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Infinite Of Infinates, whats the solution? 
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#19
Dec1908, 12:21 PM

P: 287

I'd say, simply and without appeal to calculus, that nothing is continuous... space, time, temperature, are discrete, and therefore there is no paradox in walking to the door or in adjusting the knob. You could enumerate the different temperatures, if you were quick enough.
Why not? 


#20
Dec1908, 12:44 PM

P: 107




#21
Dec1908, 12:58 PM

P: 287

I think you misunderstood my point, matticus.
The real numbers are not "real" in the general sense. There is not a onetoone correspondence between mathematics and nature. Math is an approximation... it's a language for describing nature. Mathematics is true in its context. But there's no reason to expect it to apply *exactly* to the real world. You said: "there is a onetoone correspondence between a time interval and an interval of real numbers" Prove that, and you'll be the most famous mathematician who ever walked the face of the earth. 


#22
Dec1908, 12:59 PM

P: 287

Furthermore, just for kicks, tell me about a real number you can't enumerate. Describe it for me. Give me an example... tell me what its value is.



#23
Dec1908, 01:41 PM

P: 107

i'll tell you what. enumerate them, and i'll show you one you missed :)



#24
Dec1908, 01:50 PM

P: 287

Simple.
Using the decimal expansion of every number, start by filling the ones place with 09, then the 10s and 10ths place with 09, etc. You'll never reach the end, but you wouldn't reach it if you enumerated the natural numbers either. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 00.0, 01.0, 02.0, ..., 10.0, 11.0, ... etc. Perhaps a more disturbing result of my above enumeration scheme is that it produces only a countable infinity of different numbers. (unless I missed a few...) 


#25
Dec1908, 01:58 PM

P: 107




#26
Dec1908, 05:49 PM

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#27
Dec1908, 11:09 PM

P: 287

But it is impossible to give an algorithm which produces all real numbers. The simple reason: there are only countably many algorithms, and as such, there are more real numbers than algorithms. There must exist some real numbers for which there is no algorithm which can tell you the digits... some examples of numbers for which there are algorithms to give you the decimal digits:
integers: Put the integer x into a variable k. Let n = 0. Divide k by 10. Put a zero into the place corresponding to the power 10^n. Put the remainder after division into the place corresponding to the power 10^n. Increase n by 1. Put the quotient back into k, and repeat (forever, or until k = 0). rationals: see Long Division special irrationals (roots, etc.): For square roots, for instance... 1.) Let k = 0, delta = 1, steps = 0 2.) if k^2 < n, add delta to k and go to step 1 3.) if k^2 = n, return this value of k 4.) if k^2 > n, subtract one from k 5.) divide delta by 10 6.) If steps > max, return the approximation k 7.) go to step 1, add 1 to steps The interesting thing to notice is that even if you take the natural numbers, the rationals, and the "special irrationals" (such as roots, logs, exponentials, sines and cosines, special constants, etc.) then you have a countably infinite set. How so? Well, consider an ordered 5tuple of rational numbers. (a, b, c, d, e) a can be any rational number b can be any power to raise a to... assuming it's valid for that a c can be either 0 or 1. if it's zero, then evaluate the 5tuple as b*exp(a). Otherwise, don't. d can be either 0 or 1. if it's zero, then evaluate the 5tuple as b*log(a). Otherwise, don't. e can be either 0 or 1. if it's zero, then evaluate the 5tuple as b*sin(a). otherwise, don't. I could get really, really technical, and make it an ordered 50tuple, and wipe out about any numbers you could imagine. Do you see the rub? There are as many rational numbers as there are ordered 50tuples of rational numbers. So I can't possibly compute (or even "define", if by define you mean give the value of) the majority of the real numbers. Which also means you will never see one or use one (directly). 


#28
Dec2008, 07:11 AM

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