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



## kylefoley76 (Oct 21, 2010)

a computer can't understand what infinity is, can it?


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## MrKowz (Oct 21, 2010)

It depends how Infinity is being used.  Computers operate on a mathematical and logic based system, in which case, Infinity cannot ever be attained by a computer, since Infinity is an undefined number that cannot be reached.  However, there are mathematical proofs that can determine the value of something that has been iterated an infinite number of times.

In all truths, humans can't understand what infinity is either.  It is always 1 more higher than the highest number we can think of.


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## kylefoley76 (Oct 21, 2010)

so if you ask a computer to choose a real number among an infinite amount, it can't do that, right?

I need to know this for a philosophy paper.


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## MrKowz (Oct 21, 2010)

To my knowledge, it cannot.  In order for a computer to choose a number from an infinite series of numbers, infinity must be defined.  However, infinity is unable to be defined, as once it is defined, it is no longer infinity; it becomes a real number.

A computer can choose a number between 1 and 9.9E+999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

But that is not infinity.

Also, the computer would have to return some defined amount.  But say it did choose "infinity"... how could it possibly be displayed?  It would take an infinite amount of time to calculate, an infinite amount of time to display, and an infinite amount of time to read.


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## kylefoley76 (Oct 21, 2010)

Ok, thanks.  This is what I thought, I just wanted to hear someone who sounds like he knows what he's talking about, also say it. Thank you for your input.


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## Greg Truby (Oct 21, 2010)

kylefoley76 said:


> a computer can't understand what infinity is, can it?
> ...
> I need to know this for a philosophy paper.


 
Erm, at least under present technology, a computer can't truly "understand" the warmth of a hug or the saltiness of a tear drop, for that matter.  I'd say a computer would have a better chance of "understanding" infinity [even if it cannot actually arrive at it] before it'll understand those.


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## cornflakegirl (Oct 22, 2010)

I believe that Turing did a lot of theoretical work on what a computing machine could and could not do in principle. Might be work seeing if he said anything on the subject.


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## revans (Oct 22, 2010)

If you take into account our current conception of computers and how they work, Greg has it right that computers "understand" nothing.  Even the most advanced AI can't really "understand" anything, they just have fuzzy logical routines to sort out possiblities and so forth.

This is not to say that a computer *can't* understand things.  
You should investigate the implications of "understanding" and take into account *our* understanding of our environment and of what we call computers are capable.  Kant's Evil Entity could very well be a computer, one sufficiently advanced that we have yet to see--one the Evil Entity is keeping from us.

Rich


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## DiscoPistol (Oct 25, 2010)

You could also read 'Beer Cans and Meat Machines' by John Searle


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## Expiry (Oct 25, 2010)

The oddest thing I heard about infinity, recently, was that there are just as many even numbers are there are odd and even numbers put together.

That kind of screws with your head.


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## Domski (Oct 25, 2010)

I quite enjoyed this program that was on the BBC earlier this year. The whole concept of infinity* still makes my brain hurt though. Don't think it's available to watch any more but you might be able to find it somewhere.

Dom

* Along with most other maths to be fair


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## cornflakegirl (Oct 25, 2010)

Expiry said:


> The oddest thing I heard about infinity, recently, was that there are just as many even numbers are there are odd and even numbers put together.
> 
> That kind of screws with your head.



Yeah, that would be Aleph Zero, the infinity of natural numbers. It's intuitively true, because you can set up a 1:1 mapping between the even numbers and the natural numbers, with each natural number mapping to the (even) number that is twice itself (ie 1 maps to 2, 2 maps to 4, 3 maps to 6 etc). Since you don't run out of numbers on either side, it's clearly true.

However, since the even numbers are a subset of the natural numbers, it's also quite clearly bonkers!

I love infinity.

I was listening to something on Radio 4 recently (think it was called A Brief History of Maths, with Marcus du Sautoy), and learned that there an infinite number of infinities. My degree only covered two (natural numbers and real numbers), so I really must find out what all the rest are for...


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## Greg Truby (Oct 25, 2010)

cornflakegirl said:


> ...so I really must find out what all the rest are for...


 
You might want to leave that for the afterlife - sounds like the perfect project for Eternity.

