On Thursday, January 3, 2013 1:31:30 AM UTC-8, zuhair wrote: > Call it what may you, what is there is: > > (1) ALL reals are distinguishable on finite basis > (2) Distinguishability on finite basis is COUNTABLE. > > So we conclude that: > > "The number of all reals distinguishable on finite basis must be > countable". > > Since ALL reals are distinguishable on finite basis, then: > > "The number of all reals is countable".
OK, this is going to require some handwaving. I want to challenge the notion that all reals are distinguishable on a finite basis.
I'll start by granting that all distinguished reals can be distinguished on a finite basis. By this I mean the rationals can be distinguished on a finite basis and there is only a finite set of the irrationals that have been distinguished.
I will further grant that the irrationals that can be distinguished can be ordered.
I will also grant that there are natural numbers that can't be distinguished in human space/time.
The natural numbers are defined by a successor algorithm. Ever natural number has exactly one successor. This is what we mean by countable.
It doesn't matter that in human terms we cannot distinguish all of the natural numbers less than one googol let alone one googolplex, each and every one of those natural numbers can be distinguished on a finite basis.
And so it is with the rational numbers. There is an effective algorithm to pair each and every rational number with a natural number. It doesn't matter that no human can distinguish each and every rational number for exactly the same reason no human can distinguish each and every natural number; the two sets can be paired and by the definition of countable the rationals can be counted.
Now we come to the irrationals. The irrationals are also part of the set of real numbers. If there is an effective algorith to find all of them then they are countable. If no such algorithm exists then they are not countable. It doesn't matter if the the algorithm is effective in human terms or not only that it is effective in the way that the natural number successor algorithm is effective and tells us one googolplex is a natural number.
I'm wondering how we find specific irrational numbers. How did we happen upon the square root of 2 and pi? Is there an effective algorithm to reveal all irrational numbers? Maybe there is one in the same way there is a God, that is God exists but cannot be proven to exist by humans. Or maybe God doesn't exist and it is enough to say God cannot be proven to exist or God cannot be proven to not exist. But even these claims are quite strong becuase they don't just talk about what humans can do but what the gods can do. And so it is with Cantor's claim about an effective algorithm to count all reals, Cantor's claim isn't about what humans can't do but what even the gods can't do.
What Cantor purports to show is that every effective algorith to count the reals fails to count all of them. The issue isn't that for every n in the count the Cantor diagonal differs from all reals counted up to and including n but that the diagonal doesn't appear in the list at all. Not even the gods in their infinite time and space can come up with an algorithm to count all of the reals.
This may seem counterintuitive but we live in very finite time and space. There are many finities much greater than those we will ever know.
-- as a side bar --
There are people who like to figure out the minimum number of lines on a ruler are needed to measure all items up to a given length to a given accuracy. I'm wondering what the minimum set of integers would be to distinguish all integers up to one googol. We can identify one google as 10^100. While I can't expand that number I have just identified it with two numbers I can exactly identify in human terms. I can easily count the next 10 beyond and before that number by reference. I don't mind having to measure in steps but all steps must be completed in human terms. Any ideas about how to approach this problem?