scott <sco### [at] scottcom> wrote:
> What happens if an infinite number of buses turn up at once?

Wouldn't that be the power set of natural numbers, making it a genuinely
larger set? Unless I have once again misunderstood something completely.

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                                                          - Warp

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John VanSickle <evi### [at] kosherhotmailcom> wrote:
> > 4) So the next question is: Does a countably-infinite-dimensional unit
> > cube contain the same amount of points as a unit line? If yes, can you
> > give a 1-to-1 mapping between them?

> Yes, as you can see from above.  The unit line, unit square, unit cube, 
> unit hypercube, etc., all have the same countability.

You demonstrated that an n-dimensional unit cube has the same amount of
points as a unit line. However, that's not what the question is asking.

> > 5) And the logical extreme: Does an uncountably-infinite-dimensional
> > unit cube contain the same amount of points as a unit line? Explain why.
> > (Also explain how the number of dimensions can be uncountably infinite.
> > That seems to defy the definition of "dimension".)

> The hypercube of infinite dimension may have a higher countability than 
> any finite-dimensioned cube, but it is certainly no lower in countability.

I'm not sure that answer makes sense...

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                                                          - Warp

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Orchid Win7 v1 <voi### [at] devnull> wrote:
> Given the 2D coordinates of a point on the unit square, you can 
> interleave their decimal digits, which always yields a unique point on 
> the unit line. For example,

>    0.3425
>    0.2183 -> 0. 32 41 28 53

One problem I see with this is that some real numbers can have more than
one decimal representation.

For example 1 and 0.99999... represent the exact same real number, but
their decimal representation is different.

Obviously if you had eg the point (0.5, 0.1), it would make a big difference
how you represent that 0.1. In one case you would get 0.51 while in
the other you get 0.500909090909..., which is obviously not the same value
as 0.51. Thus it wouldn't be a pure bijection.

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                                                          - Warp

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On 15/07/2013 10:12, Warp wrote:
> scott <sco### [at] scottcom> wrote:
>> What happens if an infinite number of buses turn up at once?
>
> Wouldn't that be the power set of natural numbers, making it a genuinely
> larger set? Unless I have once again misunderstood something completely.

Thinking about it a bit more I came up with this:

People arriving on a bus, give them a bus number, B, from 1 to infinity, 
and a seat number, S, from 1 to infinity
People already in the hotel, give them bus number 0, and their seat 
number is their existing room number
Everyone then has a unique B and S number.

To map 2D integers to 1D you can use a formula like the cantor pairing 
function (had to look up what that was called, but I remember it from 
school) 0.5*(B+S)*(B+S+1)+S. This then gives you their new room number 
in the hotel.

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On 15/07/2013 08:49 AM, scott wrote:
> What happens if an infinite number of buses turn up at once?

Which infinity?

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On 15/07/2013 10:47 AM, Warp wrote:
> Orchid Win7 v1<voi### [at] devnull>  wrote:
>> Given the 2D coordinates of a point on the unit square, you can
>> interleave their decimal digits, which always yields a unique point on
>> the unit line. For example,
>
>>     0.3425
>>     0.2183 ->  0. 32 41 28 53
>
> One problem I see with this is that some real numbers can have more than
> one decimal representation.

Yes. However, a real has at most two decimal expansions - one ending 
with a recurring 9, the other with a recurring 0. If you insert a rule 
that the latter is always the one to be chosen, the representation 
becomes unique.

This allows you to unambiguously transform any 2D point into a 1D point. 
It is unclear to me whether it solves the reverse transformation...

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On 14/07/2013 10:48 PM, John VanSickle wrote:
> This is why the reals are considered to have a higher countability than
> the integers. Reals are allowed to have infinite precision (such as pi,
> for which any digital representation goes indefinitely without
> repeating), but integers are defined to have finite precision.
>
> (Which brings up the question, why this imposition on the integers?)

Integers are allowed to have infinite digits before the decimal place. 
Reals are allowed them after the decimal place as well. ;-)

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On 7/13/2013 6:50 AM, Warp wrote:
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)

Taking "same amount of points" to mean "the two sets of points have the 
same cardinality", then yes.


> 2) If yes, that means there has to be a 1-to-1 mapping between those
> points. Give a function that expresses such a mapping.

An injective mapping suffices to prove #1 (since the other direction is 
trivial), and interleaving the digits achieves this.  A 1-to-1 mapping 
is trickier due to annoying edge cases with multiple decimal 
representations of the same number.  I don't see a (constructive) way to 
do a true 1-to-1 mapping off of the top of my head.


> 3) If the answer to the first question is yes, then it follows that
> the amount of points inside a unit cube is also the same as the amount
> of points on a unit line. The same for a four-dimensional hypercube,
> and so on. Can you give a generic function that gives a 1-to-1 mapping
> between a unit line and an n-dimensional unit cube?

My answer is identical to #2.


> 4) So the next question is: Does a countably-infinite-dimensional unit
> cube contain the same amount of points as a unit line? If yes, can you
> give a 1-to-1 mapping between them?

Assuming you take the "obvious" definition of an "inifinite dimensional 
unit cube" to actually be a unit cube (which I think is natural to do), 
then yes.  Again, only an injective mapping is obvious to me.  This can 
be done with a triangular interleaving type pattern.


> 5) And the logical extreme: Does an uncountably-infinite-dimensional
> unit cube contain the same amount of points as a unit line? Explain why.
> (Also explain how the number of dimensions can be uncountably infinite.
> That seems to defy the definition of "dimension".)

As you've stated it, I believe the answer to this depends on the 
continuum hypothesis.

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> Integers are allowed to have infinite digits before the decimal place.
> Reals are allowed them after the decimal place as well. ;-)

Genuine question then, why isn't the set of real numbers countable, 
given that you could represent each one with two integers (one for the 
digits before the decimal point, one for the digits after the decimal 
point)?

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> Yes. However, a real has at most two decimal expansions - one ending
> with a recurring 9, the other with a recurring 0. If you insert a rule
> that the latter is always the one to be chosen, the representation
> becomes unique.
>
> This allows you to unambiguously transform any 2D point into a 1D point.
> It is unclear to me whether it solves the reverse transformation...

If you take out all the numbers that could be represented with recurring 
0s or 9s from the 2D points, and the corresponding 1D representations of 
them, the remaining numbers have a 1-to-1 relation.

So the question comes down to is there a 1-to-1 relation between the 1D 
and 2D values you took out from the original sets.

The 2D coordinates taken out will be all possible pairs of numbers from 
this list:
0,1,0.1,0.2,0.3,...,0.9,0.01,0.02,0.03,...

The 1D numbers taken out will be the ones generated by interleaving the 
digits from all possible pairs of this list:
0.000...
0.999...
1.000...
0.099...
0.100...
0.199...
0.200...
...

It seems there is then an easy 1-to-1 mapping function for these 
remaining numbers, namely that if your 2D coordinate is the i'th and 
j'th element from the first list, then you use the i'th and j'th element 
from the 2nd list to generate your 1D coordinate.

So (0.3,0.4) is the 5th and 6th element from the first list, so you use 
the 5th and 6th from the second list (0.100...,0.199...), which gives 
your 1D value of 0.110909090909...

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