John Cowan wrote:
> Thomas Lord scripsit:
>
>   
>> Asked to compute (= +inf.0 +inf.0) Scheme should dynamically raise
>> the question: "Well, what do you mean exactly?"
>>     
>
> Why dynamically?  Why isn't static override (i.e. replacing = with a
> predicate that does what you want) good enough?
>
>   

Goedel.

Suppose you pick a static definition of = and all the other math operators.
The "meanings" of your programs using that system have to be
consistent.  That is, your definitions of the operations amount to a
set of axioms and applying those axioms to one another we must not
arrive at inconsistent results.

Therefore, your semantics must also be incomplete.  That is,
there must exist multiple, mutually-incompatible, consistent
extensions of your number system.  [Says Goedel.]

Your number semantics must be incomplete but they can always
grow in upwards compatible ways on the fly.   Hence the need
for those exceptions.

-t



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