§ 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 is a valid basis for the topology. Generate a topology from this. So we are composing 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 of a prime.
- Hence, .
- Since there a finite number of primes [for contradiction ], the right hand side must be must be closed.
- The complement of is . This set cannot be open, because it cannot be written as the union of sets of the form : any such union would have infinitely many elements. Hence, cannot be closed.