FOM: Shipman on introducing concepts

Walter Felscher walter.felscher at
Sun May 2 12:53:21 EDT 1999

It seems to me that Mr. Mayberry is a bit too rigid in requiring that
axiom systems always be presented as 'minimal and independent' (as the
ancient phrase used to be). It is, for instance, easy to see that in a
ring with unit element 1 for multiplication the commutativity of addition
can be derived from the other axioms; still, nobody would advocate to
define the class of these rings  -  and in particular the class of 
fields  -  by axioms not containing that commutativity. 

Of course, in the case of ordered pairs the situation is slightly
different as here not only an axiom, but also a concept (a term forming 
operation) is reducible to the other concepts. Yet no harm seems to
have been done when Bourbaki in his Th/eorie des Ensembles used the
ordered pair as a primitive notion and defined its use through an
axiom of unique readability.  


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