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Journal ynotds's Journal: Countable and uncountable infinities 8

This may not strictly be a "note for nowhere else" but I am test running it here first because of the frequency with which assertions turn up on /. about the likes of the works of Shakespeare turning up in the digits of pi.

The importance is that even world renouned mathematicians get this wrong way to often. I just finally started reading Roger Penrose's Shadows and quickly found him confounding the number of algorithms with a countable number (even though he had just made mention of Cantor and "diagonal" in the same breath) in his effort to make a case that there are things human minds can do that are not computable. Meanwhile, in trying to make the opposite case, Wolfram cites the "oracle" argument which contains exactly the same misunderstanding.

While there are a whole lot of technical arguments that can be got into about orders of infinity, there is really only one distinction that matters to our everyday understanding of the world, that between aleph 0, countable infinity, and aleph 1 (and higher) which is uncountable.

I could just make the point that actualities are countable and possibilities are uncountable, but that would be putting the cart before the hourse.

First we need to recognise that any form of infinity is just a mathematical construct. We are not ever going to have to deal with actual infinities, but what is very useful is understanding limiting behaviour as something or other goes towards infinity. It is here that the difference between countable and uncountable infinities matters immensely.

can be put into a one to one correspondence with the natural numbers
cannot be put into a one to one correspondence with the natural numbers

The digits of pi, the keystrokes made by a room full of monkeys, the galaxies, stars, grains of sand, atoms are all countable. Each could be given a number.

Possible texts, possible algrithms, possible macromolecules, possible species, potential humans cannot be given a number. Each is a unique combination of a long sequence of units and every time you add one to the length of the base sequence, you multiply the possibilities by some factor greater than one, often much greater but even that does not matter except in the limit.

So if you struggle to count the possibilities of a small complex system, just making that system slightly larger multiplies those possibilities out of all proportion and counting them becomes orders of magnitude harder.

Even if, as I strongly suspect, our cosmos is but a semen stain on the sheets of some grand cosmological process going on beyond our event horizons, there is no "other earth" anywhere else. There is no other you. There is no other me! Fortunately!!

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Countable and uncountable infinities

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  • I hate to sound dismissive, but I thought you where headed somewhere in attempting to prove or disprove something. Maybe you are just too subtle for me to grasp how you conclude "There is no other you. There is no other me!"

    I personally believe in an infinite number of universes solution to the explanation of how quantum mechanics works, neatly explaining things like "spooky action at a distance." By my lights there is not just another "me", but an infinite number of them. Of course by "me" I refer to

    • I can't try to match the length of your reply, but thanks anyway. It has brought out two interesting connections and brought to the table two things I strongly disagree with you about.

      The first of those disagreements is obviously your negative reaction to my original point. I guess I have to just remind myself again that even clearly intelligent people will fail to see something that is so obvious to me that I fail to put it into the right words, or maybe there just are no right words. So I guess there is
      • Very sorry if my post came off negative. I'll have to work on that. I do disagree, but that doesn't make me right of course.

        I really did think I was missing something however. I will go back later and try to pick out exactly what it is that I was failing to understand, rather than just carp that there is no explanation.

        Small world I guess, I know Conrad also, but our similar intrests shrink it no doubt. We had a couple of employees from Australia in Champaign Illinois for a time, it wouldn't surprise

        • ... was most likely Paul Abbott, who was helping at Conrad's presentation, having maintained an active association with Wolframs during his ten years back in Perth.

          I too should remember who the other Australian was, but don't. One other person I did get to know reasonably well in the early days was Theo Gray, because we shared an interest in graphical user interface issues.
          • Yep, knew them both well, and attended several parties at Theo's ranch/farm several miles outside of town.

            Theo was the surpreme overlord of the front end in much the same way Stephen was the overlord of the kernel.

      • Damn I hate posting something then realize I made a mistake.
        The number I need has 10^10000000 digits (or maybe even 10000000^10000000) to get to all 10 million encodings one has to count to the number represented by 10 million digits, not just to 10 million. We are no where close to finding Shakespear in Pi. My concept still holds, but its a way way way down stream towards infinity.

        Feel free to now just discard my ramblings as those of a kook :-(

        • I was confident there was something amiss with your first shot but didn't want to redo the calculation in a hurry in case I was equally off track.

          There are times when you can predict the next letter with great confidence, but many others where the choice is much wider, but to find the right string out of even say 2**10000000 possible strings of equivalent size, you would need to search a random string of effectively meaningless length, in effect the same length as the number of coin tosses as you would exp
    • Watch the Butterfly Effect this winter?

COMPASS [for the CDC-6000 series] is the sort of assembler one expects from a corporation whose president codes in octal. -- J.N. Gray