Originally posted by royalchicken
Can anyone solve my exercise nonconstructively, ie show that there is a nonconstructive proof of the irrationals' existence without actually formulating such a nonconstructive proof?
Claim: There exists a nonconstructive proof of the existence of irrational numbers.
Challenge: Prove the claim without formulating a nonconstructive proof of the existence of irrationals.
Consider the set T of all theorems in our axiom system.
t1: "R - Q is not the empty set."
t2: "If R-Q is not the empty set, some irrationals exist."
There exists a proof of t1 that is nonconstructive.*
There exists a proof of t2 that is nonconstructive.*
Further, T must also contain:
t3: "Some irrationals exist."
which follows syllogistically from two theorems whose proofs are nonconstructive.
Hence, there exists a nonconstructive proof for t3, which must be a nonconstructive proof for the existence of irrationals.
*If I can only demonstrate these two claims of existence by construction of the proofs, have I met the challenge?