Although this is not the "detailed proof" you seek,
still you might find Chaitin's own reasoning interesting,
as articulated in his *Scientific American* article "The Limits of Reason" ([PDF download link](http://www.umcs.maine.edu/~chaitin/sciamer3.pdf)):
<hr />&nbsp;&nbsp;&nbsp;![GCincomp][1]<hr />


  [1]: https://i.sstatic.net/yMViB.png