Re: infinity

David R Tribble said:
> Tony Orlow (aeo6) wrote:
> > Oh, somebody coined the term for my infinite numbers.
> > If you see numbers like 000...001:000...000, they're Orlovian in this
> > context. The colon separates infinite-base digits, which each consist of
> > infinitely many finite-base digits. There is still a single digital point to
> > the right of the 1's column, which acts the same as a colon, but marks where
> > the units are. preety neat huh?
>
> We'll consider your definition of what you call 'N', the first
> infinite natural number (and which you also define as the size of
> the set of natural numbers).

But it is not the "first" infinite natural, if by that you mean the smallest in
quantity. It is simply a unit infinity which may be considered the count of
whole numbers. I certainly consider the even numbers to be half that size, so
it is not the "first" in that sense. There is no smallest infinite number any
more than there is a largest finite. Those two ideas imply each other. But
anyway...

> You have frequently defined it as
> being equal to the decimal number ...999 and the binary
> number ...111. You've also stated that the decimal representation
> of 'N' has log_10(N) digits, and the binary representation has
> log_2(N) digits.
I might not have stated that clearly. There are considered to be N numbers from
000...000 to 999...999 or 111...111 binary. N is 1:000...000, the next number
up, the unit infinity. This doesn't change your following objection....
>
> So for any given base b = 2,3,4,5,..., N can be written as a number
> containing log_b(N) digits, all digits being b-1. Thus 'N' can be
> written in, say, base 3 as ...222, and in base 7 as ...666, and
> so forth, for any finite base we choose.
These representations are digital numbers. Consider in the reals that these
numbers also extend for the same number of digits to the right of the point,
and the difference disappears. So N is really 999...999.999...999, which is the
same as 1:000...000, with log10(N) digits on each side of the point.
>
> Further, for any base b that is even, all of the digits are b-1,
> which is odd. For example, ...999 (base 10, even) has digits
> that are all '9', and is obviously an odd number. Same thing
> for ...111 (base 2, even), which is also obviously odd.
>
> But for any base b that is odd, all of the digits are b-1, which
> is even. Thus ...222 (base 3, odd) has digits that are all '2',
> and is therefore obviously an even number, as is ...666
> (base 7, odd), both containing only even digits.
Yes, given an even base, N is even, and given an odd base, N is odd. Switched
only because of the addition of 1 to get 1:000...000.
>
> Thus we have a contradiction. 'N' is both an odd and even.
Apparently true, and possibly a contradiction, but perhaps more of a paradox.
Not the kind of paradox like Banach-Tarski, which is a nonsensical result, but
a true paradox, which offends certain intuition, but has no real effects. Or,
perhaps, the product of all finite primes, plus 1, makes a better unit
infinity, but I am not sure how to characterize this number. Perhaps Martin has
some suggestions? I still need to digest his bit pattern for log2(N). He has
some good interesting ideas.
>
> Which means that the assumption that 'N' is a natural number
> cannot be correct.
Well, the assumption that every infinite whole number is necessarily even or
odd may be incorrect. After all, aren't Cantorians fond of saying, "Not every
property that holds for the finite need hold for the infinite"?
>
>

--
Smiles,

Tony
.

Relevant Pages

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