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Re: is infinite countable or uncountable?
Posted:
Apr 16, 2005 8:14 PM
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Your phrasing is funny. "Infinite" is an adjective.
So it's like asking "Is hot scorching or scalding?"
If you meant the noun "infinity", then it's like
asking "Is heat scorching or scalding?" :-)
Saying that a set is finite means that you can apply
mathematical induction to the subsets. Call a property
of a set "inductive" if it holds of the empty set, and
if it also is preserved by adding one element, i.e. if
S is a set satisfying the property and x is an element
not in S, then the union of S with {x} satisfies the
property. A finite set satisfies every inductive property
and any set that satisfies every inductive property is
finite.
Infinite means "not finite".
A set is called countable if it can be put into one-to-one
correspondence with a subset of the integers, uncountable
if not. So there are three kinds of sets: finite, countably
infinite, and uncountable (and infinite). In that sense
infinity is sometimes countable and sometimes uncountable.
There are uncountably many infinite subsets of the integers.
In that sense infinity is uncountable.
Two sets are said to have the same cardinality if they can
be put into one-to-one correspondence with each other. The
cardinalities of infinite sets are sometimes described as
different "sizes" of infinity. There are uncountable
collections of infinite sets, each having a different
cardinality. In that sense infinities are uncountable.
Is one of the above things an answer to your question?
Keith Ramsay
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