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RE: Infinity.
Posted Aug 11, 2000
The previous example, though it illustrated the point, was not the full story!
Two sets can be said to have the same number of elements iff there is a bijection between them. That is, if there is a function taking each element of one set to a unique member of the other, and that everything from the second set has an element of the first set mapped onto it.
This is actually true of the natural numbers and rationals! (Which blows my previous argument out of the water 
.)
It is also true of the algebraic irrationals (numbers which are a root of a polynomial with integer coefficients).
However, try finding a bijection between the real numbers and the natural numbers, and you'll find it impossible. This is essentially what Cantor's Diagonal argument is all about - showing that the infinity of reals is larger than the infinity of naturals.
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Latest reply: Aug 11, 2000
More research topics...
Posted Aug 11, 2000
The size of infinity, or 'why there are more real numbers than natural numbers'.
Russell's Paradox.
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Latest reply: Aug 11, 2000
The size of infinity.
Posted Aug 10, 2000
A point about infinity - infinity isn't a number, its the limit of an unending sequence. It's fairly easy to show that there are different 'sized' infinities as well. Consider the natural (counting) numbers. That is an infinite series: 1,2,3,4,5,.. but between any two natural numbers there are a *finite* quantity of numbers. (eg. between 4 and 8 there are 3 numbers - the same holds for *any* two natural numbers.)
Now consider the real numbers (ie. numbers with decimal places). Again it is an infinite series - for any real number, you can think of a bigger one. However, between any two real numbers, you can find an infinite quantity of real numbers. Between 1 and 2, for example, is 1.5; between 1 and 1.5 is 1.25; between 1 and 1.25 is 1.125 etc etc...
Hence, the limit of the series of real numbers is a larger infinity than the limit of the series of natural numbers.
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Latest reply: Aug 10, 2000
The impossibility of time travel as we know it...
Posted Aug 9, 2000
I don't believe that time travel is possible in the way we think it *should* work. That is, I think that if (backwards) time travel were possible, then we could not travel backwards in time and see our grandparents etc., since they would no longer be *there*. They have travelled forward in time to the present day, and would no longer be in that 'past' time.
If this all sounds a bit weird, it's because words like 'no-longer', are referring to an extra dimension relative to which travel is possible (compare with travelling through space. This only makes sense if there is an extra dimension - time - to travel relative to. Likewise with time travel - travel through time presumes a fifth dimension to travel relative to.)
I may try and clarify what I mean by all of this. In fact I may have an old Philosophy essay I can get relevant bits from...
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Latest reply: Aug 9, 2000
Ideas for Articles
Posted Jul 13, 2000
Some ideas for articles I may write:
Newcastle-under-Lyme
Taoism
Climbing Terminology
Bruce Lee
Climbing Venues (Various - see e[version] for details)
Trivalent Logic
LaTeX - the typesetting language, *not* the rubber
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Latest reply: Jul 13, 2000
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