1) Does a unit square contain the same amount of points as a unit line?
(We are talking about real numbers here.)

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.

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?

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?

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".)

-- 
                                                          - Warp

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On 13/07/2013 2:50 PM, Warp wrote:
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)

I would say no. They are different orders of infinity.
I recommend Rudy Rucker's fiction book White Light as a good entry to 
thinking about infinity. It is a bit strange but easier to read than 
Edwin Abbott Abbott's Flatland.

http://en.wikipedia.org/wiki/White_Light_%28Rudy_Rucker_novel%29


-- 
Regards
     Stephen

Post a reply to this message

On 13/07/2013 02:50 PM, Warp wrote:
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)

Weirdly, yes.

(As an aside, if the answer were no, the rest of the questions all 
become in applicable, which kind of gives the game away...)

> 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.

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

> 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?

This amounts to interleaving N sequences of digits rather than just two.

> 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?

Now I'm lost. I can neither prove nor refute that the cardinalities match.

Post a reply to this message

Le 13/07/2013 15:50, Warp nous fit lire :
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)
> 
Yes. It must be, from Cantor, something bigger than Aleph-0. (and is not
countable)


> 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.

IIRC, Peano curve would do. Expressing the position of any point using
(x,y) can be replaced with the ratio (between 0 and 1) over the Peano
curve of the length to reach the position over the total length of the
curve (and the precision of the ratio number can be used to infer the
order of the peano curve, or you can use any very big order of Peano
curve as it suits you)

You can also consider Wunderlich curves. Same approach. All real numbers
in [0,1) maps to the points on the unit square.

> 
> 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?

Be careful, using infinite and common sense might yields some surprise.
A basic approach would be to convert the first two coordinates (both in
[0,1) ) into a single coordinate using either Peano or Wunderlich, and
then repeat.

 X,Y -> a
 X,Y,Z -> a,Z -> b
 X,Y,Z,T -> a,Z,T -> b,T -> c

 C0,C1,C2,... Cn -> (C0,C1)->p0,C2,... Cn

> 
> 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, apply the answer of point 3, once the dimension value has been
mapped (because countably-infinite) to the set of natural number , in
order to have the number of original coordinates.

The mapping (of dimension) is obvious, because it is the definition of
the countably-infinite. ("obvious" as in math, after 10 to 2000 pages of
demonstration and 40 minutes or 500 years of intense thinking)

> 
> 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".)
> 
No.
Dimension can be uncountably infinite if it is not mappable to the set
of natural number. Such case can be the set of real number in [0,1),
(see Cantor's diagonal).
The number of dimension can be uncountably infinite as soon as the
number of dimension is no more restricted to a value mappable to the set
of natural integer. Fractal object have such dimension.

Post a reply to this message

Stephen <mca### [at] aolcom> wrote:
> On 13/07/2013 2:50 PM, Warp wrote:
> > 1) Does a unit square contain the same amount of points as a unit line?
> > (We are talking about real numbers here.)

> I would say no. They are different orders of infinity.

You would have to explain that in more detail.

Remember that, for example, one could easily think that there are "more"
rational numbers than there are integers. Yet that's not correct. There
are equally many. (That's because it's possible to construct a one-to-one
relationship between every rational number and every integer.)

If you can construct a function that gives a one-to-one mapping between
the points on a unit line and the points on a unit square, that means
that both sets have the same size (as unintuitive as that might sound.)

-- 
                                                          - Warp

Post a reply to this message

Orchid Win7 v1 <voi### [at] devnull> wrote:
> > 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.

> 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

That doesn't work because it's not a one-to-one mapping. Ie. the mapping
is not unambiguous.

-- 
                                                          - Warp

Post a reply to this message

On 13/07/2013 05:44 PM, Warp wrote:
> If you can construct a function that gives a one-to-one mapping between
> the points on a unit line and the points on a unit square, that means
> that both sets have the same size (as unintuitive as that might sound.)

Nothing in set theory is intuitive. ;-)

Post a reply to this message

On 13/07/2013 05:47 PM, Warp wrote:
> Orchid Win7 v1<voi### [at] devnull>  wrote:
>>> 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.
>
>> 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
>
> That doesn't work because it's not a one-to-one mapping. Ie. the mapping
> is not unambiguous.

That would imply that two distinct 2D points exist which map to the same 
1D point. Can you provide such a counter-example?

Post a reply to this message

Le_Forgeron <jgr### [at] freefr> wrote:
> > 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".)
> > 
> No.
> Dimension can be uncountably infinite if it is not mappable to the set
> of natural number. Such case can be the set of real number in [0,1),
> (see Cantor's diagonal).
> The number of dimension can be uncountably infinite as soon as the
> number of dimension is no more restricted to a value mappable to the set
> of natural integer. Fractal object have such dimension.

Note that this might be a bit of a trick question. It's talking about
an "uncountably-infinite-dimensional *unit cube*". As you say,
dimensionality being uncountably infinite requires non-integral dimensions.
Thus answering the question correctly would first require demonstrating
whether a non-integral-dimensional unit cube can exist. (I think it can't,
because it goes contrary to the definition of "cube".)

Of course for the sake of the question we can loosen up the requirement
of the object being a unit cube and simply say that it's an arbitrary
object of a unit "volume".

-- 
                                                          - Warp

Post a reply to this message

Orchid Win7 v1 <voi### [at] devnull> wrote:
> On 13/07/2013 05:47 PM, Warp wrote:
> > Orchid Win7 v1<voi### [at] devnull>  wrote:
> >>> 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.
> >
> >> 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
> >
> > That doesn't work because it's not a one-to-one mapping. Ie. the mapping
> > is not unambiguous.

> That would imply that two distinct 2D points exist which map to the same 
> 1D point. Can you provide such a counter-example?

