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On 13/07/2013 5:44 PM, Warp wrote:
> Stephen <mca### [at] aol com> 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
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>>>> 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.
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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 ?)
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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
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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.
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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
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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
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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?
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