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In a chat on Mathematics.SE about countability of infinite sets, someone responded that regarding the 'counting', infinite in mathematics does not mean uncountable, but means 'not finite'.

Can someone explain the difference between these concepts?

[edit] Do Not Respond to The Clarification Below (It will elicit a discussion which I do not want)

Started of chat on math.se with this:

Infinity by definition means uncountable, but Cantor jumps & says he's counted it and the answer is aleph-something.

then I was told:

infinity means not finite countable means can be indexed (so, counted) by integers

and:

Well, disregarding the (philosophical?) question of whether something infinite can be counted or not, countable is a term with a specific meaning.

I then later responded with:

I'll try to be clearer: the context in which something is applied, fully defines that something for that context, and is valid only in that specific context. (it's a coffee-mug if you drink coffee from it; it's a paperweight if it prevents papers from being blown away; a paperweight is not a mug, they cannot be equated) ==> it's integers if it is used with respect to the set of integers, it's counters if it is used to count something, a counter is not an integer, integers just happen to be used (like the mug just happen to be used as a paperweight). My problem is that I cannot see how the integers-as-counters can be given all the attributes of the integers-as-a-set, they are fully different contexts. All this stem from having the cardinality of the set of integers is the same as that of the set of rational numbers: the sameness deriving from using counters and saying that the counters and the set of integers are the same thing, which it is not.

which got the following reponse:

the integers are the prototypical and canonical example of an index set for counting things. calling elements of other index sets "counters" is misleading (e.g. you can index a collection of objects with an uncountable set). you seem to want the word "count" to be more general than it actually is. the thing in set theory that allows us to compare sizes of sets is the idea of bijections.

after which I gave up on math.se...


From the accepted answer:

"An infinite set is defined as countable if it is in one to one correspodence with the natural numbers, N={1,2,3,...,n,...}. An infinite set X is uncountable if there exists no bijective map between X and the natural numbers N."

closed as off topic by DBK, stoicfury Mar 28 '12 at 17:07

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    -1 Isn't that a question about definitions of mathematical terms? Could you expand on why you are posting this on Philo.SE instead of Math.SE? – DBK Mar 28 '12 at 8:41
  • However, a quick off-topic answer to your question goes as follows: There is no difference between "infinite" and "not finite". However, there are infinite sets with different cardinality. The "smallest" infinite set is the set of natural numbers , which is countable. All infinite sets with greater cardinality are uncountable. See en.wikipedia.org/wiki/Infinite_set. – DBK Mar 28 '12 at 8:53
  • Questions are expected to generally relate to philosophy, within the scope defined in the faq. – stoicfury Mar 28 '12 at 17:10
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So in mathematics we measure the cardinality of a set with bijective functions/maps or one to one correspondences.

For example suppose you know that there are 100 seats in some movie theatre. When the movie starts, suppose it is a hit movie and fills up. In other words, there is a person for every seat in the theatre. Without counting the number of people, we can deduce that there are 100 people in the theatre. This is an example of a one to one correspondence (also known as a bijective function or map) between people and seats in the theatre, i.e. the cardinality of the people is the same as the cardinality of seats because for every seat there is one person sitting in it, and for every person there is one seat that they are sitting on.

There are two types of sets, countable and uncountable sets. Countable sets can either be finite or infinite, but uncountable sets are always infinite just a 'larger' infinite.

More precisely, A set X is finite if there is a bijection between the set X and the finite whole numbers, N_n={1,2,3,...,n}. If X is not finite, then X is infinite (they mean the same thing). Now concerning infinite sets, there are two types, countable and uncountable (here is the difference you seek). An infinite set is defined as countable if it is in one to one correspodence with the natural numbers, N={1,2,3,...,n,...}. An infinite set X is uncountable if there exists no bijective map between X and the natural numbers N. Note: Finite sets are also countable.

I think examples will be helpful here:

The set A={1,2,3,4,5} is finite and countable.

The set of integers is considered *infinite *and countable.

The set of real numbers (rational numbers and irrational numbers) is infinite and uncountable.

You can, informally, think of a countable set as a set where you are able to potentially list all of the elements of the set, and think of an uncountable set as saying there is no list that contains all the elements of the set. Naively, we can see that the real numbers are uncountable, because between any two real numbers there is another real number. Whereas there is no integer between the numbers 1 and 2.

Hope this helps!

Links for further reading:

http://gowers.wordpress.com/2011/11/28/a-short-post-on-countability-and-uncountability/

http://en.wikipedia.org/wiki/Cardinality

  • If interested, there is a wealth of resources on these topics. I can recommend books if you like, but most books on set theory or analysis will work. – Alborz Yarahmadi Mar 28 '12 at 9:46
  • This part helped: "An infinite set is defined as countable if it is in one to one correspodence with the natural numbers, N={1,2,3,...,n,...}. An infinite set X is uncountable if there exists no bijective map between X and the natural numbers N." Thanks – slashmais Mar 28 '12 at 10:25
  • Giving the definition of a finite set as a set from which every injective function to itself is also surjective is also an option, especially if we want to avoid axiom of choice. – Konformist Liberal Jan 2 at 11:47
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infinite and not finite are per se, regarding the word, the very same thing.

Regarding mathematics: There are several possible kinds of infinite sets, such as countable and uncountable sets, how ever both are infinite (and not finite, which is the same).

You'd better restate your question on Math.SE as they can help you much better. Also have a look in an encyclopedia for “countable” and “uncountable” (or “countable set” and “uncountable set”).

  • I added a clarification to my question – slashmais Mar 28 '12 at 10:22

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