Some thougts on cartesian products

Terry Reedy 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 
issue.

Terry J. Reedy






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