Re: Calculus XOR Probability

Tony Orlow wrote:
Matt Gutting said:
Tony Orlow wrote:

<snip>


Yes, you're right, over the finite naturals, of course. It seems impossible to define what the range or count is, and trying to assume any such thing leads to contradiction. Correct. Of course, in the above example, I can't see, despite the inability of calculating any such numbers, that any ball has any different chance than any other. If there is some set of balls in the vase, then whatever size that set is, each ball has the reciprocal of that size as a probability.
Assuming that one can define a reciprocal for that number, okay. But how
does one compare the probability (say) of drawing one of the first ten balls
dropped in with the probability of drawing one of the first twenty? Can you
show that the numbers you use there exist?

If we have n balls, the probability of each is 1/n, and so the probability of the ball being one of the first 10 is 10/n, as opposedto 20/n chance of it being one of the first 20. If we have n=1:000...000, a megabigulous number, then the inverse is 1/n, or 0:000...001 probability for each, 0:000...010 that it be one of the first 10, or 0:000...020 that it be in the first 20.

Now, you want me to prove these numbers exist? I think if one creates a number system, which is basically a language where each word represents a quantity, then one can determine what operations are possible with it. The possibilities are somewhat limited compared to finite arithmetic, but rational portions of these quantities can certainly be represented using the connecting repeating digit strings between limit points. Does that capability make the system "exist"?

I'm still not sure what you mean by megabigulous. In addition, you can certainly
create a symbol system obeying whatever rules you want; but you have to make
sure that the rules are (a) well-defined and (b) at least not grossly
contradictory. Given those restrictions, you need to *at least*:

1) Define "megabigulous number".
2) Show that megabigulous numbers have multiplicative inverses.
3) Show that the inverse of the megabigulous number 1:000...000 can be
represented as 0:000...001 using the same symbol scheme.
4) Show that the symbol strings "0:000...010" and "0:000...020" are well-defined
and are equal to (respectively) 10 and 20 times 0:000...001.

Then, one can begin asking about the probability that the ball is one of the
first 10 as compared with the probability that it's not, and the probability
that it's in the first 20 as compared with the probability that it's not.

Matt
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