I recognize that it has overwhelming consensus at this point, but from what I understand, it's still technically considered "probably true" instead of "definitely true".
If we go by their original definition, then it amounts to dealing with things that can be accomplished by "effective methods", meaning finite sets of instructions of specific steps that can be carried out (when applicable) in finite time. Basically, it sounds logically identical to our modern conception of an algorithm.
If that were the case, then by simply having a computer (with infinite memory, of course) enumerate every possible program and execute them in dovetail-succession so that all halting programs will eventually halt, you would be guaranteed to hit upon any more-powerful computational technique which could be carried out with an "effective method" sooner or later. However, this is clearly impossible since by definition any such method could not be carried out by a computer (or Turing machine, or whatever).
I am certain I am retreading old ground here but could not find it on search. Do I have the thrust of their thesis wrong, or is there a flaw in my find-it-with-enumeration approach?
I also realize that I have possibly oversimplified their thesis to "anything a computer can do, a computer can do", but I'm finding a hard time getting around that.