Re: Cantor Confusion

Andy Smith wrote:
Since the integers are finite, you cannot represent a real requiring an
actually infinite number of bits, is what I meant. Maybe that is too
simplistic?

To be more explicit, to represent (= address) all the reals in say
[0,1] you would need as many bits for your integers as the reals occupy.
But that would require integers with an actually infinite bit length
e.g. say, the reflection of the reals about the decimal point to give
"numbers" like ...1101

Which is where I came in (with what's the problem with enumerating the
reals?) but I am now better informed - . If all members of N are
finite, there is no prospect of addressing the reals, so no surprise
there....


Andy Smith wrote:
Are you sure you are better informed? What you wrote is nonsense. Every
real number has a binary expansion (and a decimal expansion). So, what
in the world are you trying to say?

No, I think he's almost got it. I think he's trying to say (in
computer programming terms) that any given real in [0,1]
requires a countably infinite number of bits to represent
as a binary fraction (bitstring), which is correct. And that
those infinite bitstrings cannot be mapped to finite naturals
(e.g., by reflecting the digits about the binary point),
because you'd end up with infinite-length binary integers,
which are not naturals.

So I think he's reached the (correct) conclusion that you
can't denumerate the reals (in [0,1]) using naturals,
albeit in a somewhat clumsy way of saying it.

.

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