Re: Well Ordering the Reals

Tony Orlow says...

>Randy Poe said:

>> In the countable bit strings, I label the bits with elements of N.
>
>That sounds okay. So, don't we have an uncountable number of bitstrings
>with a countable number of lengths?

Randy is talking about infinite bitstrings, all with exactly
the same "length". And yes, there are uncountably many infinite
bit strings.

>> > Which of those maps to N?
>>
>> The one that has a 1 for every bit n, where n is an element of N.
>> Every countable bit string has either a 1 or a 0 for every n in N,
>> so there's certainly one which has 1 for all n in N.
>Does that look like .....111111?

Yes.

>Ahem. Are you saying that, without knowing where the unending string of bits
>ends, you cannot make any judgements about what an infinite string of bits
>means?

For any set A, you can talk about "bit strings over A", which
are basically functions from A into {0,1}. For Randy's infinite
bitstrings, A=N, the set of all (finite) natural numbers.

>Listen, a countable number of bits, such that each is finite, is
>supposedly sufficient to produce an uncountable number of bit
>strings, ala 2^aleph_0, no?

Yes, that's right. There are uncountably many infinite bitstrings.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... and instead compare the strings ... > infinite strings. ... > bijection involves a contradiction, ... there is nothing wrong with infinite bitstrings. ...
    (sci.math)
  • Re: infinity
    ... > Cantor-infinite, and so the set of strings will also be ... Sure, Cantor-infinite, but not actually infinite. ... the Cantor-infinite set of finite naturals. ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :>: that the number of elements is "infinite" that we get into any trouble ... :>:> are in the set of strings that correspond to decimal representations of ... :>: strings representing octals are just a subset of the strings ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :>: that the number of elements is "infinite" that we get into any trouble ... :>:> are in the set of strings that correspond to decimal representations of ... :>: strings representing octals are just a subset of the strings ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... :>: that the number of elements is "infinite" that we get into any trouble ... :>:> are in the set of strings that correspond to decimal representations of ... :>: strings representing octals are just a subset of the strings ...
    (sci.physics)
Loading