Let's discuss Mathematics

[ThinkTank]

It breaks if a countably infinite number of infinitely long buses arrive, each containing a countably infinite number of guests though (I think).

I think you think wrong here. N x N has the same cardinality as N. So this case just reduces to one countably infinite bus.

I think you can accommodate them if the hotel has a countably infinite number of floors (each containing countably infinite numbers of rooms).

By the same argument, that hotel is as big as the original one.

 

[Petek]

I think that ParadigmShifter is going for

N x N x N x ...

(countably infinite number of copies of N). That set is uncountable.

 

[ThinkTank]

I think that ParadigmShifter is going for

N x N x N x ...

(countably infinite number of copies of N). That set is uncountable.

If that's what he meant then yes, that won't fit. Though his second statement is wrong then.

 

[Spoonwood]

"Countably infinite"... does there exist a more "unhappy" mathematical term? I mean, it *seems* like it would mean "it is infinite, in the manner of counting." Except counting doesn't work as infinite, ever. And the natural numbers have far more many uses than counting. Anyone like the term "countably infinite" and think it actually expresses the idea here well?

 

[Atticus]

It can be funny word in the way that you can never count to infinity.

On the other hand the word's point is probably that if you start to count the objects, they all will be included when you "go to the limit", unlike R for example which always leaves some out. Finnish word is "numeroituva", which in English is about "numberable", a set which can be rearranged by numbers, a set which you can put numbers on. I think it's a good word modulo many understandings of the word "number".

 

[Souron]

It breaks if a countably infinite number of infinitely long buses arrive, each containing a countably infinite number of guests though (I think). I think you can accommodate them if the hotel has a countably infinite number of floors (each containing countably infinite numbers of rooms).

It's easier than that. To fit infinite buses of infinite people, you can use the following scheme: People file out of each bus one at a time. The first person in the first bus goes in the first room. The second person in the first bus goes into the second room. The first person in the second bus goes third. fourth is the third person in the first bus. then second person in the second bus. This pattern continues, thus counting all the people in the buses.

People are countable. So as long as you are counting people, then the most you can have is a countable infinity.

 

[Spoonwood]

On the other hand the word's point is probably that if you start to count the objects, they all will be included when you "go to the limit", unlike R for example which always leaves some out.

That sounds even funnier, since the limit of the counting numbers as n approached infinity makes absolutely no sense.

 

[Spoonwood]

People are countable. So as long as you are counting people, then the most you can have is a countable infinity.

But there is *no* most of a countable infinity. So, how does a "most of what we have" type of query make sense, unless we know that we have a finite number of objects?

 

[ParadigmShifter]

I see countably infinite meaning that you can provide an arrangement of the set members so you can say there is a next member in the ordering (which doesn't have to be a unique ordering). You can't provide an ordering of R with this property, given a real number you cannot provide a "next" number under any circumstances (one which will enumerate all numbers in R).

 

[Atticus]

That sounds even funnier, since the limit of the counting numbers as n approached infinity makes absolutely no sense.

When you count numbers: 1,2,3,...,n, every natural number will be included as n-> infinity.

 

[ParadigmShifter]

Don't let dutchfire hear you say that He's one of those weirdos who thinks 0 is a member of N.

 

[dutchfire]

Don't let dutchfire hear you say that He's one of those weirdos who thinks 0 is a member of N.

Of course, it's the logical thing to do. Addition of natural numbers is more important than multiplication, so including 0 is the right thing to do.

 

[ParadigmShifter]

So if someone puts 3 apples in front of you and asks you to count them you go "0, 1, 2, 3 apples"?

EDIT: The Count disagrees


Link to video.

 

[Spoonwood]

I see countably infinite meaning that you can provide an arrangement of the set members so you can say there is a next member in the ordering (which doesn't have to be a unique ordering). You can't provide an ordering of R with this property, given a real number you cannot provide a "next" number under any circumstances (one which will enumerate all numbers in R).

Put another way, I interpret "countably infinite" as saying that a unary function F on the (infinite) set can get provided such that F(s) never equals "s". Does that qualify as a sufficient condition for a set to qualify as countably infinite, or just a necessary condition, or neither? I'll guess necessary, since I can't see what such a function might exist on the reals... though that doesn't seem quite right, so maybe it's sufficient.

