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Thought for today

Nigel Browne
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Joined: May 15, 2001
Posts: 673
If you have two sets of numbers and one set has all integers and the other has the squares of all numbers possible then which set is larger?
The set with the squares of all numbers is missing an infinite number of numbers (for example, 3, 5, 6, 7, 8, 10, etc) and so the one with all possible integers is larger since it contains all numbers.
However, since all numbers have a square then they are the same size. How can the be they same size and one infinitely larger than the other at the same time?
Nigel Browne
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Joined: May 15, 2001
Posts: 673
Are there highly repetitious situations which occur in our lives time and time again, and which we handle in the identical stupid way each time, because we don't have enough of an overview to perceive their sameness? [Hofstadter: G�del, Escher, Bach, p. 614].
Ashok Mash
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Joined: Oct 13, 2000
Posts: 1936
Hmm, interesting.
I think we can safely assume that there is a difference of infinity between these two sets. But since the difference is a ?very small infinity? (since the value of these sets would be really really big infinity), we can even ignore the tiny difference and say both sets are actually very same (ie, infinity).


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Tom Hughes
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Joined: Feb 09, 2002
Posts: 86
The set of all squares is an infinite subset of the infinite set of all integers. let me try and explain.
The size of the set of all squares is infinity
The size of the antiset (I don't think that's the right word, A-Level maths was a long time ago !) of that set, ie. the set of all the numbers that aren't squares (3,5,6,7 etc...) is also infinity.
The size of the set of all integers equals the sum of the sizes of the two sets above i.e. infinity + infinity == infinity
Does that make sense ?
Tom
[ October 09, 2002: Message edited by: Tom Hughes ]
Nigel Browne
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Joined: May 15, 2001
Posts: 673
Originally posted by Tom Hughes:
[QB]The set of all squares is an infinite subset of the infinite set of all integers.
Does that make sense ?
QB]

Err, yes and no, the problem is that as infinity can not be defined there is inconsistancy in your answer.
To explain a bit better, if we call the set of all prime numbers x and the set of all squares y.
We can then write ther equation:
x+y = z
Where z is the set of all whole numbers(infinity)
However the very nature of infinity is that it can not be definitivly measured. Thus disallowing the following:
z + 1 = z (this would lead to an infinite loop)
So if Z can never be known neither can its subsets.
The same is true also in the negative realm where the smallest number can never be known. So is this proof of infinity or is it a case of our minds being limited, in that they can not disprove infinity?
Tom Hughes
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Joined: Feb 09, 2002
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Infinity cannot be defined ? - You learn something new everyday.
Tom
Nigel Browne
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Joined: May 15, 2001
Posts: 673
Originally posted by Tom Hughes:
Infinity cannot be defined ? - You learn something new everyday.
Tom

Touch�
In the dictionary infinity is defined as something which is boundless or endless.
However mathmatically there is inconsistancy in the proof of infinity, because infinity can never be definitively measured.
Tom Hughes
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Joined: Feb 09, 2002
Posts: 86
so, you're saying that mathematically, infinity cannot be proved ?
Ashok Mash
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Joined: Oct 13, 2000
Posts: 1936
I think mathematically we can prove that infinity is 'indefinable'.
If we ever 'define' infinity to 'definite' value or position, every single law in mathematics will fail. )
Tom Hughes
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Joined: Feb 09, 2002
Posts: 86
Can't you define infinity like this ?
infinity = lim(1/x) when x -> 0
Nigel Browne
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Joined: May 15, 2001
Posts: 673
The symbol for infinity is a sideways 8, which is not symmetrical in the loops. The infinity symbol is a simplified Mobius Band.
In geometry they teach that any line of any length consists of an infinite number of points. This is of course, foolish nonsense, and leads to an inconsistent, self-collapsing, paradox-ridden geometry. The Greek thinker Xeno showed with paradox that motion was 'kinema', or a discontinuous series of frozen still positions in space, similar to the illusion of motion produced by cinematic projection of a series of still pictures.
Infinity is like a virus in mathematics. If you subtract any number from infinity it is still infinity; in fact if you subtract infinity you still have infinity and if you add infinity you still have infinity. Consequently, mathematicians consider that there must be "magnitudes of infinity". For example, if the integers are infinite then the real numbers must be of a higher magnitude of infinity because between each of these infinite integers there are an infinite set of fractions. Once introducing infinity, number loses all meaning, and you can never get it out. The virus is such that if you add in infinity, when you subtract it back out the number does not return to it's original state. Infinity makes multiplication and addition to any number have identical results, eliminating the basic axioms of mathematics. Dividing by infinity does not produce zero, it is an invalid operation, just as dividing by zero (which is not only invalid but also a meaningless operation).
Magnitudes of infinity are paradoxical consequences of an invalid concept. Infinity has impossible properties. The true condition must be that every line of every length contains a finite set of points, and the number of points must be proportional to the length of the line. There can be no infinite lines. All lines must wrap around to their origin eventually, in a loop that disappears in the region of "potential infinity". Aristotle expressed the theorem of potential infinity with the phrase "For every number there exists a larger number." He could have benefited from my theorem that "Everything real in the Cosmos is finite." Numbers are concepts, and not real of themselves. I would amend his theorem with the phrase "Every larger number is still finite." Mathematics of Infinity by Thomas Gilmore
Randall Twede
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Joined: Oct 21, 2000
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    2


