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.
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.