Can all real numbers be written down with an infinite amount of time and people?

[cragar]

Could I write down all the real numbers from zero to 1? I know this sounds crazy. But let's say I have a person for every number between 0 and 1 . and then I tell them to write down a number different from every one else. Suppose they have the largest infinite amount of time to do this. Could I put all the numbers from 0 to 1 into bins, each number in a different bin. I have seen Cantors diagonal argument and it makes sense. My idea probably won't work. But I am just wondering If I had enough people could I do this. If this won't work for the reals could It work for the integers?

 

[HallsofIvy]

The difficulty is that the number of "people" you have to do the writing (or bins in which to put numbers) is necessarily countable while the set of all real numbers is not.

 

[Floid]

Could I write down all the real numbers from zero to 1?

No, there are an infinite number of those. No matter how many numbers you wrote down, there would always be more that you could write.

But let's say I have a person for every number between 0 and 1 . and then I tell them to write down a number different from every one else. Suppose they have the largest infinite amount of time to do this.

1.) You would never have enough people, because you are looking for an infinite amount of people.

2.) There is no "largest infinite amount". Infinite is infinite, there are no degrees of infinite.

If this won't work for the reals could It work for the integers?

Yes, you could write down all the integers between 0 and 1 and you could do it yourself in a matter of seconds.

 

[Hootenanny]

Infinite is infinite, there are no degrees of infinite.

Yet, some infinities are bigger than others ;)

 

[tiny-tim]

hi cragar!

Suppose they have the largest infinite amount of time to do this.

i think you mean the opposite …

the first one takes half an hour, the second one takes a quarter of an hour, the third one one takes an eighth of an hour, and so on, so they're finished after an hour …

let's be practical about this! ​

I have seen Cantors diagonal argument and it makes sense.

it sure does …

at the end of the hour, these people come to you and say "here's a complete list", and you use Cantor's diagonal argument to find a number, and reply "it's not complete, because you missed this one!"

 

[momentweaver]

I have a question to the OP, I am interested what was the motive behind this idea?
What is your idea of infinity?

 

[cragar]

thanks for all the answers . I got into an argument with a kid about if it was possible to write down all the reals from 0 to 1 , and I said you could have as many people as you want and as much time as you want. And we were seeing if it was possible. Like what Halls of ivy said , the people I had to do the writing would be considered countable while the reals are uncountable. Its interesting to think about uncountably infinite and countable infinite. And I always found cantors ideas of different infinities to be very interesting because it challenges common sense and gives a whole new way of looking at things.

 

[SteveL27]

thanks for all the answers . I got into an argument with a kid about if it was possible to write down all the reals from 0 to 1 , and I said you could have as many people as you want and as much time as you want. And we were seeing if it was possible. Like what Halls of ivy said , the people I had to do the writing would be considered countable while the reals are uncountable. Its interesting to think about uncountably infinite and countable infinite.

Since these are hypothetical people, we're free to imagine anything that's permitted by the rules of mathematics. So I think we could write down all the reals if we just used an uncountable number of people. They could each write down one real number, and all the reals would then get written down in less than a second.

Another way to approach this question is to ask what is time? If time is visualized as a continuum, then we can write down real x at time x. In other words at time t = 5 we write down 5. At time t = pi we write down the number pi. In this way one person can write down all the reals, assuming that each real can be written in one instant of time. [The negative reals are a technicality. At time t we'd write down both the reals t and -t. Maybe that would take two people!]

If you can have "as much time as you want," then you could do it.

Another thing you could do would be to use a uncountable ordinal. Such a thing is given to us by the Axiom of Choice, which in fact says that you can well-order the real numbers. So you could write down the first real, then the second real, ... and eventually write them all down. Of course uncountable ordinals are very hard to imagine, but the Axiom of Choice is a well-accepted axiom these days, and if you accept AC, then you can well-order the reals.

Just wanted to stretch out the boundaries of this thought experiment a bit. Countability is all well and good, but there's more than one way to skin a transfinite cat!

 

[cragar]

interesting approach steveL27, but when you say write down the first real then the second real, there is no second real how would you know what is next? And I still don't know if we can get around cantors diagonal argument.

 

[cragar]

@ steveL27: How can we say that when can well-order the reals if there is no least element.

 

[Hurkyl]

@ steveL27: How can we say that when can well-order the reals if there is no least element.

There is a least element in a well-order. You're observing that the usual ordering is not a well-order, but steveL27 didn't say "the usual ordering on the reals is a well-ordering".

 

[cragar]

He said we could well order the reals but I probably don't understand.
Hurkyl what do you mean by usual order?

 

[SteveL27]

He said we could well order the reals but I probably don't understand.
Hurkyl what do you mean by usual order?

The usual order is the standard order of the reals on the number line.

If you assume the Axiom of Choice (AC), you can well-order the reals. That means you can rearrange the reals so that any nonempty subset of the reals has a smallest element. So then you could pick out the first one, the second one, the third one, etc.

http://en.wikipedia.org/wiki/Well-order

In addition to the usual Wiki articles, I found this PF thread.

https://www.physicsforums.com/showthread.php?t=214418

Needless to say it's impossible to visualize such an order; but mathematically it exists if you accept AC. Its order type would be an uncountable ordinal.

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

So, since the original question was just a thought experiment, we could well-order the reals and write them down one by one. It would still take a long time ... I don't think I solved that part yet, now that you mention it ... because you could NOT use a trick like you can with a countably infinite set of writing down the first element in 1/2 second, the next element in 1/4 second, etc.

It's not possible for an uncountable series to have a finite sum unless all but countably many elements are zero. So the only way to do this is to use AC to write them all down at once!

Of course you can't really do this physically.

