Physical significance of the fact that the cardinality of the real number line is the same as a finite interval of the real number line

Question

It is known that the cardinality of the real number line is the same as a finite interval of the real number line. Is there a physical meaning of this apparently conter-intuitive statement?

Answer 1

Assuming that by "physical meaning" you are referring to "helpful analogies to world of physical things", I would say you are barking up the wrong tree. While it is not too difficult to conceive of an infinite number of things that map to the counting numbers, how can one conceive of an uncountably infinite number of physical things? Can't these things always be mapped to the counting numbers? Think of counting stars, grains of sand, etc. The more you think about this, the more you will realize that this result that you mention makes no sense in the "ordinary world". When we run in to such frustration, we often call these notions "counter-intuitive".

Mathematics often demands we release our "naive intuition" (by "naive" I mean nonmathematical) and rely on the mathematics alone, i.e. understanding the precise definitions and arguments in detail. Once the truth of the mathematics begins to sink in, you may find your intuition transformed.