The problem with your method is that you're thinking in terms of finite numbers. There's a problem when you deal with arbitrarily large (i.e., infinite) amounts of decimal places. You can prove that your way doesn't work using diagonalization, the same method Cantor used. We must find p, a real number that isn't in your set. Take your first number, .1, and make the first decimal place of p 0. Your second number, .2, or .20, gives us the second decimal place of p 1. So now we have p = .01 . Continue this for all numbers of your set and we will have p, a real number that isn't in your set. Therefore, the reals between zero and one are more numerous than the integers. Q.E.D.
The problem is that while you can find an infinite number of integers with only threes, you cannot find an integer with infinitely many 3s. Think of it this way. The decimal portion of pi is 0.14159265... While you could find an integer that matches this number up to the 100th digit, or any arbitrary digit; you could not match all of them. Such an integer would be infinitely large, unbounded, and would itself be ∞ (infinity). So you would need infinity many integers for one real number. This is not a formal argument and would never hold as a proof (it proves nothing). The basic idea of the linked proof is: lets say integers and reals are the same size, every integer for every real. Once we've matched them all up, can we make a real that's not matched to any integer (remember they are all taken). We can, and that proves that our assumption was wrong.
This is why mathematics gets more interesting as you progress: it no longer is about NUMBERS, it's about STRUCTURE. So creating an example in very discrete terms does not really help to do a proper assessment regarding uncountable and countable infinites. Really, the question is : is there a way to construct a mapping so that every natural number is accounted for by a subset of the reals?
This reminds me of the DIrichlet Discontinuous function. Although i do agree with rootneg2 above in that i believe you need to show that there exists elements in [0,1] that are not accountable in any fashion by a mapping of [0,1] -> N.
[HEADDESK]If you don't understand mathematics, please do not comment on what you *think* the definition of "infinity" is.Arguing a point without understanding the field is just nonsense.
Cantor's arguments are false. In fact the real set (0,1) is countably infinite according to Cantor's definition. Visit my personal blog for proof of this:<a class="user" href="http://mathphile.blogspot.com/">http://mathphile.blogspot.com/</a>I am not surprised that so many comments are made by people who are evidently clueless as to what they are talking about. :-(
That was by far the worst book on infinity I've ever read - full of errors, more footnotes than text, disjointed, and trying to be way to "cute". If you'd like an excellent read on infinity, pick up AW Moore's "The Infinite".And by the way, Godel proved that transfinite arithmetic is, in fact, NOT provable. The Lowenheim-Skolem Theorem does basically the same. Don't believe everything mathematicians tell you (esp. Georg Cantor).
What this Crossegg has shown is that a countable subset of the reals is no larger that the naturals. This is true, but rather silly, since countable means "no bigger than the naturals".
yeyuiJul 22, 2007
They all have the same cardinality as long as you stick with a finite number of dimensions.
quentinmcalmottJul 22, 2007
The problem with your method is that you're thinking in terms of finite numbers. There's a problem when you deal with arbitrarily large (i.e., infinite) amounts of decimal places. You can prove that your way doesn't work using diagonalization, the same method Cantor used. We must find p, a real number that isn't in your set. Take your first number, .1, and make the first decimal place of p 0. Your second number, .2, or .20, gives us the second decimal place of p 1. So now we have p = .01 . Continue this for all numbers of your set and we will have p, a real number that isn't in your set. Therefore, the reals between zero and one are more numerous than the integers. Q.E.D.
jeeumJul 23, 2007
Yes.
jeeumJul 23, 2007
This is not a worthless brainteaser. It is part of the foundation of probability theory and numerical analysis.
gbladeclJul 23, 2007
The problem is that while you can find an infinite number of integers with only threes, you cannot find an integer with infinitely many 3s. Think of it this way. The decimal portion of pi is 0.14159265... While you could find an integer that matches this number up to the 100th digit, or any arbitrary digit; you could not match all of them. Such an integer would be infinitely large, unbounded, and would itself be ∞ (infinity). So you would need infinity many integers for one real number. This is not a formal argument and would never hold as a proof (it proves nothing). The basic idea of the linked proof is: lets say integers and reals are the same size, every integer for every real. Once we've matched them all up, can we make a real that's not matched to any integer (remember they are all taken). We can, and that proves that our assumption was wrong.
evilninjaxJul 27, 2007
This is why mathematics gets more interesting as you progress: it no longer is about NUMBERS, it's about STRUCTURE. So creating an example in very discrete terms does not really help to do a proper assessment regarding uncountable and countable infinites. Really, the question is : is there a way to construct a mapping so that every natural number is accounted for by a subset of the reals?
evilninjaxJul 27, 2007
This reminds me of the DIrichlet Discontinuous function. Although i do agree with rootneg2 above in that i believe you need to show that there exists elements in [0,1] that are not accountable in any fashion by a mapping of [0,1] -> N.
padraic2112Aug 1, 2007
[HEADDESK]If you don't understand mathematics, please do not comment on what you *think* the definition of "infinity" is.Arguing a point without understanding the field is just nonsense.
johngabrielFeb 1, 2008
Cantor's arguments are false. In fact the real set (0,1) is countably infinite according to Cantor's definition. Visit my personal blog for proof of this:<a class="user" href="http://mathphile.blogspot.com/">http://mathphile.blogspot.com/</a>I am not surprised that so many comments are made by people who are evidently clueless as to what they are talking about. :-(
girardstreetJun 6, 2008
That was by far the worst book on infinity I've ever read - full of errors, more footnotes than text, disjointed, and trying to be way to "cute". If you'd like an excellent read on infinity, pick up AW Moore's "The Infinite".And by the way, Godel proved that transfinite arithmetic is, in fact, NOT provable. The Lowenheim-Skolem Theorem does basically the same. Don't believe everything mathematicians tell you (esp. Georg Cantor).
yeyuiJun 22, 2009
What this Crossegg has shown is that a countable subset of the reals is no larger that the naturals. This is true, but rather silly, since countable means "no bigger than the naturals".
yeyuiJun 22, 2009
You are thinking of measure, not cardinality.
yeyuiJun 22, 2009
The funny thing is that there ARE the same number of primes as composites.