 

[pmsrw3]

Of course you can't really do this physically.

Rats.

 

[cragar]

Even if we could well order the reals. How could we get around cantors diagonal argument?
Dont worry about time being a problem we could use time dilation to our advantage .
Even if we could write down the next number in half the time of the previous one I still think we couldn't get around cantors powerful diagonal argument.

 

[moa_osen]

Just wanted to stretch out the boundaries of this thought experiment a bit. Countability is all well and good, but there's more than one way to skin a transfinite cat!

I cannot wait to say that to anyone. That is so darn adorable and geeky! I love it.

And I agree, but I also think that a "No" works in a practical non-unlimited time scenario. This is a fun argument, but also incredibly all-encompassing. The question of physical boundaries comes into play and if there is an interchange between extremely large real numbers and 0 and where the change in relation happens.

 

[SteveL27]

Even if we could well order the reals. How could we get around cantors diagonal argument?
Dont worry about time being a problem we could use time dilation to our advantage .
Even if we could write down the next number in half the time of the previous one I still think we couldn't get around cantors powerful diagonal argument.

Uncountable sets have a get-out-of-Cantor-free card.

Cantor's diagonal argument says there's no countable list of reals. An uncountable ordinals is uncountable. How would the diagonal argument concern you?

 

[Hurkyl]

Even if we could well order the reals. How could we get around cantors diagonal argument?
Dont worry about time being a problem we could use time dilation to our advantage .
Even if we could write down the next number in half the time of the previous one I still think we couldn't get around cantors powerful diagonal argument.

(To others:) I think he's trying to argue "you can't order the reals, because they're uncountable".

What is true is that any ordering of the reals cannot be an (ordinary) sequence -- a "list" of objects and we can say "that is object #3, that is object #17, that is object #1000381, and so forth", where the numbers are natural numbers.


However, orderings need not be sequences.

The usual order on the integers are not a sequence, because it has no smallest element, but you can still say things like "3 is the integer after 2, and 1 is the integer before 2".

The usual order on the rational numbers is even worse. It doesn't make sense to ask "what is the rational number after 2/3?" or "what is the rational number before 1?"

Any ordering of the real numbers is not a sequence, because it's "too long" (i.e. uncountable). But we can still have orderings, and even well-orderings (depending on AoC).

 

[cragar]

Uncountable sets have a get-out-of-Cantor-free card.

Cantor's diagonal argument says there's no countable list of reals. An uncountable ordinals is uncountable. How would the diagonal argument concern you?

It bothers me because even if we have a huge list of the reals and an infinite amount of time, Cantor says that we could create a real number that is not on our list no matter how big it is. Even if we could order the reals I still don't think we could write them all down even if we have an infinite amount of time.

 

[disregardthat]

It bothers me because even if we have a huge list of the reals and an infinite amount of time, Cantor says that we could create a real number that is not on our list no matter how big it is. Even if we could order the reals I still don't think we could write them all down even if we have an infinite amount of time.

That doesn't make much sense, we can't write down all the integers either given infinite time. At no point in time every integer would be written.

 

[tiny-tim]

At no point in time every integer would be written.

yes it would, see post #5

 

[SteveL27]

It bothers me because even if we have a huge list of the reals and an infinite amount of time, Cantor says that we could create a real number that is not on our list no matter how big it is. Even if we could order the reals I still don't think we could write them all down even if we have an infinite amount of time.

You're right. A convergent sequence of time intervals only works with countable sets.

So you would have to write them down all at once. Perhaps just drawing the number line counts as "writing" them. Since there are no names or symbols for most of the reals, it's difficult to imagine what writing them all down would mean anyway.

Maybe you can get uncountably many people to each write one number. But again, that's problematic, because you only have names for countably many numbers! So what does it mean to write down a number?

 

[disregardthat]

yes it would, see post #5

Well if we allow such quick hands, let's say you do this:

In the time interval (-pi/2,pi/2) measured in seconds you write down arctan(t) at time t on a piece of paper. The paper includes a line, of length (-pi/2,pi/2), and at time t you point out where on this line the number is, write it down, and draw an arrow towards where on the line it fits.

Now we have listed the reals in pi seconds. It's easy when one has quick hands.

 

[tiny-tim]

… you write down arctan(t) at time t on a piece of paper …

you can't do that at time t, you need a non-zero time interval to do it

 

[disregardthat]

you can't do that at time t, you need a non-zero time interval to do it

Obviously we are not concerned with physical limitations here, since you just proposed writing down every integer in finite time!

 

[SteveL27]

Obviously we are not concerned with physical limitations here, since you just proposed writing down every integer in finite time!

Well, there really is a valid point here. Even as a thought experiment, if we agree that it takes nonzero time to "write down" a real, then we can only write down countably many reals via the 1/2 + 1/4 + ... trick.

We can NOT do that for an uncountable set. Any uncountable set of positive numbers whose sum is finite must consist of at most countably many nonzero terms. So if we are to write down all the reals, we must either use uncountably many writers, or we must be able to write down a real in a zero-length instant of time.

 

[disregardthat]

Well, there really is a valid point here. Even as a thought experiment, if we agree that it takes nonzero time to "write down" a real, then we can only write down countably many reals via the 1/2 + 1/4 + ... trick.

We can NOT do that for an uncountable set. Any uncountable set of positive numbers whose sum is finite must consist of at most countably many nonzero terms. So if we are to write down all the reals, we must either use uncountably many writers, or we must be able to write down a real in a zero-length instant of time.

But why do we insist on there taking a non-zero time writing down a number?

The problem here is not the physical limitations, and it has nothing to do with time. The reason why everyone (including me) would say that my method is not valid is because it is not actually a method at all.