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Sometimes I talk about math to my friends, and some (with an engineering background) aren't used to the idea that in math, we have to define stuff. For example, they may not be used to the idea that $\sum_{n=0}^{\infty} a_n$ doesn't have an a priori meaning.

To get the point across, I would like to have an example to give these friends about infinite sets. Under the standard (and very natural) definition, we say that two sets have the same cardinality if they can be put into a bijection to one another. Is there any other reasonable sounding definition of cardinality which still says that (for example) $\{a, b, c\}$ has $3$ elements, but says that (for example) $\mathbb{N}$ and $\mathbb{Q}$ have different cardinalities?

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  • $\begingroup$ "Reasonable" is a subjective thing. What sounds reasonable to you might not to me. $\endgroup$ – Robert Israel Jan 6 '19 at 15:48
  • $\begingroup$ @RobertIsrael Of course; perhaps I should've tagged it as a soft question. But I hope that approximately everyone will approximately agree on what is approximately reasonable. $\endgroup$ – Ovi Jan 6 '19 at 15:50
  • $\begingroup$ I feel like the concept you are trying to demonstrate is much easier to do with series rather than the cardinality of sets. For example, $\sum_{n=0}^\infty (-1)^n$ is not a convergent sum if you define convergence as the limit of the partial sums, but does converge under the Cesaro sum, which is the limit as $n\to\infty$ of the arithmetic mean of the first $n$ partial sums. This shows how, although both of these definitions of convergence are somewhat reasonable, they lead to very different results. $\endgroup$ – Noble Mushtak Jan 6 '19 at 16:04
  • $\begingroup$ @NobleMushtak Yes. But I'm also curious for myself if there are any other alternative definitions for cardinality out there. $\endgroup$ – Ovi Jan 6 '19 at 16:12
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