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."