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Why are countable sets considered as small?

It is something that I never really understood. In fact, I would rather say they are big since they contain infinitely many elements. Any explanation would be welcome.

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    $\begingroup$ They're small compared to other infinite sets. $\endgroup$ – Riley Nov 16 '17 at 18:24
  • $\begingroup$ Is this notion of "small" an elaborate concept, or is it just something that was thrown there in a conversation without context? $\endgroup$ – user228113 Nov 16 '17 at 18:32
  • $\begingroup$ In some contexts, they are small in quantifiable ways. For example, any countable subset of $\mathbb R$ has Lebesgue measure zero, meaning zero "length". $\endgroup$ – Bungo Nov 16 '17 at 18:43
  • $\begingroup$ Countable sets are considered to be "small" compared to their "uncountably infinite" counterparts. For instance: $N$ can be said to be "small" compared to $R$ $\endgroup$ – Shatabdi Sinha Nov 16 '17 at 19:06
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The term "small" is subjective. If you are comparing countably infinite sets to finite sets, then yes, they are big. But if you are comparing countably infinite sets to other infinite sets, then they are small. For example, the set of real numbers is uncountable. This is shown by Cantor's Diagonal Argument.

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Riley called it subjective; my point of view is that size is relative.

Compared to a mountain, I'm small. Compared to bacteria, I'm huge. It's therefore reasonable to expect that a geologist considers me to be small while a microbiologist may think of me as rather large.

To a set theorist, anything countable is small - countable objects are the smallest nontrivial objects they interact with on a regular basis. And likewise, a mathematician working with the finite may consider anything infinite to be large.

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  • $\begingroup$ Cantor's Attic actually has a nice overview that may provide some inside into the kind of structures set theorists consider. The 'lower attic' considers countable objects and it's no accident that this is the lowest level of Cantor's attic whereas everything else lives below it - think of the rest as (mostly) outside the interest of set theory. $\endgroup$ – Stefan Mesken Nov 16 '17 at 18:44
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I've never really heard anyone call a countably infinite set "small". I've only heard people call uncountable sets "bigger". Which makes sense in the intuitive sense that now matter how you try to shove an uncountable set into a countable number of slots you'll always have an uncountable number that don't fit so it makes sense to assume that an uncountable set has "more" elements.

Of course all of this language is casual and based an intuition developed in a context that simply no longer applies. It'd be best to avoid using terms like "bigger" or "more" or "small" or "large" unless you specifically make it clear that a set $A$ being "larger" than $B$ means that no injective function from $B$ to $A$ nor surjective function from $A$ to $B$ can possibly exist.

In that sense calling countable sets "small" does make perfect sense as among all the possible cardinalities of infinity, countable is the smallest. The only sets that are smaller than a countable set would be finite sets.

Which I suppose sounds a bit like saying the only things slower than the speed of sound are, pffft, subsonic activity, and scoff they barely count.

Everything is relative.

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There are times that I convince myself that "more" is not the issue but that "thicker" is more appropriate. If I take a bag of integers and dump them onto a geometric plane or line or number line... whoo-ee... there sure are a lot of them all over the place like a bunch of specks of dust. But they sure are thin. I can poke my fingers between them and push them apart. But the points in the plane or line.... They are THICK. So thick I can stand on a plane and not fall through.

(Ain't language fun?)

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  • $\begingroup$ So, don't you view $\omega_1$ as discrete? That's certainly my view which is why I'm a bit conflicted to consider uncountable objects to be "thick" since "thickness", with regards to my intuition, has nothing to do with size but with "density". $\endgroup$ – Stefan Mesken Nov 16 '17 at 21:36
  • $\begingroup$ Of course thickness has nothing to do with size but with density. That was my point. Sometimes it seems to me that infinity is more about density than about size. It's a subjective thought I'm throwing out there. $\endgroup$ – fleablood Nov 17 '17 at 3:36

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