I've sometimes wondered, if, indeed infinity is actually needed.  At some point, you run out of things to count.  If we take the smallest of the subatomic particles and figure out however many there are of these in the universe; wouldn't that be, from a pragmatist's POV, the highest number we should ever need?


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## Expiry (Oct 25, 2010)

Greg Truby said:


> You might want to leave that for the afterlife - sounds like the perfect project for Eternity.
> 
> I've sometimes wondered, if, indeed infinity is actually needed.  At some point, you run out of things to count.  If we take the smallest of the subatomic particles and figure out however many there are of these in the universe; wouldn't that be, from a pragmatist's POV, the highest number we should ever need?



I think if I counted  my wife's shoes, I would need a higher number than that.


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## Mark O'Brien (Oct 25, 2010)

Greg Truby said:


> I've sometimes wondered, if, indeed infinity is actually needed.  At some point, you run out of things to count.  If we take the smallest of the subatomic particles and figure out however many there are of these in the universe; wouldn't that be, from a pragmatist's POV, the highest number we should ever need?



You always need more than infinity just to get it over the edge.  To paraphrase Nigel Tufnel, "This is one louder.  These go to Infinity plus one."


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## MrKowz (Oct 25, 2010)

Expiry said:


> I think if I counted my wife's shoes, I would need a higher number than that.


 
This


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## RoryA (Oct 25, 2010)

My computer understands infinity quite well - that's what it loops to when I'm not paying enough attention.


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## Easy-XL Support (Oct 25, 2010)

My new Core I7 is so fast it can process an infinite loop in under 10 seconds!


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## DBA (Oct 26, 2010)

Greg Truby said:


> If we take the smallest of the subatomic particles and figure out however many there are of these in the universe; wouldn't that be, from a pragmatist's POV, the highest number we should ever need?


So what you're saying is - "It's the little things in life that count"


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## DiscoPistol (Oct 26, 2010)

Chuck Norris counted to infinity. Twice.


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## PaddyD (Oct 26, 2010)

"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>
<o></o>
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?"
<o></o> 
<o></o>
<o></o>
(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>


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## Lewiy (Oct 27, 2010)

PaddyD said:


> 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


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## Taul (Oct 27, 2010)

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


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## mikerickson (Oct 27, 2010)

"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.


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## kylefoley76 (Oct 27, 2010)

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.


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## mikerickson (Oct 27, 2010)

kylefoley76 said:


> ...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.


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## kylefoley76 (Oct 27, 2010)

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.


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## Lewiy (Oct 27, 2010)

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.


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## cornflakegirl (Oct 27, 2010)

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


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## Taul (Oct 27, 2010)

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)


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## Domski (Oct 27, 2010)

Paul-H said:


> There are a few things that spring to mind.



- Why am I here after 5:00pm?
- Should I go for a pint?
- Steak or pizza for tea?

That's about all that springs to my mind at the moment.

Dom


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## kylefoley76 (Oct 27, 2010)

“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."

you misunderstand me.  if you create something, let's say, a language, you can attach an infinite number of arbitrary symbols to it.  you could come up with a 1000 types of horse, a 100 types of snow, etc.  you're looking at something after it has been done and saying it's limited. 

someone mentioned the anthropic principle.  they're misunderstanding it.  you could say the same thing about stonehenge:

stonehenge exists, therefore it could come about at random. 

just because something exists, does not follow that it is random


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## mikerickson (Oct 27, 2010)

kylefoley76 said:


> 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.



The references at http://en.wikipedia.org/wiki/Chomsky_normal_form should give you somewhere to go.


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## mikerickson (Oct 27, 2010)

"A Human can utter an infinite number of sentences" is different than "There are an infinite number of sentences that can be uttered by a human."

But both are wrong. Both for the same reason.
The first avers that a human can utter, but the death thing shows that this is false.
The second avers that the set of sentences that can be uttered (once) by a human is infinte. 
But..If we define length of a sentence to be the number of seconds it takes to utter, then the second hypothisis is that there are in infinite number of utterable sentences i.e. that there are an infinite number of sentences that can be uttered in less than "3 score and 10" years.

I question that implication. (If there are a fintite number of sounds that the human mouth can utter, then it is false.)


