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I introduced friends to the idea of infinite sets of different sizes/cardinalities. They get it, but think it's silly. Where is this concept used in every day life (or in technology that is used in everyday life)?

T. Brian Jones
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    https://math.stackexchange.com/q/154234/622, https://math.stackexchange.com/q/2347121/622, https://math.stackexchange.com/q/323971/622 – Asaf Karagila Jun 25 '18 at 17:28

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The number of functions from natural numbers to natural numbers has a greater cardinality than the number of computable functions. This implies that there are functions that cannot be computed, no matter how powerful a computer is.

For a layperson, I would think this is a pretty interesting result that they should be able to understand. Roughly put, there are more problems than you can solve.

Admittedly, is more of a negative result than a positive one, though I suppose it makes a 'too hard to solve' verdict a little more palatable.

Bram28
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You won't find examples of infinite sets in everyday life. There are only finitely many atoms in the universe for example. However the distinction between countably infinite and uncountable sets is very important in modern mathematics. For example it is essential in the theory of probability, which is used very heavily in all sorts of things.

saulspatz
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  • Do you have a good reference for its use in the theory of probability? – T. Brian Jones Jun 25 '18 at 17:44
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    Yes, but it's for a course generally taken by beginning graduate students in math, which I assume is way over your head. The need for infinite sets in math didn't become apparent until early in the twentieth century, so you have to take a good deal of math before you understand the need in "practical" subjects. Most scientists and engineers get along perfectly well without understanding the details of this. – saulspatz Jun 25 '18 at 18:00