Re: Multiple infinities - one more look

"Venkat Reddy" <vreddyp@xxxxxxxxx> wrote in message
news:052a8b26-92db-46a9-8caf-1ba9c7300927@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

I see. I think order is related to how you choose to represent the
number. If the real number is represented by a digit sequence of
infinite length, we immediately have an order for all reals, since
digit position have an order. For example, for the length of 6 digits,
one can generate these sequences by writing all possible permutations
and combinations of digit sequences in an orderly fashion. This can be
continued for lager length of digit sequences without limit. All
possible digit sequences are guaranteed to appear somewhere in the
list.

No. This process only generates finite digit sequences. Real numbers have
infinite digit sequences...


Instead, if you choose to represent a real number by an algebraic
equation, here also we have an order in representing the equation
itself, so the resulting reals have an order. Likewise for
transcendental as well - once you have a way of identifying the
number, we have a way of ordering them.

It seems you are thinking of something like "computable numbers", and these
are indeed countable. Real numbers need not be computable in this sense
(i.e. having a finite program to output their digits in sequence).


You may say you want to represent the real number as a point on the
line such as the interval [0,1]. Here also we have an order. Cut the
line into half and mark the 0.5 as your first real number. Cut the two
pieces in to two equal parts, mark 0.25 and 0.75 as new real numbers.
Continue the process for ever and you have an ordering of all points.

No. Real numbers need not be one of these points you construct, although
your process will construct real numbers arbitrarily close to any given real
number.


One might be quick to point that this process can only hit the real
numbers of the form of multiples of 1/(2^n), and misses the numbers
like 1/3, 1/pi, 1/sqrt(2). My answer for this is, have a
representation of these numbers as points on the real line and show me
why my process can't hit these points, if you really continue the
process for ever.

Better yet, have a list of all possible representations - digit
sequences, algebraic equations, geometric models etc. For each of
these representations, have a process to generate all possible numbers
in that representation as I have shown above. That should cover all
real numbers in an orderly fashion, though in a 2-dimensional order.

Well, you've given 3 suggestions above, and all 3 suggestions are wrong and
putting them together doesn't make things any better! :-)

Ask for any number, one of these processes is guaranteed to have it in
its list (I would even say every process have that number in its list,
but at different positions, but I don't need to depend on this
anyway).

Also, since each of these processes have a countable list, and the
number of processes is also countable, the reals must be countable.
Did I miss anything?

Each of your processes covers only a proper subset of the real numbers.

Regards,
Mike.



- venkat


.

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