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Date: Fri, 7 Jul 2000 05:08:49 EDT
Subject: Re: [lojban] 2 maths questions
To: lojban@egroups.com
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From: pycyn@aol.com
It is a bit of a mistake to think of infinity as a number and then say that
natural numbers and even natural numbers have the same number of members,
but, since we do it all the time, let's try to sort out what it means. In
one sense, there is a boundary cardinal, aleph null, which is he cardinal of
both the set of natural numbers and the set of even natural numbers (and the
integers and the primes and.....). That means, ultimately, that, for each of
these sets, there is a one-one mapping between that set and the canonical set
that "is" (some funny mathematical sense) the cardinal. The mapping are
different for the different sets, but can be combined to give one-one mapping
diirectly between the different sets (naturals<=> evens, say). To say that
there are twice is many naturals as evens is to say that there is a different
mapping, this time between discrete pairs of naturals and evens (e with {e,
e+1}, say). (Doing this for cases like the primes is a lot harder, for the
integers or the rationals a lot fancier but not harder). Now, the fact that
two things are related in one way does not usually mean they cannot be
related in another as well; it seems paradoxical only in the case of size.
But it is not a real paradox -- it just says there is a relation such
that.... and another relation such that ... (and, in fact, it is not too hard
to show that there are three of seventy-seven -- not hard but tedious --
times as many naturals as evens -- or, come to that, evens as naturals). It
is like saying that somthing is green and tall: there are two standards where
by the one.... and by the other ....