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## PaddyD (Oct 27, 2010)

It is not clear to me why "There are an infinite number of sentences that can be uttered by a human" should take the limited interpretation "...the set of sentences that can be uttered (once) by a human is infinite."<?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o>
<o></o>
If this is taken to mean something like "It is possible to construct a grammatically correct sentence of arbitrary length." then the set of possible sentence is straightforwardly infinite (E.g. Keep on adding a he said / she said to a sentence of the form "He said that she said that he said that she said it was raining"). The fact that only a finite number of them could ever be uttered in a finite universe is true, but a different issue surely. (Or at least if you don't think the set of possible sentences is infinite then you can't think the natural numbers are either). <o></o>
<o></o>
But all of this is moot given what the OP’s is apparently trying to do:<o></o>
“…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.”<o></o>
There is a confusion here, especially given where OP’s trying to head re ‘coordination’. Randomness is, as a starting point, about the distribution of things. How you choose among a (possibly infinite) number of sets with randomly distributed characteristics is something totally different. One obvious solution has been implied by earlier post from Lewiy – simple allow there to exist an infinite number of universes and just stipulate that you happen to live in this one. <o></o>
p.s. thanks Mike for picking up the ‘reals are countably infinite’ slip J<o></o>
p.p.s. “…but I think this is heading towards one of the 3 subjects that forums generally try to avoid (race, religion & politics)” Concur. Let’s be careful here – any fights over creationism etc and this thread gets shut down J. <o></o>


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## Taul (Oct 27, 2010)

mikerickson said:


> 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.




Original question was; “a computer can't understand what infinity is, can it?”

Based on what I have read in the threads and in particular Mike’s comment:-

Computers Understand infinity = no (not wishing to imply computers can undersatand)
Work with infinity = yes


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## Greg Truby (Oct 27, 2010)

PaddyD said:


> “…but I think this is heading towards one of the 3 subjects that forums generally try to avoid (race, religion & politics)” Concur. Let’s be careful here – any fights over creationism etc and this thread gets shut down J. <?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o>


 
Thus far this thread has inspired a lot of good posts regarding math theory and some general philosophical thoughts on computers and "comprehension". If it does veer off into religion then one of us moderators will just move any nettlesome posts off the thread. But so far I think we've done a great job of staying out of the brambles. So, back to discussion at hand:

I would say that - if one goes with the ever-expanding universe, i.e. no Restaurant at the End of the Universe dinner, and one assumes that humankind can indeed conquer interstellar travel to defeat the sun's eventual burnout and one assumes that language will never cease to evolve, then yeah, I can buy into the number of sentences that humankind can create could indeed be infinite while still obeying some form of grammatical structure.  But wouldn't that still qualify for the "there are as many even numbers as there are numbers" thing, i.e. there are as many sensical sentences as there are sensical + gibberish sentences?


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## arkusM (Oct 27, 2010)

Not to drag this down to the knuckle draggers (I have enjoyed this where I can follow it) but I do think that people can utter an infinite amount of ...strange.. things. Generally, take a look at our respective counties politicians, there seems no end to the most mind numbing drivel the spews forth from their mouths. (Sorry to dance near one of the 3, but no names, and none blameless)

Didn't Einstain say something about mans foolishness....


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## arkusM (Oct 27, 2010)

Ahh got it.
"Only two things are infinite, the universe and human stupidity, and I'm not sure about the former."


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## ZVI (Oct 27, 2010)

It is possible to program infinity in terms of its declaration.

Operations with infinite values are defined in IEEE floating point standard.

Some mathematical software uses the special values to represent of positive and negative infinity. 

For example MATLAB represents infinity by the special value inf. Infinity results from operations like division by zero and overflow, which lead to results too large to represent as conventional floating-point values. MATLAB also provides a function called inf that returns the IEEE arithmetic representation for positive infinity as a double scalar value. There are some other functions which operate with infinite values.

MAPPLE supports operations with infinite values as well:


infinity is a name in Maple which has several special properties. 

infinity is used to denote a mathematical infinity, and hence it is usually used as a symbol by itself or as -infinity.

Several functions accept infinity as a parameter or produce it as a result of computation. The to part of the for loop statement accepts infinity as an argument, which will cause it to loop forever. 
The kernel can compare infinity or -infinity to any other numerical value. Hence, infinity can be used in boolean expressions (for example, max, min, etc.). 