By using a decimal notation you are equating the set of real numbers
with the set of integers, thus making the assumption that the set of
real numbers is countable. Not all real numbers can be represented with
digits, because digits can only be used to represent a countably infinite
set, which the set of reals isn't.

-- 
                                                          - Warp

Post a reply to this message

On 13/07/2013 5:44 PM, Warp wrote:
> Stephen <mca### [at] aolcom> wrote:
>> On 13/07/2013 2:50 PM, Warp wrote:
>>> 1) Does a unit square contain the same amount of points as a unit line?
>>> (We are talking about real numbers here.)
>
>> I would say no. They are different orders of infinity.
>
> You would have to explain that in more detail.
>
> Remember that, for example, one could easily think that there are "more"
> rational numbers than there are integers. Yet that's not correct. There
> are equally many. (That's because it's possible to construct a one-to-one
> relationship between every rational number and every integer.)
>
> If you can construct a function that gives a one-to-one mapping between
> the points on a unit line and the points on a unit square, that means
> that both sets have the same size (as unintuitive as that might sound.)
>

I got it wrong. It is not multiplying or raising the dimension that 
changes the size/order of an infinitive set. It is raising it by itself. 
E.g. multiplying the number of real numbers on a line by itself, 
itself's times. (Badly explained but I'm sure you can work out what I mean.)

I had a really bad journey yesterday, sitting on a grounded plane at the 
wrong airport for four hours. So I am not going to hurt my brain any 
more by reading about the perverted going ons of trans-infinities.

(If fractions were good enough for Alfred E Neuman then 11 dimensions, 
or is that 10, are good enough for me. :-) )

-- 
Regards
     Stephen

Post a reply to this message

>>>>      0.3425
>>>>      0.2183 ->   0. 32 41 28 53
>>>
>>> That doesn't work because it's not a one-to-one mapping. Ie. the mapping
>>> is not unambiguous.
>
>> That would imply that two distinct 2D points exist which map to the same
>> 1D point. Can you provide such a counter-example?
>
> By using a decimal notation you are equating the set of real numbers
> with the set of integers, thus making the assumption that the set of
> real numbers is countable. Not all real numbers can be represented with
> digits, because digits can only be used to represent a countably infinite
> set, which the set of reals isn't.

I was under the impression that if you allow infinitely long decimal 
expansions then all reals are representable.

Post a reply to this message

Le 13/07/2013 18:55, Warp nous fit lire :
> Le_Forgeron <jgr### [at] freefr> wrote:
>>> 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".)
>>>
>> No.
>> Dimension can be uncountably infinite if it is not mappable to the set
>> of natural number. Such case can be the set of real number in [0,1),
>> (see Cantor's diagonal).
>> The number of dimension can be uncountably infinite as soon as the
>> number of dimension is no more restricted to a value mappable to the set
>> of natural integer. Fractal object have such dimension.
> 
> Note that this might be a bit of a trick question. It's talking about
> an "uncountably-infinite-dimensional *unit cube*". As you say,
> dimensionality being uncountably infinite requires non-integral dimensions.
> Thus answering the question correctly would first require demonstrating
> whether a non-integral-dimensional unit cube can exist. (I think it can't,
> because it goes contrary to the definition of "cube".)

Nope, the question is similar to asking: can a white crow swim ?
The answer does not have to prove the existence of a white crow. But
using the properties of such white crow, answer the question of the
capability to swim or not.

Nevertheless, my answer is bit out of scope, regarding the question: it
just asserts that the previously used method of mapping to a single
value cannot be used with a non-integral-dimensional unit-cube. Maybe
there is a different method for mapping even transcendental numbers.

> 
> Of course for the sake of the question we can loosen up the requirement
> of the object being a unit cube and simply say that it's an arbitrary
> object of a unit "volume".
> 
We "just" need to define the notion of distance in
non-integral-dimension, and from that just cut any cube out of such
setting. But that's not the question. (cutting a sphere is easier, but
cube are just sphere too)

If we immerse the non-integral-dimension cube into an integral-dimension
space (of higher dimension), it is obvious that we can map. But then the
question becomes: is there a way to immerse a non-integral-dimension
into an higher-integral dimension. (and then, yes, does such white crow
exists ?)

Post a reply to this message

Orchid Win7 v1 <voi### [at] devnull> wrote:
> I was under the impression that if you allow infinitely long decimal 
> expansions then all reals are representable.

The question would thus be: Can the set of real numbers be represented
with digits chosen from a finite set?

Or even: Does an infinite amount of digits (chosen from a finite set)
become uncountably infinite? Doesn't that go contrary to the notion of
countable sets?

-- 
                                                          - Warp

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On 13/07/2013 09:16 PM, Warp wrote:
> Orchid Win7 v1<voi### [at] devnull>  wrote:
>> I was under the impression that if you allow infinitely long decimal
>> expansions then all reals are representable.
>
> The question would thus be: Can the set of real numbers be represented
> with digits chosen from a finite set?
>
> Or even: Does an infinite amount of digits (chosen from a finite set)
> become uncountably infinite? Doesn't that go contrary to the notion of
> countable sets?

No.

Consider for a moment the set of all natural numbers, N. This is (by 
definition) a countable set. Yet the set of all possible subsets of N is 
*uncountable*. And that's just plain ordinary subsets; the decimal 
expansion of a number is not merely a subset of N but an *ordered 
sequence* of digits. Intuitively it sounds like there should be *more* 
of these. (But, as with many things in set theory, it turns out actually 
the cardinalities are the same.)

Wikipedia explicitly mentions this:

http://en.wikipedia.org/wiki/Cardinality_of_the_continuum#Alternative_explanation_for

Post a reply to this message

Orchid Win7 v1 <voi### [at] devnull> wrote:
> Consider for a moment the set of all natural numbers, N. This is (by 
> definition) a countable set. Yet the set of all possible subsets of N is 
> *uncountable*. And that's just plain ordinary subsets; the decimal 
> expansion of a number is not merely a subset of N but an *ordered 
> sequence* of digits. Intuitively it sounds like there should be *more* 
> of these. (But, as with many things in set theory, it turns out actually 
> the cardinalities are the same.)

