Some thougts on cartesian products
tjreedy at udel.edu
Wed Jan 25 19:20:56 CET 2006
"Christoph Zwerschke" <cito at online.de> wrote in message
news:dr5opf$d51$1 at online.de...
> Bryan Olson wrote:
>> The claim "everything is a set" falls into the category of
>> 'not even wrong'.
> No, it falls into the category of the most fundamental Mathematical
> concepts. You actually *define* tuples as sets, or functions as sets or
> relations as sets, or even all kinds of numbers and other things which
> exist in the heads of Mathematicians as sets.
You might so define, but Bryan and others might not. The
philosophical/methodological idea that 'everything is a set' has been very
fruitful but it is not a fact. Alternative ideas are 'everything is a
function' and 'everything is defined by axioms'. As for set theory itself,
there are multiple nonequivalent consistent theories for uncountable sets
and therefore multiple definitions of what a set it. And to me, Russel's
'paradox' is a proof by negation that there is at least one collection that
is *not* a set.
Back to the thread topic: The cross-catenation operator (a generalization
of 'cartesian product' to sequences) is sometimes very useful. Recursive
definitions of combinatorial functions provide several examples. Whether
it should be built into the language or left to subclassers is a different
Terry J. Reedy
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