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Post by abacus9900 on Sept 15, 2010 10:31:15 GMT 1
No, it is not, which is why it is infinity.
I can see what eamonn is getting at but...
I think eamonn is making a mathematical point and I am making a philosophical one - this is probably where the disagreement arises from.
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Post by speakertoanimals on Sept 15, 2010 13:05:52 GMT 1
No, it is not, which is why it is infinity. I can see what eamonn is getting at but... I think eamonn is making a mathematical point and I am making a philosophical one - this is probably where the disagreement arises from. Nope, you're just plain wrong. Counting is after all a mathematical noiton, hence claiming you are talking philosophy is just trying to evade the fact that you are wrong. So, what is counting (define your terms before you start!). If I have the SAME number of apples and oranges, that means that you can pair them, one apple to one orange, with none left over. That is what it means when we say we have the SAME number of each. There is NOTHING in the definition that says the number of apples can't be infinite So, if we have a set of something that can be matched one to one with the integers, then that is countable. If it takes the whole infinite set of the integers to do it, then we have aleph-0, the smallest infinity. Hence there are as many even numbers as integers, and as many odd numbers as integers, hence we see we have a new arithmetic for these infinite numbers, where twice a thing can be the same as the thing. If you dismiss this perfectly sensible definition of counting, which even a supposed philosopher should be able to get their head round, then you can discover exciting things such as there beingvarious types of infinity, some bigger than others. Just saying , all infinite, can't be counted, is just boring (and incorrect), however intuitive it may seem. |
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Post by abacus9900 on Sept 15, 2010 13:21:16 GMT 1
It's not intuitive, it's perfectly logical.
The point to make here is no infinite series can be counted and therefore cannot be tested. You can begin to count an infinity of things till the end of the universe but you will not have even made the tiniest of progress since it goes on forever. Consequently, to assert that there exists different 'infinities' of numbers is completely wrong. The concept of 'bigger' sets is only useful when dealing with finite sets, otherwise the idea has no possible meaning. Get it now?
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Post by speakertoanimals on Sept 15, 2010 13:59:40 GMT 1
It's not intuitive, it's perfectly logical. The point to make here is no infinite series can be counted and therefore cannot be tested. You can begin to count an infinity of things till the end of the universe but you will not have even made the tiniest of progress since it goes on forever. Consequently, to assert that there exists different 'infinities' of numbers is completely wrong. The concept of 'bigger' sets is only useful when dealing with finite sets, otherwise the idea has no possible meaning. Get it now? Totally wrong. Take the even numbers, of the form 2n. I can very easily match them up, one-to-one with the integers n. Hence by any sensible definition, there are as many integers as even numbers. How many even numbers? Well, I can perfectly well define that a aleph-0 (just a name). Okay, it isn't FINITE number, but who cares? Most useful stuff in maths is either infinite or infinitesimal. So, I have just shown a simple test to compare the number of integers and the number of even numbers. You DON'T test statements about infinite things (be they infinite decimal expansions or infinite sets), by going through one by one -- that is just plain daft, and shows someone who can't get much further in maths than arithmetic.................... Okay, so now I will show, by strict logical argumnet, that the number of decimals between 0 and 1 is greater than the number of integers. So, suppose there were the same number. That means I can make a list of decimals, labelled 1,2, 3 and so on, where each decimal number occurs once and only once, somewhere in that list. Now consider the following -- I take the first number in that list, look at the first digit of that first number, and create a new number whose first digit is different. I now take the second number and second digit, and so on. This is a perefctly well-defined procedure, it generates a new decimal, that by construction is DIFFERENT to every number on the list. Hence no such list can contain ALL possible decimals. Hence (since one-to-one match with integers means such a list exists), this means that number of decimals is greater than number of integers, since for any such list, some are left out. That's logic. Saying infinity doesn't exist because you can't coun it is like saying that the number pi doesn't exist because no one could ever calculate or write down EVERY digit in the decimal expansion of pi. Yet still it exists, and so do countable infinities. |
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Post by mak2 on Sept 15, 2010 19:39:39 GMT 1
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Post by abacus9900 on Sept 15, 2010 19:57:53 GMT 1
Well, what do you expect? They live in a world of their own. |
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Post by mak2 on Sept 17, 2010 9:49:32 GMT 1
Why make do with one infinity when you can have an infinite number of them?
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Post by abacus9900 on Sept 17, 2010 10:50:54 GMT 1
Why make do with one infinity when you can have an infinite number of them? Exactly, it gets kind of silly, doesn't it? |
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Post by mak2 on Sept 17, 2010 15:04:48 GMT 1
You used the phrase "Mathematical Dyslexia"!
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Post by abacus9900 on Sept 17, 2010 15:42:32 GMT 1
You used the phrase "Mathematical Dyslexia"! I think it's a case of some mathematicians convincing themselves maths is real - it isn't, it's just about ideas. |
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Post by mak2 on Sept 18, 2010 10:25:10 GMT 1
Mathematics is a creation of the human mind but it is remarkable how much of it is applicable to the real world. I guess this is because the human brain evolved to cope with reality.
Having said that, I can't think of any practical application of Cantor's ideas of transfinite numbers. Are there any?
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Post by abacus9900 on Sept 18, 2010 10:40:17 GMT 1
Mathematics is a creation of the human mind but it is remarkable how much of it is applicable to the real world. I guess this is because the human brain evolved to cope with reality. Having said that, I can't think of any practical application of Cantor's ideas of transfinite numbers. Are there any? Not really, but that's the thing about maths - you never know whether or not it might come in handy one day. |
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Post by oldcofe on Sept 18, 2010 12:02:29 GMT 1
webplodder, why have you changed your login to abacus9900?
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Post by StuartG on Sept 18, 2010 14:17:29 GMT 1
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Post by abacus9900 on Sept 18, 2010 15:30:56 GMT 1
webplodder, why have you changed your login to abacus9900? We all need a change. |
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