[FOM] Replacement
Lawrence Stout
lstout at iwu.edu
Sat Aug 18 16:06:55 EDT 2007
In general I dislike the axiom scheme of replacement. I'm not
claiming that it is false for sets in the cumulative heirarchy (which
I also don't particularly) but that it somewhat misrepresents the
typing which is fundamental in the practice of mathematics when
defining relations and functions. When I teach calculus it is
important for my students to learn that a function consists of three
pieces of information: a domain, a codomain, and a rule for assigning
a unique member of the codomain to each element of the domain. It is
important to know that you change the properties of the function if
you change any of the parts. The axiom of replacement only uses the
rule part of this definition. If you insist on functions (including
the information about domain and codomain) then the specification of
the range only needs bounded comprehension, not full replacement.
Bounded comprehension is a key concept; full replacement strikes me
as an over-generalization for mathematical practice.
As for needing replacement to get Cartesian products: I'd rather
have an axiom of ordered pairing instead of unordered pairing, making
the Cartesian closedness of the category of Sets one of the basic
axioms (as is done when thinking of Sets as just one of many
interesting topoi).
Larry Stout
On Aug 13, 2007, at 2:08 AM, Thomas Forster wrote:
>
>
> I know there are lots of people who dislike the axiom scheme of
> replacement. They say things like ``it has no consequence for
> ordinary mathematics'' and the like. Unfortunately i have none
> of them handy at the moment, so i have to ask: do any of them
> think that the axiom scheme is actually *false*? Or do they
> merely think that it shouldn't be a core axiom?
>
> tf
>
>
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