§ Topological proof of infinitude of primes
We take the topological proof and try to view it from the topology as
- Choose a basis for the topology as the basic open sets.This set is indeed semi-decidable. Given a number , I can check if. So this is our basic decidability test.
- By definition, is open, and . Thus it isa valid basis for the topology. Generate a topology from this. So we arecomposing machines that can check in parallel if for some , for some index.
- The basis is clopen, hence the theory is decidable.
- Every number other than the units is a multiple ofa prime.
- Hence, .
- Since there a finite number of primes [for contradiction], the right hand sidemust be must be closed.
- The complement of is . Thisset cannot be open, because it cannot be written as the unionof sets of the form : any such union would have infinitelymany elements. Hence, cannot be closed.