Failure of the computable weak Kőnig's lemma 

Constructing a decidable binary tree with infinitely many vertices that does not have a computable infinite path. In analysis contexts such as RCA0 or CZF, the non-constructive WKL is more or less as strong as Brouwer's fixed point theorem or restricted forms of Brouwer's fan theorem. Text with links and code: gist.github.co...
Btw. I find the UTM Wikipedia article to be a bit concise, e.g. not mentioning that all codes will pop up often in the enumeration \phi. I encourage everybody to extend them.