Re: infinity

Tony Orlow wrote:
>> [...]. If Virgil had actually
>> tried to remember TO's position, Virgil might have recalled that 111...111
>> is the largest whole number (in binary), but that the largest countable real
>> number is 111...111.111...111, approximately 1 greater in quantity. He might
>> have recalled my revelation that this extra 1 explains the addition of 1
>> when finding the negative of a number in 2's complement.

Tony Orlow wrote:
> Actually, I have addressed this very issue, and it boils down to there being
> two perspectives on the number system: the number line, and the number
> circle. I don't expect you to understand this. If you have questions,
> ask, but don't be snide.

I suspect that you're talking about treating the number line as a ring,
so that it wraps around from the "largest" number to the smallest.
That's not a new concept. I believe Rucker mentioned it in his
book "Infinity and the Mind", among various other places.


David Tribble:
>> Your slip-up is claiming that the real number ...111.111... is
>> (almost) one greater than the integer ...111, when in fact they
>> are the same value (oo). The fraction .111... is exactly equal
>> to 1, and oo + 1 is simply oo.

Tony Orlow:
> The fraction 0.1111... is essentially 1, with an infinitesimal difference
> which doesn't matter at the finite level.

Actually, 0.111... (binary) is exactly equal to 1, since:

s = 0.111...
s = sum{1 to oo} 2**-i
s = lim{n->oo} sum{1 to n} 2**-i
s = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
s = 1

Writing the number 0.111... is the same as denoting a repeating
sequence of digits, each one being 2**-i, whose sum is the rather
well-defined limit of 1.


> Besides, when it comes to 2's complement, 0.111... IS treated as equal to 1,
> since you are working on a finite integer scale, which was what I had related
> about the explanation of 2's complement, which I haven't given in this thread
> anyway.

But you were talking about a binary number with an infinite number
of digits. Which is an ill-defined concept, since N contains only
finite natural numbers; there is no infinite natural "number".

-drt

.

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