the infinity sign(∞) is often used casually but it is very abstract concept and ill-defined... when there are 'infinite' natural numbers and aleph-zero is cardinality of a set of natural numbers.. is ∞ bigger than aleph-zero? or smaller? or it can't be compared?
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If in your case $\infty$ denotes the cardinal number of the reals, then it's bigger than $\aleph_0$. – implicati0n Sep 17 '15 at 16:06
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4That is the most poignant example of comparing apples to dead snails saturated with salt, that I have ever seen. – Asaf Karagila Sep 17 '15 at 16:07
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@T_M: How do you figure? – Asaf Karagila Sep 17 '15 at 16:07
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He said "the infinity sign". There is absolutely no way I could tell what he means by that, so IF he means that $\infty$ is continuum, then it's bigger. @AsafKaragila – implicati0n Sep 17 '15 at 16:09
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@T_M: And if the OP means the cardinality of the natural numbers? – Asaf Karagila Sep 17 '15 at 16:11
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Well I really don't see how I could have written "If in your case..." any better. That's also the reason that I posted it as a comment, not an answer, I was assuming. @AsafKaragila – implicati0n Sep 17 '15 at 16:12
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1"...but it is very abstract concept and ill-defined..." With all due respect, and without intention to be inflammatory, I think it's your understanding of infinity that's "ill-defined." – daOnlyBG Sep 17 '15 at 16:24
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Infinity has a context. Like everything else in mathematics. The $\infty$ sign in calculus is more order theoretic than anything else, signifying that something grows unbound. This is why it's not an issue using it for limits of functions of the reals, and limits of sequences of the natural numbers.
Cardinal numbers are something completely different. They come to give some notion of a size of a set.
Asking which one is bigger is missing the fact they don't even "share the same meaning" as far as infinities go.
You might also be interested in The Aleph numbers and infinity in calculus.
Asaf Karagila
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