§ Topological proof of infinitude of primes
We take the topological proof and try to view it from the topology as
semidecidability perspective.
- Choose a basis for the topology as the basic open setsS(a,b)={an+b:n∈Z}={mathbbZ+b}.This set is indeed semi-decidable. Given a number k, I can check if(k−b). So this is our basic decidability test.
- By definition, ∅ is open, and Z=S(1,0). 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 i,(k−b[i]) for some index.
- The basis S(a,b) is clopen, hence the theory is decidable.
- Every number other than the units {+1,−1} is a multiple ofa prime.
- Hence, Z∖{−1,+1}=∪pprimeS(p,0).
- Since there a finite number of primes [for contradiction], the right hand sidemust be must be closed.
- The complement of Z∖{−1,+1} is {−1,+1}. Thisset cannot be open, because it cannot be written as the unionof sets of the form {an+b}: any such union would have infinitelymany elements. Hence, Z∖{−1,+1} cannot be closed.