a question about the general definition of the words "less" and "fewer," it is a question about whether the mathematical definitions of "countable" and "uncountable" will hold for the standard English definitions of "countable" and "uncountable."