Re: performance and eloquence

Maciek Pasternacki wrote:

On Pungenday, Bureaucracy 59, 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.

.

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