In geometry they teach that any line of any length consists of an infinite number of points. This is of course, foolish nonsense

i dont care how famous Thomas Gilmore is, i have no trouble seeing that any line has an infinite number of points


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Nigel Browne
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Joined: May 15, 2001
Posts: 673
Originally posted by Randall Twede:

i have no trouble seeing that any line has an infinite number of points

If a line doesn't have a start point and an end point how do you draw it?
Tom Hughes
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Joined: Feb 09, 2002
Posts: 86
why can't a line have a start point and end point and an infinite number of points inbetween ?
Nigel Browne
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Joined: May 15, 2001
Posts: 673
In mathematical theory there are an infinite amount of points in a line segment. So when travelling from point A to point Z a particle would have to go through an infinite series of co-ordinates. In other words it would have to pass through an infinite number of points in a fixed segment of time. Firstly, the distances between points would have to be infinitely small and infinitely small = 0, so in moving from point A to point B it would not have moved at all. Secondly an infinitely small amount of time would have passed i.e.. no time. It is not getting anywhere!
Tom Hughes
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Joined: Feb 09, 2002
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but an inifinitely small amount of distance != 0 it's an infintessimally small distance, so In moving from point A to point B the particle would have moved an infintessimally small distance in an infintessimally small amount of time.
Nigel Browne
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Joined: May 15, 2001
Posts: 673
In mathematical theory there are an infinite amount of points in a line segment. So when travelling from point A to point Z a particle would have to go through an infinite series of co-ordinates. In other words it would have to pass through an infinite number of points in a fixed segment of time. Firstly, the distances between points would have to be infinitely small and infinitely small = 0, so in moving from point A to point B it would not have moved at all. Secondly an infinitely small amount of time would have passed i.e.. no time. It is not getting anywhere!
R K Singh
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Joined: Oct 15, 2001
Posts: 5371
Originally posted by Tom Hughes:
infinity = lim(1/x) when x -> 0

As x is tending towards zero value is tending towards infinity but x can never be zero and value can not be infinity.
Zero & infinity both are undefined.
R K Singh
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Joined: Oct 15, 2001
Posts: 5371
Originally posted by Nigel Browne:
If a line doesn't have a start point and an end point how do you draw it?

Line does have a start point and end point.
By definition:
Shortest distence between two points is straight line
And one more ..
Two parallel lines meet at infinity
Tom Hughes
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Joined: Feb 09, 2002
Posts: 86
you guys are mad . Isn't zero defined as the integer > -1 and < 1 ??
and surely parallel lines never meet ..?
R K Singh
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Joined: Oct 15, 2001
Posts: 5371
Originally posted by Tom Hughes:
you guys are mad . Isn't zero defined as the integer > -1 and < 1 ??
and surely parallel lines never meet ..?

search google OR check any maths book ..
parallel lines do meet at infinity..
Tom Hughes
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Joined: Feb 09, 2002
Posts: 86
by definition parallel lines never meet :
GLOSSARY PROGRAMME OF STUDY
Glossary Contents

