On May 3, 8:15 pm, fom <fomJ...@nyms.net> wrote: > On 5/3/2013 2:43 AM, Graham Cooper wrote: > > > > > Its not possible to test Equality by Extension in the inf. case. > > That is correct Herc. > > On the other hand, it is not possible to interpret > the universal quantifier as a universal statement if > it is interpreted as a course-of-values.
Humans do it.
How do you know:
nats /\ odds = odds ?
Somewhere in the calculations induction is going on...
element-n matches -> element-n+1 matches
> > Aristotle wrote this. It is ignored by a certain > contingent of the mathematical community who merely > argues on the basis of beliefs concerning infinity. > > You know well that any computer system balances > choices that affect performance. Relational databases > run faster on logic chips optimized for integral > arithmetic as opposed to floating point. The analogy > applies here. > > Brouwer had been clear concerning how the effectiveness > of working with finite sets differed from working > with infinite sets. But, the reason infinity enters > mathematics is because it is how the identity relation > is extended to convey the geometric completeness of a > line when used to represent the real number system. > > Infinity does not arise because of testability. It > arises because of the nature of the identity relation. >
Well its fine to call it all philosophy anyway, but there are real world problems here!
Most of ZFC axioms port across to PROLOG...
AXIOM OF PAIRING
e( A , union( S1, S2 ) ) :- e( A , S1 ). e( A , union( S1, S2) ) :- e(A , S2 ).
Now you can WRITE EXPRESSIONS with UNION
eq( union( odds, evens) , nats ) ? > YES
but the AXIOM OF EXTENSIONALITY
(set equality) is completely useless on infinite sets.