One to the power of infinity
- Thread starter embargiel
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So? You just wait until "the monkeys" have finished doing the complete works of Shakespeare and then ask them to look amongst the discarded sheets of paper and worn-out typewriters for the documentation QEDBasic said:.....
Now, what's the probability that the full design and documentation of R500 is present somewhere in a sample from the random number from said source?
(Since the full design and doc of R500 likely isn't done yet, we'll have to wait for the verification of course.)
I read "Godel, Escher, Bach. The Eternal Golden [EDIT]Braid[/EDIT]" over a year ago and I vaguely remember the discussion on Godel's incompleteness theorem but I thought it was in the discussion of the enumeration of Turing machines the diagonal argument was used. I could easily be wrong.Nathan said:
For a start, Cantor's proof (well, as presented in my 1st year lectures (20 years ago!!)) was to try to develop a 1:1 mapping of the natural numbers [1..infinity) to the reals in the range [0..1) and then show that you get a contradiction, i.e. that there are always numbers that you've missed out.
So what you were arguing above is covered by his proof. I know who I'd rather believe
Damn! You've stumbled across our secret technology... we don't have CPU farms.... just rubber duck farms with keyboards.Basic said:Simon: Yep, it's as simple as that.
(Or you could try a horde of higly trained rubber ducks.)
Keeping them fed is a pain but it does have a very useful by-product... polystyrene filling for bean-bags.
We're both right!
http://www.exploratorium.edu/complexity/CompLexicon/godel.html
This has really got me interested. I think I'm going to have to find book on the subject. It's so hard to get a complete view of things like this from the interweb. All the sites I looked at last night only gave very a bridged versions of the proof and none of them talked about 1:1 mappings. :?
arjan de lumens
Veteran
Beyond the 3 space and 1 time dimensions, the easiest way to visualize additional dimensions that I can think of is to use colors to represent the extra dimensions - you should be able to get 1-3 extra dimensions that way (albeit with severely limited range). For 4 space dimensions, you could also try to project objects onto lower-dimensional "surfaces", but this gets unwieldy for more than 4 dimensions. Beyond this point, you will probably have to dive into mathematic formalism to get any further understanding of how additional dimensions work.ByteMe said:
It gets even worse when the dimensions themselves have curvature, such as in Einstein's General Relativity and superstring theories.
I thought it was Reihman who came up with the relationship between real and complex numbers. This relationship showed how the reals could be thought of as a sphere, upon which there are two poles (zero and infinity, south and north respectively) where the south pole would make contact with an orgin on the complex plane. AFAIK, negative numbers can be thought of as another sphere on the other side of the plane.
So you are trying to map a natural number N to the rational number 1/N ? That obviously won't find all real numbers.Nathan said:
It doesn't matter if you try some other mapping - Cantor's diagonal argument will still beat it.
Actually it's really easy to explain his proof - which is by contradiction (i.e. assuming you have something and then showing that this assumption is rubbish).
Imagine that you do have a list (in any order) of all the real numbers between 0 and 1 eg.
Code:
1: 0.0000000000......
2: 0.21254563822.....
3: 0.872517223833.....
4; ... etcIt's now really easy to show that you've got holes in your list. You pick another real number 0.{digit1}{digit2}{digit3}... where
you choose digitN by looking at the Nth decimal place of Number N and selecting something else, eg if it was a 7 choose 3.
Your new number is not in your "complete" list because it's not the same as the Number 1 ('cause digit1 is different) nor 2, nor 3, etc.
Therefore you can't have a list of the reals.
Simon's reference to Cantor's diagonal argument is enough as a proof. But it might not be enough to explain exactly what is wrong with your argument, so I'll elaborate.
I agree that you could have a mapping where 1.000... and ...0001 is mapped to each other. But as I said, that mapping only works for a small subset of the real numbers. That's the set of numbers that can be written E+2^M, where E and M are finite integers. (You can't even get all rational numbers in there.) As I said, the problem is with numbers with an infinite amount of 0's and 1's (infinite amount of each of them).
How would you map PI with your mapping? (Using decimals here for readability.) 3.1415... => ...51413 ?
Notice that the elipsis before 51413 reperesent an infinite amount of digits not equal to 0. That's not an integer!
Remember that even though [1 inf] includes values larger than any given number, each individual value in the range is still finite.
And most real numbers are just like PI, with an infinite number of decimals/binals of different values (not just 0's).
I agree that you could have a mapping where 1.000... and ...0001 is mapped to each other. But as I said, that mapping only works for a small subset of the real numbers. That's the set of numbers that can be written E+2^M, where E and M are finite integers. (You can't even get all rational numbers in there.) As I said, the problem is with numbers with an infinite amount of 0's and 1's (infinite amount of each of them).
How would you map PI with your mapping? (Using decimals here for readability.) 3.1415... => ...51413 ?
Notice that the elipsis before 51413 reperesent an infinite amount of digits not equal to 0. That's not an integer!
Remember that even though [1 inf] includes values larger than any given number, each individual value in the range is still finite.
And most real numbers are just like PI, with an infinite number of decimals/binals of different values (not just 0's).