Parallel
Two straight lines that stay the same distance apart.
They will never meet.
Parallel lines are indicated with an arrow on each line.
[ October 10, 2002: Message edited by: Tom Hughes ]
R K Singh
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Joined: Oct 15, 2001
Posts: 5371
How lazy you are Tom
http://www.nrich.maths.org.uk/mathsf/journalf/aams/q51.html
Tom Hughes
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Joined: Feb 09, 2002
Posts: 86
Nope, I just don't believe what I read ...
His explanation seems to hinge on the fact that railway tracks appear to converge on the horizon. The keyword is appear. I grant you that parallel lines will appear to meet at infinity but I don't believe that they actually do.
Unconvinced. Maybe we should agree to disagree ?
[ October 10, 2002: Message edited by: Tom Hughes ]
[ October 10, 2002: Message edited by: Tom Hughes ]
R K Singh
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Joined: Oct 15, 2001
Posts: 5371
Search on google man ..
even Einstein relative theory more or less cover it ..
no more spoon feeding ... search google.
Paavam Payyan
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Joined: Jun 25, 2002
Posts: 86
I dont understand the point here! Because Ravish and Tom, both are saying the same thing, are they not? 'Will meet at Infinity', exactly means 'Will never meet'.
Come on big kids! Stop thinking over it now - after all, it was a topic of thought for 'yesterday', not today!


<i>All that is gold does not glitter, not all those who wander are lost - <b>Gandalf</b></i>
Tom Hughes
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Joined: Feb 09, 2002
Posts: 86
but Ravish started it ! Nevermind, I'll exact my revenge by not inviting him to my birthday party
R K Singh
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Joined: Oct 15, 2001
Posts: 5371
When is your b'day?
Tom Hughes
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Joined: Feb 09, 2002
Posts: 86
It's undefined
Paul Stevens
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Joined: May 17, 2001
Posts: 2823
Paavam
2 months.

Ref. your signature.
R K Singh
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Joined: Oct 15, 2001
Posts: 5371
Oh your b'date is infinity
Only parallel lines will be there to say HBD
I like you
Paavam Payyan
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Joined: Jun 25, 2002
Posts: 86
Yup, TWO months to TWO Towers!
I am only worried about the average viewers, who might get really bored when they sit thru Two Towers, watch another 3 hours of CGA, and then still read 'Journey Continues'!
Michael Ernest
High Plains Drifter
Sheriff

Joined: Oct 25, 2000
Posts: 7292

FWIW, I'm reading John L. Casti's Searching for Certainty this week and he uses the terms "smaller infinity" and "larger infinity" quite blithely. It sounds to me like the terms might be quite commonly used in math chat.
IEEE 754 defines a few categories of infinite reduction (infinitesimal) and expansion. Although not sauteed in the argot of pure mathematics myself, different kinds of boundlessness doesn't seem all that surprising a concept.
[ October 14, 2002: Message edited by: Michael Ernest ]

Make visible what, without you, might perhaps never have been seen.
- Robert Bresson
Mapraputa Is
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Sheriff

Joined: Aug 26, 2000
Posts: 10065
This book is good also.


Uncontrolled vocabularies
"I try my best to make *all* my posts nice, even when I feel upset" -- Philippe Maquet
Rick Hightower
Author
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Joined: Feb 20, 2002
Posts: 350
Smoke is coming out of my ears... I am going to bed now....


Rick Hightower is CTO of Mammatus which focuses on Cloud Computing, EC2, etc. Rick is invovled in Java CDI and Java EE as well. linkedin,twitter,blog
Mapraputa Is
Leverager of our synergies
Sheriff

Joined: Aug 26, 2000
Posts: 10065
One of the most fascinating intellectual ideas is the idea of Continuum -- that not only your axis doesn't have the end, but also there is infinitely many points between any two points of "real number" axis.
Barry Gaunt
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Joined: Aug 03, 2002
Posts: 7729
Regarding the original array problem...
Both [ i | i<-[1..] ] and [ i*i | i<-[1..] ] are generated from the same generator [1..] so they have the same number of elements.
Functional programming forum? Haskell?
-Barry
PS:
you guys are mad

No, they are just proving the existence of "Meanless Drivel".
[ October 12, 2002: Message edited by: Barry Gaunt ]

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subject: Thought for today