Understanding the No Free Lunch Theorem
You're asking about optimization and universal search, BUT machine-learning is tagged and you're wondering about "a uniform distribution on an infinite" discrete set so perhaps this will be helpful. There's a classic No-Free-Lunch result in PAC learning which says that you cannot learn the concept class of all boolean functions on the integers in a distribution-free fashion. Meaning: there is no function $m=m(\epsilon,\delta)$ such that if you request $m$ or more iid labeled examples from any unknown distribution, you'll be able to guarantee an $\epsilon$-accurate classifier with $\delta$-confidence. The proof proceeds by taking a uniform distribution over a large enough support size -- big enough to thwart any given purported sample complexity $m(\epsilon,\delta)$. A shorthand for saying this might be "take the uniform distribution over the integers". Now we all know that's absurd, and the explanation above is the only way I know to make sense out of it (and also jibes with the No-Free-Lunch context).