*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.

Direct Proof:

Consider the set T of all theorems in our axiom system.

T contains:

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.

QED

Dr. S

*If I can only demonstrate these two claims of existence by construction of the proofs, have I met the challenge?