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I searched but couldn't really find an answer for that, so sorry if its a duplicate or anything else.

My question is why a set, for example A, that has a subset/equal set which is infinite(N for example), isn't infinite by definition? The formal definition for an infinite set is "A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number."

It makes sense of course, and its easy to prove that A is also infinite, but then why wouldn't N ⊆ A also mean that A is infinite by definition? Is there an opposite example when it doesn't happen? I'm not very familiar with math definitions in english, so sorry if I wrote something wrong. Thanks.

Edit - @Asaf Karagila answered it perfectly. Thanks for the replies.

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    $\begingroup$ You can't use the word "infinite set" in the definition of an infinite set. That makes no sense. $\endgroup$ – Mark Sep 25 '18 at 13:15
  • $\begingroup$ What definition are you proposing? "A set is infinite if it contains an infinite subset" does not make any sense. $\endgroup$ – lulu Sep 25 '18 at 13:21
  • $\begingroup$ Where did you get that definition? Also, definitions are most of the time arbitrary, they are good because they are useful in the sense they give the same consequences. For example, you can define an odd number as an even number plus one, or as an odd number minus one. By the first definition, $2x-1$ is not an odd immediately by definition, but this is fine because you can proof that it is one by using the axioms. $\endgroup$ – Shiranai Sep 25 '18 at 13:21
  • $\begingroup$ @lulu: From the question: "The formal definition for an infinite set is "A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number." $\endgroup$ – hmakholm left over Monica Sep 25 '18 at 13:22
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    $\begingroup$ @HenningMakholm Yes...but I thought the OP was objecting to that definition. I suppose one could say "a set is infinite iff it has a subset which is isomorphic to the integers." Seems awkward, though. $\endgroup$ – lulu Sep 25 '18 at 13:24
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Yes. A set with an infinite subset is infinite. But this is not "by definition", but rather a theorem (or a proposition).

In mathematics we have definitions, and we have consequences. The definition of "an infinite set" should not refer to "infinite subsets". In fact, the standard definition of "infinite" is simply "not finite".

But we can prove that $X$ is infinite if and only if it has arbitrarily large finite sets. And then we can easily prove that if $X$ is infinite and $X\subseteq Y$, then $Y$ is infinite.

The thing is that when you say "this holds by definition", then you mean that this is the literal definition (perhaps with a minor and obvious modification). Whereas the definition of an infinite set is not that $\Bbb N$ is a subset of that set, or so on.

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  • $\begingroup$ Thanks for the reply, I did not mean that the definition for an infinite set should be that way, but in a lot of aspects there are many sentences for different cases. If for example someone told you to prove that N ⊆ A, you'd need to prove it in a way(which again, is not very hard), but if N is an infinite set, why wouldn't A considered an infinite set as well without the need for deeper prove? (hope its understanble). Thanks :) $\endgroup$ – msacco Sep 25 '18 at 13:25
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    $\begingroup$ If I am writing a research paper, I wouldn't think twice. Yes, $\Bbb N\subseteq A$ immediately implies that $A$ is infinite, case closed. If I am proving something to a classroom full of freshmen who never saw mathematics before, then I will absolutely prove this statement in details. $\endgroup$ – Asaf Karagila Sep 25 '18 at 13:27
  • $\begingroup$ I see, Its just that I'm taking a math course atm, and they gave this question that appeared in one of the tests, there are many sentences similar to "if X then Y", and it didn't really made sense to me. Again, obviously its not something hard to prove, but it just feels weird to me that you actually need to prove it instead of simply having that sentence. But as you explained, that's just part of the course I guess, so its understandable. Thanks for the reply :) $\endgroup$ – msacco Sep 25 '18 at 13:31
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    $\begingroup$ If something is not hard to prove, that means you shouldn't have a hard time proving it. But one of the harder things in freshmen mathematics is understanding what is the statement that you even need to prove. For that kind of case, this question is perfect. $\endgroup$ – Asaf Karagila Sep 25 '18 at 13:32
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So your definition of an infinite set would be

A set is infinite if it contains an infinite subset.

This definition cannot be used to prove that any set is infinite. For example, we cannot prove that $\mathbb N$ is infinite, because to prove it is infinite, we need to find an infinite subset of $\mathbb N$. Ok, maybe $A_1=\mathbb N\setminus\{1\}$?

But how can we prove $A_1$ is infinite? Well, we have to find $A_2$, an infinite subset of $A_1$. And how can we prove $A_2$ is infinite? Well, by finding some subset $A_3$ of $A_2$ which is infinite. OK, but how do we prove $A_3$ is infinite? $\color{red}{\text{and on, and on, and on...}}$

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  • $\begingroup$ No, the question contains a precise definition: The formal definition for an infinite set is "A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number." The OP merely appears to be confused about the difference between saying "this is so by definition" and "this is so as a consequence of the definition". $\endgroup$ – hmakholm left over Monica Sep 25 '18 at 13:23

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