> Wikipedia explicitly mentions this:

>
http://en.wikipedia.org/wiki/Cardinality_of_the_continuum#Alternative_explanation_for

Perhaps the mapping you proposed between the points on a unit line and
the ones in a unit square is indeed valid.

Although proving that it indeed is would be better than by something else
than "can you show me a counter-example?" :)

-- 
                                                          - Warp

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Le 14/07/2013 00:28, Warp nous fit lire :
> Orchid Win7 v1 <voi### [at] devnull> wrote:
>> Consider for a moment the set of all natural numbers, N. This is (by 
>> definition) a countable set. Yet the set of all possible subsets of N is 
>> *uncountable*. And that's just plain ordinary subsets; the decimal 
>> expansion of a number is not merely a subset of N but an *ordered 
>> sequence* of digits. Intuitively it sounds like there should be *more* 
>> of these. (But, as with many things in set theory, it turns out actually 
>> the cardinalities are the same.)
> 
>> Wikipedia explicitly mentions this:
> 
>>
http://en.wikipedia.org/wiki/Cardinality_of_the_continuum#Alternative_explanation_for
> 
> Perhaps the mapping you proposed between the points on a unit line and
> the ones in a unit square is indeed valid.
> 
> Although proving that it indeed is would be better than by something else
> than "can you show me a counter-example?" :)
> 
That's remind me of the old metaphor of the hotel of infinite rooms used
for dealing with infinity.

1. the hotel is empty
2. a bus of infinite capacity (Aleph0 only, countable infinity) arrives.
Each passager get the room number from its place in the bus.
3. a single guy came in. Current customers get moved from room N to room
N+1. The guy get the first room.
4. another bus of same infinite capacity... Current customers get moved
from room N to room 2N, passagers get the room 2s+1, where s is the seat
number in the second bus.

and so on.

Post a reply to this message

On 7/13/2013 8:50 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.)

Saying that they have the same number of points is not a meaningful 
statement.  However, we can say that they have the same countability.

Consider any ordered pair (u,v) in the unit square.  These two values 
can be presented as

0.u_0 u_1 u_2 u_3 u_4 u_5 ...

and

0.v_0 v_1 v_2 v_3 v_4 v_5 ...

Where the different u_n and v_n are digits of whatever radix you care to 
select.

 From these two numbers you can construct a third number thusly:

0. u_0 v_0 u_1 v_1 u_2 v_2 u_3 v_3 ...

Essentially you can interleave the digits of u and v to produce a third 
number w that falls in the unit line.

> 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.

Done.

> 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?

I'm sure you can extrapolate from the above example to any higher dimension.

> 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.

> 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.

Regards,
John

Post a reply to this message

On 7/13/2013 3:16 PM, Warp wrote:
> Orchid Win7 v1 <voi### [at] devnull> wrote:
>> I was under the impression that if you allow infinitely long decimal
>> expansions then all reals are representable.
>
> The question would thus be: Can the set of real numbers be represented
> with digits chosen from a finite set?
>
> Or even: Does an infinite amount of digits (chosen from a finite set)
> become uncountably infinite? Doesn't that go contrary to the notion of
> countable sets?

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?)

Regards,
John

Post a reply to this message

On 14/07/2013 08:51, Le_Forgeron wrote:
> Le 14/07/2013 00:28, Warp nous fit lire :
>> Orchid Win7 v1 <voi### [at] devnull> wrote:
>>> Consider for a moment the set of all natural numbers, N. This is (by
>>> definition) a countable set. Yet the set of all possible subsets of N is
>>> *uncountable*. And that's just plain ordinary subsets; the decimal
>>> expansion of a number is not merely a subset of N but an *ordered
>>> sequence* of digits. Intuitively it sounds like there should be *more*
>>> of these. (But, as with many things in set theory, it turns out actually
>>> the cardinalities are the same.)
>>
>>> Wikipedia explicitly mentions this:
>>
>>>
http://en.wikipedia.org/wiki/Cardinality_of_the_continuum#Alternative_explanation_for
>>
>> Perhaps the mapping you proposed between the points on a unit line and
>> the ones in a unit square is indeed valid.
>>
>> Although proving that it indeed is would be better than by something else
>> than "can you show me a counter-example?" :)
>>
> That's remind me of the old metaphor of the hotel of infinite rooms used
> for dealing with infinity.
>
> 1. the hotel is empty
> 2. a bus of infinite capacity (Aleph0 only, countable infinity) arrives.
> Each passager get the room number from its place in the bus.
> 3. a single guy came in. Current customers get moved from room N to room
> N+1. The guy get the first room.
> 4. another bus of same infinite capacity... Current customers get moved
> from room N to room 2N, passagers get the room 2s+1, where s is the seat
> number in the second bus.
>
> and so on.

What happens if an infinite number of buses turn up at once?

Post a reply to this message

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.

-- 
                                                          - Warp

Post a reply to this message

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...

-- 
                                                          - Warp

Post a reply to this message

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.

-- 
                                                          - Warp

Post a reply to this message

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.

Post a reply to this message

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

Which infinity?

Post a reply to this message

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...

Post a reply to this message

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. ;-)

Post a reply to this message

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.

Post a reply to this message

> 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)?

Post a reply to this message

> 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...

Post a reply to this message

scott <sco### [at] scottcom> wrote:
> > 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)?

The set of all strings that contain all possible combinations of two
or more digits (in other words, the set of all possible decimal
representations) is uncountable. In other words, it's not possible to
enumerate them all. This might be a bit surprising (it was to me...)

Cantor's diagonal argument is used to demonstrate this. Note, though,
that said argument only proves that there are sets that are uncountable
(ie. cannot be enumerated with integers), not that the set of reals is
uncountable.

