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It is known that $nZ$ is a strict subset of $Z$, but we can construct an isomorphism between them, showing that they are equinumerous. How come a strict subset, which contains less element, is equinumerous to the larger set containing it? This phenomenon intuitively does not make much sense to me. Is it one of the prices of accepting the infinity axiom of Zermelo-Fraenkel set theory?

Thuc Hoang
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  • Mapping $x\mapsto nx$ is a bijection between $\Bbb Z$ and $n\Bbb Z$. – Wuestenfux Dec 23 '21 at 09:02
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    Working with infinity is frequently counterintuitive. And, it used to be believed that if you stared to hard at infinity it would drive a person to madness. Two infinite sets can have the same cardinality even though one is a proper subset of the other. – Doug M Dec 23 '21 at 09:04
  • See so-called Galileo's paradox: "a demonstration of one of the surprising properties of infinite sets." – Mauro ALLEGRANZA Dec 23 '21 at 09:04
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    So "the prices of accepting to "manage" mathematically the infinity" is that we have to give up some of our intuitions about "size" that we have developed dealing with finite collections. – Mauro ALLEGRANZA Dec 23 '21 at 09:08
  • @MauroALLEGRANZA Do you mean that to accept the existence of infinity, we just have to come to terms with this phenomenon? – Thuc Hoang Dec 23 '21 at 09:13
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    More or less... This is what is needed IF we accept the treatment of infinity according to modern mathematical theory of sets. – Mauro ALLEGRANZA Dec 23 '21 at 09:16
  • Thank you for all comments made here. I think infinities make our lives much easier, so why not let them exist? Extending the notion of 'equal size' to infinite sets, which might cause some counter-intuitive phenomena, is a small price compared to the benefits we get out of having infinities. – Thuc Hoang Dec 23 '21 at 09:33

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