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?

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).

[ flickr ]

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 ]

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?

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.

If we ever 'define' infinity to 'definite' value or position, every single law in mathematics will fail. )

[ flickr ]

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

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

SCJP

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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 ]

http://www.nrich.maths.org.uk/mathsf/journalf/aams/q51.html

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 ]

even Einstein relative theory more or less cover it ..

no more spoon feeding ... search google.

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>

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'!

<i>All that is gold does not glitter, not all those who wander are lost - <b>Gandalf</b></i>

*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

Sheriff

Uncontrolled vocabularies

"I try my best to make *all* my posts nice, even when I feel upset" -- Philippe Maquet

Ranch Hand

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

Sheriff

*any*two points of "real number" axis.

Uncontrolled vocabularies

"I try my best to make *all* my posts nice, even when I feel upset" -- Philippe Maquet

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 ]

Ask a Meaningful Question and HowToAskQuestionsOnJavaRanch

Getting someone to think and try something out is much more useful than just telling them the answer.