In the floating-point computation domain, infinity is represented as Float(infinity), and -infinity is represented as Float(-infinity) or -Float(infinity). (The exponent fields are both the symbol infinity, while the mantissas are 1 and -1, respectively.) For completeness, these can be entered as Float(n, infinity), where n is a non-0 integer. Float(n, infinity) automatically simplifies to sgn * Float(infinity). 

Float(infinity) cannot be assumed to represent the mathematical concept of infinity, as it may also arise from non-infinity operations, such as overflows. Care must therefore be taken when converting between computation domains (e.g., via round()) that spurious information is not artificially created. 

The quantities infinity, -infinity, infinity*I, -infinity*I, infinity + y*I, -infinity + y*I, x + infinity*I and x - infinity*I, where x and y are finite, are all considered to be distinct in Maple. However, all 2-component complex numerics in which both components are infinity are considered to be the same (representing the single point at the "north pole" of the Riemann sphere). 

Similarly, different floating-point infinities are generally considered to represent distinct entities, except that the four infinities Float(+-infinity+-infinity*I) are considered to be the same.


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## PaddyD (Oct 27, 2010)

"But wouldn't that still qualify for the "there are as many even numbers as there are numbers" thing, i.e. there are as many sensical sentences as there are sensical + gibberish sentences? "<?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o></o>
 <o></o>
Hmmm.  That one's a bit more complicated.  Thinking out loud:<o></o>
 <o></o>
- let us assume a finite dictionary of meaningful words (he, she, said, it, that, was raining) and a defined grammar that allows recursion (i.e. you can only generate gibberish by grammatical errors, not meaningless words, and sentences of the form "He said that she said that he said...." are OK). <o></o>
- you will exhaust the list of grammatically well-formed, non-recursive sentences pretty quickly (let's call this set the set of first order sentences – things like ‘He said it was raining.’)<o></o>
- add the first layer of recursion to each of the sentences in the first set to get second order set and repeat indefinitely (‘He said she said it was raining’).  <o></o>
- the recursive process for generating new sentences is exactly the same as the recursive process for generating new numbers (n+1).  So the size of the set of sentences in this language will be the same as the size as the set of the natural numbers (or odd numbers, or even numbers, or numbers ending in 7 or whatever).  <o></o>
- it feels like the non-grammatical sentence set should also be countably infinite, because the sentences are generated either (a) by a one-off breach of the grammar (an atomic or first order item again – e.g. ‘He it was raining’), or (b) wrapping an atomic error in an otherwise grammatically well-formed recursive structure that is not well formed as a whole (‘She said he it was raining’), or variations on those two approaches <o></o>
- so the question is then (a) does expanding the dictionary make any difference and (b) can you run a version of Cantor’s diagonalisation argument across this?<o></o>
- and I don’t think either would change the outcome, because (a) dictionary expansion would only (countably) increase the size of first order sentences (or that sort of thing) and (b) I can only imagine diagonalisation for natural language as, effectively, changing the spelling of words in the dictionary, in which case back to (a).<o></o>
<o> </o>
<o> </o>
But probably not   <o></o>


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## SuperFerret (Oct 28, 2010)

You'd also have to account for the non-grammatically correct, yet frequently used sentences such as "Going pub" (instead of going _to the _pub) that I frequently hear uttered by family and friends. 

They could potentially add to the finite/infinite sentences utterable


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## Lewiy (Oct 28, 2010)

kylefoley76 said:


> someone mentioned the anthropic principle. they're misunderstanding it. you could say the same thing about stonehenge:
> 
> stonehenge exists, therefore it could come about at random.
> 
> just because something exists, does not follow that it is random


 
I think you’ve missed the point of the anthropic principle.
<?xml:namespace prefix = o ns = "urn:schemas-microsoft-comfficeffice" /><o> </o>
Stonehenge was created for a purpose by human beings.  The concept of purpose is unique to things that are created by or used (in ways other than for their original “design” – for want of a better phrase) by living organisms.  As such, Stonehenge almost certainly wasn’t created at random (although the argument that it could have been created at random still holds to some degree).
<o> </o>
The anthropic principle covers the argument that the conditions for life MUST exist on Earth in our Universe because here we are pondering its existence.  This in turn suggests a much higher possibility that the properties of the Universe are random than that Stonehenge came into being at random.  I don’t believe that the anthropic principle has any connection at all with the concept of “man-made” objects.


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