Cantor's diagonal argument is a proof by contradiction. It goes something
like this:

Let's assume that it is possible to enumerate every single possible infinite
string of digits. Thus we have a(n infinite) list of numbered strings (its
order doesn't really matter, as long as it contains *all* of them.) For
example like:

1: 1001101110...
2: 0010110111...
3: 1100011110...
4: 0111001101...
...

If the assumption is true, that means that this list is complete, ie. it
contains every single possible string using those two digits. Which would
mean that it's impossible to construct a string that does not appear in
that list.

However, we can construct such a string: Take the first digit of the first
string and invert it, the second digit of the second string and invert it,
and so on. In other words, with the example above the string would be
0110... etc.

Said string is different from every single one in that list (because it
necessarily differs from each one by at least one of the digits.) This is
a contradiction because the list was supposed to contain *all* of them.
(It doesn't help if you just "add" this new string to the list. You can
then construct another string that's not in the list.)

Because this is a contradiction, the initial premise is necessarily false:
It's *not* possible to enumerate all of those strings. Therefore the set
of all those strings is genuinely larger than the set of integers.

(But as said, AFAIK this only proves that the set of all possible strings
of two digits is larger than the set of integers, it doesn't yet prove
that the set of real numbers is also. However, it's probably getting there,
and may only require a few steps more to prove also that.)

-- 
                                                          - Warp

Post a reply to this message

>> 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)?
>
> The set of all strings that contain all possible combinations of two
> or more digits (in other words, the set of all possible decimal
> representations) is uncountable. In other words, it's not possible to
> enumerate them all. This might be a bit surprising (it was to me...)

This is surprising to me too, I would have thought that because every 
string of digits (without a decimal place) represents an integer then it 
would be countable.

> Let's assume that it is possible to enumerate every single possible infinite
> string of digits. Thus we have a(n infinite) list of numbered strings (its
> order doesn't really matter, as long as it contains *all* of them.) For
> example like:
>
> 1: 1001101110...
> 2: 0010110111...
> 3: 1100011110...
> 4: 0111001101...
> ...
>
> If the assumption is true, that means that this list is complete, ie. it
> contains every single possible string using those two digits. Which would
> mean that it's impossible to construct a string that does not appear in
> that list.
>
> However, we can construct such a string: Take the first digit of the first
> string and invert it, the second digit of the second string and invert it,
> and so on. In other words, with the example above the string would be
> 0110... etc.

I understand that, but don't understand why that proves it is 
uncountable. Because you can do a similar thing with a list of all 
natural numbers, ie calculate the sum and then you get a new natural 
number that wasn;t in the list to start with.

Post a reply to this message

scott <sco### [at] scottcom> wrote:
> >> 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)?
> >
> > The set of all strings that contain all possible combinations of two
> > or more digits (in other words, the set of all possible decimal
> > representations) is uncountable. In other words, it's not possible to
> > enumerate them all. This might be a bit surprising (it was to me...)

> This is surprising to me too, I would have thought that because every 
> string of digits (without a decimal place) represents an integer then it 
> would be countable.

Note that an infinite string of digits does not represent an integer
(because no integer is infinite.) That's probably what causes the
confusion.

("No real number is infinite either." That's correct. But the decimal
representation of a real number can be.)

That's not to say that no set containing infinite decimal representations
is countable. For example the set of rational numbers contains infinitely
many values which decimal representation is infinite, yet this set is
countable. (It just that, obviously, the set of rationals does not
contain *all* possible infinite decimal representations.)

"Decimal representation" in itself might be a problematic thing to rely
on when dealing with infinite sets, because it's not very intuitive.

> > However, we can construct such a string: Take the first digit of the first
> > string and invert it, the second digit of the second string and invert it,
> > and so on. In other words, with the example above the string would be
> > 0110... etc.

> I understand that, but don't understand why that proves it is 
> uncountable.

Because there's no way to enumerate them. (More technically, there exists
no bijective function between the two sets.)

> Because you can do a similar thing with a list of all 
> natural numbers, ie calculate the sum and then you get a new natural 
> number that wasn;t in the list to start with.

You cannot sum all the natural numbers because there's an infinite
amount of them.

Any sum of a finite amount of natural numbers gives a number that was
already in the set.

-- 
                                                          - Warp

Post a reply to this message

> Note that an infinite string of digits does not represent an integer
> (because no integer is infinite.) That's probably what causes the
> confusion.

True.

> ("No real number is infinite either." That's correct. But the decimal
> representation of a real number can be.)

That also probably explains why you can't represent every real number 
with two integers (one for the fractional part and one for the integral 
part).

>> Because you can do a similar thing with a list of all
>> natural numbers, ie calculate the sum and then you get a new natural
>> number that wasn;t in the list to start with.
>
> You cannot sum all the natural numbers because there's an infinite
> amount of them.

I guess the bit I got confused with / didn't understand is that you can 
have a set with an infinite number of natural numbers, yet none of the 
natural numbers themselves are infinite in size.

Post a reply to this message

On 7/16/2013 12:35 AM, scott wrote:
>> 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)?

I didn't comment on Orchid's quip since I assume he was joking, but you 
are rightly confused because it is completely incorrect.  Real numbers 
do indeed have an "integer before the decimal place", but critically 
*not* necessarily "after the decimal place".  The reason is that there 
may be infinitely many digits after the decimal place, all integers only 
have a finite number of digits.

Post a reply to this message

> I didn't comment on Orchid's quip since I assume he was joking, but you
> are rightly confused because it is completely incorrect.  Real numbers
> do indeed have an "integer before the decimal place", but critically
> *not* necessarily "after the decimal place".  The reason is that there
> may be infinitely many digits after the decimal place, all integers only
> have a finite number of digits.

How do I better understand that all integers have a finite number of 
digits, but there are infinite number of them? That's the bit I'm 
struggling with.

