Re: performance and eloquence

On Sweetmorn, Bureaucracy 62, 3172 YOLD, Spiros Bousbouras wrote:

Most of the time when we say a set is "infinite", we mean "countably
infinite", which states that there can be a 1-1 correspondence between
that set and the set of natural numbers. Integers, rationals, pairs of
rationals, and even sequences of rationals fall into this category.

The set of **finite** sequences of rationals is countable. But if
you're thinking infinite sequences then there are continuum many
of them.

Rationals are countable (|Q|=aleph null). Set of indices of infinite
sequence is also aleph null-sized (indices are natural numbers). So,
infinite sequence of rationals *is* aleph null-sized

Your premises are correct but the conclusion doesn't follow.
Consider that for every real number there is a sequence of
rationals converging to that number. If the set of all infinite
sequences of rationals were countable as you're claiming then
the set of converging sequences in particular would be countable
hence the set of reals would be countable. Since this isn't the case
the set of all infinite sequences of rationals is not countable.

(aleph null times
aleph null is aleph null).

Indeed but the set of all sequences is given by aleph-null
raised to the power aleph-null and that's the continuum.

I see now, thanks for clearing this up.

--
__ Maciek Pasternacki <maciekp@xxxxxxxxxxxxxxx> [ http://japhy.fnord.org/ ]
`| _ |_\ / { ...so I talked about conscience, and I talked about pain,
,|{-}|}| }\/ and he looked out of window, and it started to rain, and
\/ |____/ I thought, maybe - I've already gone crazy... } ( Fish ) -><-
.

Relevant Pages

  • Re: Problems I have with 1.999...=2
    ... we come to the problem of what these infinite decimal ... we know that rationals are dense on the real number ... infinite lists only contain rational numbers for our purposes here so ... As far as I understand these special sequences are ...
    (sci.math)
  • Re: Another set with cardinality |Z|
    ... > infinite binary expansion. ... Not all trees have such limits, but yours does, and they *are* the reals ... there are more limits of sequences of computations than there ... All rationals are computable, so any Cauchy ...
    (comp.theory)
  • Re: Another set with cardinality |Z|
    ... > infinite binary expansion. ... Not all trees have such limits, but yours does, and they *are* the reals ... there are more limits of sequences of computations than there ... All rationals are computable, so any Cauchy ...
    (sci.math)
  • Re: performance and eloquence
    ... infinite", which states that there can be a 1-1 correspondence between ... rationals, and even sequences of rationals fall into this category. ... raised to the power aleph-null and that's the continuum. ...
    (comp.lang.lisp)
  • Re: An uncountable countable set
    ... > But we could apply the transpositions even without any law, ... well-order by size too. ... the rationals is *not* a sequential process. ... I said "sequences". ...
    (sci.math)
Loading