computational philosophy of mathematics

[gusl]

I only believe in mathematical objects with finite information (i.e. with a finite expression). Things can expressed in any way you wish, intensionally, extensionally, whatever. So, while the natural numbers exist, not all subsets of it do.

While the set of real numbers exists (R can be expressed as the power set of N), not all of the traditional real numbers exist (only the computable ones do). So, in my definition, the cardinality of R is aleph_0 (i.e. the same as N). In fact, no set can have greater cardinality, for that would imply it had non-existing elements (since only computable things exist).

Can we design a set theory this way? How much of traditional set theory can be translated? Does we lose any good mathematics this way?

Whereas people seem to view uncomputability as a fundamental property of a problem, I tend to view it as another form of self-reference paradoxes. All "well-defined" problems are computable. Again, this is not a theorem or mathematical insight, but a re-definition, just like my "R only has countably many existing numbers" is a definition. But the point isn't just to change names and keep everything the same... it's to change the intuition that goes with the names.

Comments

[easwaran.livejournal.com]

I should have chosen as an example some sort of existential statement for real numbers, rather than natural numbers, because of course if you prove that such a number exists, there is an effective search procedure for the naturals.

Perhaps even the least upper bound property for sets of real numbers might not turn out to be true. Certainly for recursive sets of real numbers one can construct a least upper bound, but even just recursively enumerable sets might not give an effective least upper bound.