The way I see it, the number of positive integers with N or fewer digits 
is 10^N, so if N is finite then 10^N has got to be finite too?

Post a reply to this message

On 7/16/2013 8:06 AM, scott wrote:
>
> How do I better understand that all integers have a finite number of
> digits, but there are infinite number of them? That's the bit I'm
> struggling with.
>
> The way I see it, the number of positive integers with N or fewer digits
> is 10^N, so if N is finite then 10^N has got to be finite too?
>

Ahh I see.  This is one of many counter intuitive things which happens 
with infinities.  Perhaps the issue here is that there are two different 
ways that we tend to think about sizes of things at play here, and it's 
not always made clear which one we're talking about:

1) finite vs. infinite
2) bounded vs. unbounded

#1 here refers the the number of elements within a set, for instance the 
set {3,7,9} has three elements, the set of integers has infinitely many 
elements, and the set of reals has "even more infinitely" many elements 
in a certain well defined sense.

#2 instead relates to some notion of the "size" of the different 
elements within a set.  This is tricky because depending on what you're 
doing you might care about a different definition of what "size" is. 
For example with positive integers you might care about the number of 
digits, but with real numbers you might care about the magnitude of the 
number with relation to the < and > relations (which is certainly *not* 
the same as the number of digits in it!).  Often you just figure it out 
from context.

So this tells you a little bit about the concepts, but I think that some 
examples are useful in getting the mathematical intuitions.  For instance:

* If you have a set of integers, where the number of digits in each 
integer is bounded, then there are only finitely many integers in that 
set (as you said, at most 10^N).

* If you have a set of integers where the number of digits in each 
integer is unbounded, then there are infinitely many integers in that 
set.  For although each particular integer has only a finite number of 
digits, for any particular number N, you can always "look further" and 
find an integer with more than N digits.

I think this last one is probably where your confusion comes from. 
There is more that you can say about the relationships between the 
concepts of finite/infinite and bounded/unbounded, but hopefully the 
small bit I've covered this helps clarify things a little.

You can, by the way, give precise mathematical definitions of these 
concepts, but I've decided not to do so since I wasn't sure it would 
actually make the issue more clear.

Post a reply to this message

> Ahh I see.  This is one of many counter intuitive things which happens
> with infinities.  Perhaps the issue here is that there are two different
> ways that we tend to think about sizes of things at play here, and it's
> not always made clear which one we're talking about:
>
> 1) finite vs. infinite
> 2) bounded vs. unbounded
>
> #1 here refers the the number of elements within a set, for instance the
> set {3,7,9} has three elements, the set of integers has infinitely many
> elements, and the set of reals has "even more infinitely" many elements
> in a certain well defined sense.
>
> #2 instead relates to some notion of the "size" of the different
> elements within a set.  This is tricky because depending on what you're
> doing you might care about a different definition of what "size" is. For
> example with positive integers you might care about the number of
> digits, but with real numbers you might care about the magnitude of the
> number with relation to the < and > relations (which is certainly *not*
> the same as the number of digits in it!).  Often you just figure it out
> from context.
>
> So this tells you a little bit about the concepts, but I think that some
> examples are useful in getting the mathematical intuitions.  For instance:
>
> * If you have a set of integers, where the number of digits in each
> integer is bounded, then there are only finitely many integers in that
> set (as you said, at most 10^N).
>
> * If you have a set of integers where the number of digits in each
> integer is unbounded, then there are infinitely many integers in that
> set.  For although each particular integer has only a finite number of
> digits, for any particular number N, you can always "look further" and
> find an integer with more than N digits.

I guess the problem is I'm thinking of each "integer" as a set of 
digits. This set has a finite yet unbounded number of elements. But then 
if you make a set of all possible "integers" you don't get a set with a 
finite yet unbounded number of elements, you get a set with an infinite 
number of elements.

Post a reply to this message

On 7/17/2013 12:44 AM, scott wrote:
>
> I guess the problem is I'm thinking of each "integer" as a set of
> digits. This set has a finite yet unbounded number of elements. But then
> if you make a set of all possible "integers" you don't get a set with a
> finite yet unbounded number of elements, you get a set with an infinite
> number of elements.

It is a strange concept, and it's not for nothing that some people 
reject it entirely.  I don't really have an explanation which will make 
it less strange, except to mention that mathematically it might be best 
to think of "infinite" as meaning "not finite".

Let me explain a bit more.  If a set S has a finite number of elements, 
then there is some integer n such that S has n elements (this is pretty 
much the definition of "finite").  Therefore, if you can show that no 
matter which number n you pick, S will always have more then n elements, 
they s is not finite, and therefore infinite.

You probably already understand this reasoning, if which case I doubt it 
did much to make it "feel" like it makes sense.  The real strange thing 
that's going on is that you're grouping all the integers into a single 
set.  This is sort of the difference between unbounded and infinite in 
this case (i.e. with positive integers).  The individual integers in 
this set increase without bound, but it's the while set of all of them 
that's infinite.  With relation to the above definition of "finite" you 
can think that for any particular integer i, there's some n such that i 
<= n (obviously), as soon as you consider *all* the integers as a single 
thing this stops being true.

I don't know if this ever starts to make sense so much as you just start 
to get used to it.

Post a reply to this message

On Sat, 13 Jul 2013 15:50:42 +0200, Warp <war### [at] tagpovrayorg> wrote:

> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)
>
> 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.
>
> 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?
>
> 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?
>
> 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".)
>

I don't know.

Maybe I should start with something simpler.

Infinity + Infinity = ?

Is there any other answer than 2(Infinity)?

If so please explain.

-Unlike Warp I don't know the answer and really want to know :)

-- 
-Nekar Xenos-

Post a reply to this message

Am 19.07.2013 20:54, schrieb Nekar Xenos:

> Maybe I should start with something simpler.
>
> Infinity + Infinity = ?
>
> Is there any other answer than 2(Infinity)?

