Re: Well Ordering the Reals

Tony Orlow wrote:
>> As in all regular digital systems, the leftmost nonzero digit is most
>> significant. You knew that.
>

David R Tribble said:
>> But what about those infinite numbers that don't have a leftmost
>> nonzero digit? You know, those infinite numbers that are infinitely
>> long sequences of digits?
>>
>> Two examples (which I gave in another post):
>> a = ...101010101010,
>> which is just '01' repeated forever, and
>> b = ...110110110110,
>> which is '110' repeated forever. Neither of these infinite numbers has
>> a most-significant digit, so we can't tell which one is larger by
>> comparing their "leftmost" digits.
>>
>> Then there is the next problem of defining what it means to add
>> infinite numbers. What is a+b?
>

Tony Orlow wrote:
> Let's see if these can be added. Not all adic numbers will work, but some do.
> Let's start with the smallest common repetition, six digits, and see if it
> involves a carry bit:
>
> 101010
> +110110
> ---------
> 1100000
>
> Well, it does involve a carry bit to the left, but the rightmost bit is a 0, so
> we can just replace it with a 1 for all but the first bit, and say, going left,
> it is the string 100000, with 10001 repeated forever to the left.

You seem to be saying that
a+b = ...1000110001100000
but that's obviously wrong.

Wouldn't it be more logical to say that
...101010101010
+ ...110110110110
------------------
= ...000000000000
with a carry of 1 that goes, um, where?

Of course, you object to the lack of a most significant digit, even
though that's pretty much the definition of an infinite number:
an infinite number of nonzero digits, so no "last" digit.

At any rate, we could try a similar (but different) addition using
your notation (where c and d are not the same as a and b above):
c = 0:101010...101010
d = 0:110110...110110
+ ------------------
s = 0:1000000...000000

Hmm. It looks like s is one bit longer than c and d, even though
they are all (supposedly) infinitely long numbers. Or perhaps
we've simply just run out of bits, even though each number has an
infinite supply of them?

Or maybe, just maybe, digital representations just don't work for
infinite numbers?

.

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