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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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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].

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).

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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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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Infinity cannot be defined ? - You learn something new everyday. Tom

Nigel Browne
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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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so, you're saying that mathematically, infinity cannot be proved ?

Ashok Mash
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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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Can't you define infinity like this ? infinity = lim(1/x) when x -> 0

Nigel Browne
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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

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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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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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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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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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!

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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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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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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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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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 ]

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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Search on google man .. even Einstein relative theory more or less cover it .. no more spoon feeding ... search google.

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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but Ravish started it ! Nevermind, I'll exact my revenge by not inviting him to my birthday party

Oh your b'date is infinity Only parallel lines will be there to say HBD I like you

Paavam Payyan
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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'!

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

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.

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 ]