Yes: Infinity + Infinity = Infinity.

> If so please explain.

I can't. I guess it's just the way infinity is defined.

Note that there's no such thing as infinity in real life, so there's no 
"natural" way of dealing with infinity. It's an artificial construct, 
and as such is defined in a way that applying the "natural" rules of 
arithmetic to it does't yield results too useless for too many 
applications too much of the time.

Post a reply to this message

On Fri, 19 Jul 2013 21:13:39 +0200, clipka <ano### [at] anonymousorg> wrote:

> Am 19.07.2013 20:54, schrieb Nekar Xenos:
>
>> Maybe I should start with something simpler.
>>
>> Infinity + Infinity = ?
>>
>> Is there any other answer than 2(Infinity)?
>
> Yes: Infinity + Infinity = Infinity.

This is what I thought, but didn't have any evidence for it.

-- 
-Nekar Xenos-

Post a reply to this message

Le 19/07/2013 20:54, Nekar Xenos nous fit lire :
> I don't know.
> 
> Maybe I should start with something simpler.
> 
> Infinity + Infinity = ?
> 
> Is there any other answer than 2(Infinity)?

Well, if anything else, that's never the answer.

Infinity + Infinity = Infinity.

> 
> If so please explain.

You have to be aware that some Infinity are bigger than some other
Infinity. (rahh... not now!)

To ease the understanding, the smallest infinity is called Aleph-0.
(well, that still a controversial subject, linked to the Axiom of Choice)

The natural integer (0, 1, 2, .. and so on) are in the N set, whose
cardinality is indeed Aleph-0.

Imagine one hotel of infinite room, each room numbered uniquely from N
set. And now you build another one next to it, of same size.
You then renumber the room of the first hotel x--> 2x, and of the second
hotel y--> 2y+1. You now have a single hotel-company of the same size as
one of the original.

You can also split it in multiple (finite number) hotels, all of
infinite size... they were already there!

Post a reply to this message

On 19/07/2013 07:54 PM, Nekar Xenos wrote:
> I don't know.
>
> Maybe I should start with something simpler.
>
> Infinity + Infinity = ?
>
> Is there any other answer than 2(Infinity)?
>
> If so please explain.
>
> -Unlike Warp I don't know the answer and really want to know :)

"Infinity" is rather vague.

Let's talk specifics. Suppose we have the set of all natural numbers 
(i.e., all positive whole numbers). There are infinity of these.

This set is commonly denoted by the latter N. I'm going to refer to the 
size of this set as being A0.

Now, suppose we compare N, the set of all *positive* integers, to Z, the 
set of *all* integers, negative and positive. Clearly there are twice as 
many of these. For every item in N, there are two corresponding items in 
Z, one negative and one positive. So the size of Z is A0 + A0.

But wait...

I can map every odd number in N to a negative number in Z, and every 
even number in N to a positive number in Z. In this way, every element 
of N is mapped to *one* element of Z, and every element of Z is mapped 
to *one* element of N.

The existence of this mapping _proves_ that the sets N and Z are 
ACTUALLY THE SAME SIZE! O_O

In other words, A0 + A0 = A0, exactly unchanged.



This seems utterly bizarre. But it's a consequence of infinitely large 
quantities. To understand, try doing the same trick to prove that the 
set {1 .. 5} is the same size as the set {-5 .. +5}. If you try this, 
you'll discover the following:

   {1 .. 5}   {-5 .. +5}
      1    <->    -1
      2    <->    +1
      3    <->    -2
      4    <->    +2
      5    <->    -3
                  +3
                  -4
   Oh bugger!     +4
                  -5
                  +5

It doesn't work. You can't match them up. Because - AS IS FRIGGING 
OBVIOUS - the sets aren't the same size. Basically you run out of 
elements of the left set before you've matched up all the elements of 
the right set.

The ONLY reason this works with infinite sets is that, because they're 
infinite, you will never "run out" of stuff to match up. Because of 
this, even "doubling" the size of an infinite set actually leaves its 
size unchanged - bizarre as that is!



If adding a set to itself once does nothing, doing this twice also does 
nothing. That is,

   A0 + A0 + A0 = A0

In general,

   x * A0 = A0

This is clearly true for any finite x. But what if x ISN'T FINITE? What 
happens if we do this:

   A0 * A0 = ???

Remember that A0 is the size of N, the set of all integers. So if we 
construct the set of all PAIRS of integers, like this...

    (1, 1)
    (1, 2)
    (1, 3)
      .
      .
      .
   infinity
    (2, 1)
    (2, 2)
    (2, 3)
      .
      .
      .
   infinity
    (3, 1)
    (3, 2)
    (3, 3)
      .
      .
      .
   infinity

Surely, the size of this set is CLEARLY equal to A0 * A0.

But wait...

If I take both numbers, add zeros to pad them out to the same length, 
and then interleave their digits, I get this:

      ...           ...
      ...           ...
   (123, 456) <-> 142536
   (123, 457) <-> 142537
   (123, 458) <-> 142538
      ...           ...
      ...           ...

Every number on the right matches exactly one pair on the left. Every 
pair matches on number on the right. THE SETS HAVE THE SAME SIZE! We 
just proved that A0 * A0 = A0.



Just when it's starting to look like A0 is A0 no matter what you do to 
it, something interesting happens.

The size of N is A0. If we construct all possible subsets of N, then for 
each element of N, a given subset of N either DOES or DOES NOT contain 
that element. That's 2 possibilities, independently for each element of 
N. So that's a total of 2 ^ A0 possible subsets.

It turns out that (2 ^ A0) > A0.

