eternal youth"; that is, we will have to live forever. But we might begin to look pretty old along the

way. And just think of the writing muscles we'd grow doing all that writing. Might look like Popeye!

to to to :mad to ?????]]>

If one "counts" the points at 1 unit, 1/2 unit, 1/4 unit, 1/8 unit ... by simply passing across each of

these points on a number line (starting at 1 and proceeding towards zero) then it would take infinitely long to cross all these points IF there is a pause of 1 second at each point. However if one walks a CONSTANT RATE of 1 unit per second going from 1 to 0, then it takes only one second to "cross" all these points and hence finish "counting" all the positive integers. So it takes hardly any time at all (an infinitesimal amount of time) to "count" a point by this scheme. Of course "counting" the points simply means to put them in a 1 to 1 correspondence with the positive integers, which is done theoretically without consideration of time.

So the reason most folks have a problem with this "counting" is that they don't take into consideration the time element perhaps intuitively thinking that there must be some pause when

counting each point. Hence they think of it as a paradox.

I guess mathematicians "plink away" or "plunk away" at Planck's constant to some extent.

Ah! isn't math wonderful?

]]>There are a few paradoxes yes (potentially anyway). I think that makes it rather more fun

]]>Start with 1 and add 1 to that at the end of the 1st second to get 2. Then add 1 to 2 at the end of one half second to get 3. Then add 1 to 3 at the end of 1/4 second to get 4. Then add 1 to 4 at the end of 1/8 second to get 5. Continue this for a total of two seconds. Then you have counted all the natural numbers by the end of second number two. To reach and count the number k it occurs at the end of the 1/(2^(k-2)) time period. The "sum" of all of these time periods is two seconds, so by the time two seconds passes doing this process we have counted all the natural numbers.

If someone says, "But no, we didn't count the number, say m, then we counter that at the end of the 1/(2^(m-2)) time period we counted m. So there were no natural numbers left out of the counting. 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2 (That is, the limit is 2). But 2 is a finite number so we can "continue this process for two seconds" to count all the natural numbers.

Infinity is a strange thing! Who can understand it? There are a number of interesting paradoxes dealing with infinity.

]]>1, 2, 3, 4, 5, 6, 7, 8, 9...

We add one to each number to get the next number. We can go on counting:

...1,000, 1,001, 1,002, 1,003, 1,004, 1,005...

And on:

...1,000,000, 1,000,001, 1,000,002, 1,000,003...

If we kept on and on counting (although, if we counted one number per second, it would take us over 31.6 years of non-stop counting to reach one billion, so it would not be practically possible to do this, just theoretically) we can reach numbers of immense size, such as Graham's number, which far far outstrips the number of atoms in the universe. In fact, I don't think it would even be possible to imagine how much bigger Graham's number is than that. And we can carry on and what it's pretty easy to realise here is that it can go on forever, we can always keep adding one to make the next bigger number, with no end to the counting. This is where the concept of ∞ comes in.

In the theories of calculus and many other areas of mathematics, we use the symbol to denote unboundedness, really, doing it over and over without end. It is not really treated as a number. If we want to try to treat it as a number and do operations on it, we really need something like Cantor's cardinal arithmetic. But, without going into all of that, we can make it a little more intuitive and just say that ∞ is the end of counting. This is not all that precise, really, but if we just want to get our heads around what this symbol: ∞ means, we have to look back to counting. Essentially ∞ says that I can go on adding 1 for ever and ever and ever and it will not end (in-finite), so I will use this symbol ∞ to mean the point beyond which I cannot keep counting. For this reason 1 + ∞= ∞ and 1 - ∞ = -∞

]]>say 1- infinity = ?

Thanks

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