 

[ParadigmShifter]

What about the hyperreals? Define w as an infinitessimal number. Then F(s) = s + w satisfies that condition, but the hyperreals are bigger than the reals.

EDIT: And F(s) = s + 1 works for the reals too, unless I'm not understanding your argument due to being on beer #3

 

[Spoonwood]

When you count numbers: 1,2,3,...,n, every natural number will be included as n-> infinity.

But you can't count as n->infinity. The notation I've seen, usually uses "1, 2, 3, ..., n" to indicate n as finite, not as infinite, nor as n->infinity.

Whether 0 belongs to N or not just qualifies as a matter of definition, or a consequence from axioms/definitions that you make. I don't see "The Count" example... though QUITE AMUSING props to ParadigmShifter... as relevant, since the set N isn't quite the same concept as the numbers we count with, which have magnitude, while the members of N just have an order relation (which corresponds to magnitude, but conceptually comes as distinct). Also, if you select axioms for the natural numbers that you want to hold under some operation to characterize them, then 0 may appear, it may not. Without 0, positional notation simply comes as too difficult or non-existent... and without positional notation, working concretely with natural numbers has its difficulties.

 

[Spoonwood]

What about the hyperreals? Define w as an infinitessimal number. Then F(s) = s + w satisfies that condition, but the hyperreals are bigger than the reals.

EDIT: And F(s) = s + 1 works for the reals too, unless I'm not understanding your argument due to being on beer #3

No, you understood it. How about this then...

"countably infinite" means that if we select some initial starting point "i" of the set, then a unary function F on the (infinite) set S can get provided such that repeated applications of F(s) exhaust the rest of the set (or repeated applications of F(s) gives us every element of S-i). Actually, that definition (IF it works) sounds like it would imply that mathematical induction can get used on an countably infinite set via F(s). Could, for example, use mathematical induction on the positive rational numbers under the ordering which shows them countable (1/1, 1/2, 2/1, 3/1, 2/2, 1/3, ...)?

 

[ParadigmShifter]

Yeah ok. Well there isn't such a function on the reals anyway.

Weird fact #1: Between every 2 rational numbers there is an irrational number
Weird fact #2: Between every 2 irrational numbers there is a rational number

Not that weird those facts

But weird fact #3: There are infinitely many more irrationals than rationals

EDIT: Ok, add "distinct" to the above. Beer #4

 

[ThinkTank]

The normal definition of countably infinite is this (this one is from Wikipedia but you'll find it anywhere):

"A set S is called countable if there exists an injective function f from S to the natural numbers N. If f is also surjective and therefore bijective, then S is called countably infinite."

I think the phrase "countably infinite" is hard to improve: there's counting:

1,2,3, ... , n

and when you realize there is no mathematical reason why you would ever have to stop counting, you get:

1,2,3, ...

which is a.k.a. N.

Put another way, I interpret "countably infinite" as saying that a unary function F on the (infinite) set can get provided such that F(s) never equals "s". Does that qualify as a sufficient condition for a set to qualify as countably infinite, or just a necessary condition, or neither? I'll guess necessary, since I can't see what such a function might exist on the reals... though that doesn't seem quite right, so maybe it's sufficient.

Not sufficient. Consider F(0) = 1 and F(1)=0 on {0,1}. In general among the permutations on a finite ordered set you'll find many with for no x, F(x)=x.

If your intention was to have the function F respect some ordering < then you would still have to postulate the relevant properties of the ordering in order to make an interesting claim

... sounds like it would imply that mathematical induction can get used on an countably infinite set via F(s) ...

Good intuition! Any well-ordered set gives rise to an induction principle, and N is sort of the prototype of well orders. See Wikipedia on Well-founded relation and Transfinite induction.

 

[Atticus]

"countably infinite" means that if we select some initial starting point "i" of the set, then a unary function F on the (infinite) set S can get provided such that repeated applications of F(s) exhaust the rest of the set (or repeated applications of F(s) gives us every element of S-i).

Isn't this pretty much the same what I said about a set being "counted" when n->infinity?

When you count numbers: 1,2,3,...,n, every natural number will be included as n-> infinity.