Yes sir, there are actually DIFFERENT SIZES of infinity. Let us write

   A1 = 2 ^ A0

   A1 > A0

But, by the same algorithm, 2 ^ A1 > A1, so we actually have

   A(n+1) = 2 ^ A(n)

   A(n+1) > A(n)

In summary, infinity comes in different sizes. (Now you see why the 
original question "what is infinity plus infinity?" is vague; there's 
one than one infinity you may ask about.)

In general,

   A(x) + A(y) = A(y)   iff y >= x

That is, if you add two infinities, the result is the size of whichever 
is the largest infinity. (Thus, when you add a particular infinity to 
itself, nothing happens.)

Post a reply to this message

On 7/19/2013 11:54 AM, Nekar Xenos wrote:
>
> I don't know.
>
> Maybe I should start with something simpler.
>
> Infinity + Infinity = ?
>
> Is there any other answer than 2(Infinity)?
>
> If so please explain.
>
> -Unlike Warp I don't know the answer and really want to know :)
>

If you've even hung around people majoring in math in undergrad, you may 
have heard stories of a set theory class in which a they actually spend 
a day's class proving that 1+1 = 2.  On the face of it this seems like a 
huge waste of time, why would you bother to construct a proof of such an 
obvious fact?  Well, it's in answering questions such as yours in which 
such effort pays off.

There are a few rather subtle points to a simple question like "what is 
1+1":

1) What is a number?
2) What is addition?

And in addition, for your question:

3) What do we mean by infinity?

For the most part your answers to these questions don't matter for 
ordinary finite numbers, but as soon as you start treating infinity like 
a number these subtle points start to matter a great deal.

In terms of your question "Infinity + Infinity = ??" you most commonly 
see one of two answers:

a) Infinity * 2   (note: not 2 * Infinity, the order often matters)
b) Infinity

And this entirely glosses over issues that for some answers to questions 
1-3 you can get more than one infinite number, at which point it matters 
which infinities you were adding!

Sorry I don't have a simpler answer.  But basically the only real way to 
answer it is "tell me what you really mean by Infinity, and only then 
can I tell you what Infinity+Infinity is".  The answers given by clipka 
and Le_Forgeron/Orchid are but two of many possibilities (although they 
both get the same result, they are using different interpretations of 
"infinity" to do it).

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On Fri, 19 Jul 2013 22:10:11 +0200, Orchid Win7 v1 <voi### [at] devnull> wrote:

> On 19/07/2013 07:54 PM, Nekar Xenos wrote:
>> I don't know.
>>
>> Maybe I should start with something simpler.
>>
>> Infinity + Infinity = ?
>>
>> Is there any other answer than 2(Infinity)?
>>
>> If so please explain.
>>
>> -Unlike Warp I don't know the answer and really want to know :)
>
> "Infinity" is rather vague.
>
> Let's talk specifics. Suppose we have the set of all natural numbers  
> (i.e., all positive whole numbers). There are infinity of these.
>
> This set is commonly denoted by the latter N. I'm going to refer to the  
> size of this set as being A0.
>
> Now, suppose we compare N, the set of all *positive* integers, to Z, the  
> set of *all* integers, negative and positive. Clearly there are twice as  
> many of these. For every item in N, there are two corresponding items in  
> Z, one negative and one positive. So the size of Z is A0 + A0.
>
> But wait...
>
> I can map every odd number in N to a negative number in Z, and every  
> even number in N to a positive number in Z. In this way, every element  
> of N is mapped to *one* element of Z, and every element of Z is mapped  
> to *one* element of N.
>
> The existence of this mapping _proves_ that the sets N and Z are  
> ACTUALLY THE SAME SIZE! O_O
>
> In other words, A0 + A0 = A0, exactly unchanged.
>
>
>
> This seems utterly bizarre. But it's a consequence of infinitely large  
> quantities. To understand, try doing the same trick to prove that the  
> set {1 .. 5} is the same size as the set {-5 .. +5}. If you try this,  
> you'll discover the following:
>
>    {1 .. 5}   {-5 .. +5}
>       1    <->    -1
>       2    <->    +1
>       3    <->    -2
>       4    <->    +2
>       5    <->    -3
>                   +3
>                   -4
>    Oh bugger!     +4
>                   -5
>                   +5
>
> It doesn't work. You can't match them up. Because - AS IS FRIGGING  
> OBVIOUS - the sets aren't the same size. Basically you run out of  
> elements of the left set before you've matched up all the elements of  
> the right set.
>
> The ONLY reason this works with infinite sets is that, because they're  
> infinite, you will never "run out" of stuff to match up. Because of  
> this, even "doubling" the size of an infinite set actually leaves its  
> size unchanged - bizarre as that is!
>
>
>
> If adding a set to itself once does nothing, doing this twice also does  
> nothing. That is,
>
>    A0 + A0 + A0 = A0
>
> In general,
>
>    x * A0 = A0
>
> This is clearly true for any finite x. But what if x ISN'T FINITE? What  
> happens if we do this:
>
>    A0 * A0 = ???
>
> Remember that A0 is the size of N, the set of all integers. So if we  
> construct the set of all PAIRS of integers, like this...
>
>     (1, 1)
>     (1, 2)
>     (1, 3)
>       .
>       .
>       .
>    infinity
>     (2, 1)
>     (2, 2)
>     (2, 3)
>       .
>       .
>       .
>    infinity
>     (3, 1)
>     (3, 2)
>     (3, 3)
>       .
>       .
>       .
>    infinity
>
> Surely, the size of this set is CLEARLY equal to A0 * A0.
>
> But wait...
>
> If I take both numbers, add zeros to pad them out to the same length,  
> and then interleave their digits, I get this:
>
>       ...           ...
>       ...           ...
>    (123, 456) <-> 142536
>    (123, 457) <-> 142537
>    (123, 458) <-> 142538
>       ...           ...
>       ...           ...
>
> Every number on the right matches exactly one pair on the left. Every  
> pair matches on number on the right. THE SETS HAVE THE SAME SIZE! We  
> just proved that A0 * A0 = A0.
>
>
>
> Just when it's starting to look like A0 is A0 no matter what you do to  
> it, something interesting happens.
>
> The size of N is A0. If we construct all possible subsets of N, then for  
> each element of N, a given subset of N either DOES or DOES NOT contain  
> that element. That's 2 possibilities, independently for each element of  
> N. So that's a total of 2 ^ A0 possible subsets.
>
> It turns out that (2 ^ A0) > A0.
>
> Yes sir, there are actually DIFFERENT SIZES of infinity. Let us write
>
>    A1 = 2 ^ A0
>
>    A1 > A0
>
> But, by the same algorithm, 2 ^ A1 > A1, so we actually have
>
>    A(n+1) = 2 ^ A(n)
>
>    A(n+1) > A(n)
>
> In summary, infinity comes in different sizes. (Now you see why the  
> original question "what is infinity plus infinity?" is vague; there's  
> one than one infinity you may ask about.)
>
> In general,
>
>    A(x) + A(y) = A(y)   iff y >= x
>
> That is, if you add two infinities, the result is the size of whichever  
> is the largest infinity. (Thus, when you add a particular infinity to  
> itself, nothing happens.)

