it's not possible to program infinity, is it?

[PaddyD]

"so if you ask a computer to choose a real number among an infinite amount, it can't do that, right"<?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o>
<o></o>
You could construct an argument either way. It depends how you operationalise the problem, whether or not you're a constructivist with respect to definitions of infinity, what you think is meant by 'choosing among an infinite amount' etc etc. <o></o>
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Consider: <o></o>
<o></o>
=randbetween(1,2)<o></o>
<o></o>
...the result is either one or two, both of which are clearly numbers in the set of real numbers. The set of real numbers is a countably infinite set. Excel can implement the formula. Excel is implemented in a computer. q.e.d. a computer can select a number that is in the infinte set of real numbers. If you think this is the same as "choose among an infinite amount', then the answer to the question is clearly yes. If you don't then you're going to have to work out which definitions you want to explore further. <o></o>
<o></o>
For example, what you clearly can't do is:<o></o>
<o></o>
=randbetween(1,∞ )<o></o>
<o></o>
...because infinity is not a number, it is (in the context of what computers can implement) a property of sets (and even if it were a number, the finite limits of a computer's memory will never permit it to be represented). <o></o>
<o></o>
FWIW, this "limitation" is not restricted to 'computers', given that you can't implement randbetween(1, ∞) either: when you "choose among an infinite amount", you do not, for example, first construct the infinite set of all reals, then pick one.

Given the above, perhaps a more revealing question would be "What definition of 'infinity' would be required, such that randbetween(1,∞) could be implemented in a physical device?"
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(By the way, anyone want some fun demonstrating that some infinite sets are bigger than others, have a hunt for "cantor's diagonalisation")<o></o>

 

[Lewiy]

For example, what you clearly can't do is:<?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o>
<o></o>
=randbetween(1,∞ )

That's surely because infinity isn't a valid parameter of the RANDBETWEEN() function

 

[Taul]

We have a number system that doesn’t have an end; we therefore need to call the end (or the thing in the far distance) something. It’s convenient to call it infinity (arguably also necessary) but it isn’t real in any tangible sense.

If you start to treat it as something real, there is a danger we (or the OP philosopher) could start to believe and treat it as real and draw the wrong conclusions.

Based on the link below, the infinity symbol was first used by John Wallis in 1657 to indicate an infinite process, (a mathematical fiddle) and the idea caught on.


http://www.math.tamu.edu/~dallen/history/infinity.pdf

 

[mikerickson]

"infinity" is not a number and can't be treated as such. Arithmetizations of infinity end up with the situation where x+x = x for x other than 0, hence they don't form fields, hence all kinds of stuff goes away.

The Real numbers do NOT form a countable set.
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

The Rationals do form a countable set:
proof:
1) every positive integer can be expressed as the (unique) product of primes,
2^C1 * 3^C2 * .... * Pn^Cn
where Pi is the i'th prime and Ci is a non-negative integer.
[Fundamental Theorem of Arthmetic]

2) every positive rational number can be can be expressed as the (unique) product of primes,
2^C1 * 3^C2 * ...... * Pn^Cn
where Pi is the i'th prime and Ci is an integer (possibly negative).
[Every positive rational number can be expressed as the unique ratio of two relatively prime positive integers. a.k.a. expressing a quotient in lowest terms.
Let A and B be natural numbers such that they are relatively prime where
A= 2^a1 * 3^a2 *...* Pn^an, and
B = 2^b1 * 3^b2 * ...Pn^bn
(Note that either (ai=0 for all i> some k < n ) or (bi=0 for all i > some k <n))
Then A/B = 2^(a1-b1) * 3^(A2-B2) * ... * Pn^(an-bn)
Since a1...an and b1...bn are unique, so are a1-b1...an-bn.]

3)The mapping, t, from the non-negative integers to all integers
t(k) = k/2 if k is even
t(k) = 0-((k+1)/2) if k is odd
is a bijection

4) Thus, the mapping, M, from the natural numbers (non-negative integers) to the rationals
M (2^a1 * 3^a2 * .... Pn^an) = 2^t(a1) * 3^t(a2) * ... * Pn^t(an)
is a bijection.

QED.

Can a computer "understand" infinity?
MatLab can find the limits of functions as x>>infinity, so apparently computers can work with infinity as well as humans can.

 

[kylefoley76]

well, i didn't expect this to generate such discussion but i'm glad it has. let me try to explain how i came onto this issue and let's see what people think. i was trying to prove a point in philosophy, that randomness cannot coordinate properties chosen from an infinite set. let me explain. the amount of utterances a human can utter are infinite. a human can utter an infinite amount of sentences and yet roughly 99.5% of the time his sentences are grammatically correct, in other words, they coordinate amongst each other. when creating a universe the properties of material objects have to coordinate amongst each other. hydrogen has to bond with oxygen in order to form water. if you create a universe and if you have the power to do it, you can assign any property to any object and the choices at your disposal are infinite. so i'm very skeptical that the properties of the objects in our universe were assigned to them at random.

randomness can only chose the right number from a finite set some of the time, randomness cannot choose the right numbers among an infinite set. when we human speak language the sentences we can utter are infinite and we choose the right sentence pretty much all the time.

in order to create universe where organic life is possible a stunning array of properties must be attached to objects, randomness cannot do this.

 

[mikerickson]

...the amount of utterances a human can utter are infinite. a human can utter an infinite amount of sentences ...

Two wrong premises. (Sentences take a non-infanitessimal time to utter. Humans die.)

Have you read Chomsky about recursive grammars? It sounds like that is the direction you are heading.

 

[kylefoley76]

i've listening to about 200 chomsky lectures in my life, but only about 5 of them were about language. so i just have the watered down version of his language theories. but nevertheless, the sentence: "human can utter an infinite amount of sentences" is straight from chomsky."

i'm not saying that humans can utter sentences for an infinite amount of time, but their choices are infinite.

 

[Lewiy]

In order to create a universe where organic life is possible certain conditions must certainly be true. But if a universe existed that could not support organic life, we would not be here to ponder the question in the first place. Therefore, it is entirely plausible that the properties of “elements” of the universe were created completely at random and it just so happened that organic life was one of the results.

 

[cornflakegirl]

I've just been mulling over the anthropic principle myself. It sounds so convincing, and yet I'm never convinced.

 

[Taul]

There are a few things that spring to mind.

There is a theory that random does not exist, there are only patterns but if the pattern is too large for us to identify, we call it random.

quote form chomsky "human can utter an infinite amount of sentences"
There are a finite number of words from which sentences can be constructed; grammar rules also limit the combinations. Therefore there must be a finite number of correct sentences that can be uttered before duplication occurs, and as Mike says, provided you stay alive long enough to say them.

“if you create a universe and if you have the power to do it, you can assign any property to any object and the choices at your disposal are infinite.”
Certain elements are attracted to each other to form bonds, there isn’t an infinite number of combinations.

“in order to create universe where organic life is possible a stunning array of properties must be attached to objects, randomness cannot do this”
I agree – but I think this is heading towards one of the 3 subjects that forums generally try to avoid (race, religion & politics)