This is all very interesting. :)

I was thinking of adding any particular infinity to itself.

-- 
-Nekar Xenos-

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On Fri, 19 Jul 2013 22:34:29 +0200, Kevin Wampler <nob### [at] nowherenet>  
wrote:

> On 7/19/2013 11:54 AM, Nekar Xenos wrote:
>>
>> I don't know.
>>
>> Maybe I should start with something simpler.
>>
>> Infinity + Infinity = ?
>>
>> Is there any other answer than 2(Infinity)?
>>
>> If so please explain.
>>
>> -Unlike Warp I don't know the answer and really want to know :)
>>
>
> If you've even hung around people majoring in math in undergrad, you may  
> have heard stories of a set theory class in which a they actually spend  
> a day's class proving that 1+1 = 2.  On the face of it this seems like a  
> huge waste of time, why would you bother to construct a proof of such an  
> obvious fact?  Well, it's in answering questions such as yours in which  
> such effort pays off.
>
> There are a few rather subtle points to a simple question like "what is  
> 1+1":
>
> 1) What is a number?
> 2) What is addition?
>
> And in addition, for your question:
>
> 3) What do we mean by infinity?
>
> For the most part your answers to these questions don't matter for  
> ordinary finite numbers, but as soon as you start treating infinity like  
> a number these subtle points start to matter a great deal.
>
> In terms of your question "Infinity + Infinity = ??" you most commonly  
> see one of two answers:
>
> a) Infinity * 2   (note: not 2 * Infinity, the order often matters)
> b) Infinity
>
> And this entirely glosses over issues that for some answers to questions  
> 1-3 you can get more than one infinite number, at which point it matters  
> which infinities you were adding!
>
> Sorry I don't have a simpler answer.  But basically the only real way to  
> answer it is "tell me what you really mean by Infinity, and only then  
> can I tell you what Infinity+Infinity is".  The answers given by clipka  
> and Le_Forgeron/Orchid are but two of many possibilities (although they  
> both get the same result, they are using different interpretations of  
> "infinity" to do it).
>

I think I could say the specific infinity I'm thinking of would be the  
biggest type of infinity.
What would that be?
Complex Infinity? (if that could be considered)


-- 
-Nekar Xenos-

Post a reply to this message

On 19/07/2013 10:48 PM, Nekar Xenos wrote:
> I think I could say the specific infinity I'm thinking of would be the
> biggest type of infinity.
> What would that be?


You will be sorry you asked that. :-P

-- 
Regards
     Stephen

Post a reply to this message

On 19/07/2013 8:54 PM, Nekar Xenos wrote:
> Is there any other answer

How about and we may have an infinity of authors spinning in their graves:

The sum of all the odd numbers is infinity.
The sum of all even numbers is infinity.
The sum of all numbers odd and even is infinity.

So by doing ordinary sums:
infinity + infinity = infinity
and
infinity = infinity * 2

Can't be right. And that is because infinity is not a number but an idea.

But if you make a mathematics using infinity as an object it opens a 
whole new way for mathematicians to think.

It was at this point that I thought that I would leave it for minds that 
want to see things with out drugs.

Give the book that I mentioned at the beginning of this thread "White 
Light" a try if you can find it.


-- 
Regards
     Stephen

Post a reply to this message

On 7/19/2013 1:48 PM, Nekar Xenos wrote:
>
> I think I could say the specific infinity I'm thinking of would be the
> biggest type of infinity.
> What would that be?
> Complex Infinity? (if that could be considered)
>

There are two different ways I'm tempted to interpret your question, but 
only one of them makes sense.  I'll try to answer them both anyway:

Q1) "Out of all the different ways you can define Infinity, what's the 
biggest?

A1) Because the different definitions of Infinity use different 
definitions of what a "number" is, there is no way to compare them at 
all to say which is bigger -- they are just completely different things. 
  It's like asking "which is bigger, 4 or fish?".

Q2) "You mentioned that for some definitions you get multiple different 
types of infinity, what's the biggest of those?"

A2) The answers depends on what particular definition of Infinity you're 
talking about, but the most common answers is that there is no biggest 
Infinity -- just like there's no biggest finite integer.  Sometimes 
people will try to add a "biggest infinity" to things, but you don't 
generally allow addition with it anyway.

---

As I mentioned in (A1), it doesn't make sense to ask if "complex 
Infinity" is bigger than another definition of infinity.

As an aside, a notion of "complex Infinity" is actually extremely useful 
in some areas mathematics.  Arguably much more useful than "real 
Infinity" is.  The standard definition of complex Infinity does not 
allow Infinity+Infinity though (it treats it as undefined, much like 1/0 
is commonly treated as undefined for the reals).

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