Right. That sort of mirrors the Busy Beaver variant of the Gödelian argument as well as its counter.
The BB variant goes: given enough time, we can proceed to find ever more complex busy beavers. A Turing machine can't, therefore we're doing hypercomputation.
The counter goes: but at any given point, there's a pre-prepared TM that can solve all BBs up to the point you've discovered. So there's no reason to presume that you can keep on solving ever more complex BBs; perhaps you're just a very complex TM and haven't reached your limit yet.
(At least that's how I understand it.) So the Busy Beaver variant can only be inductive: it can say something to the effect of "we've surpassed ourselves this many times and we keep doing so, so the evidence that there's some hidden limit is starting to look awfully specific whereas the alternate explanation is much simpler". It can't deductively prove anything by puzzle-solving alone since the bar can always be set higher than whatever has been solved so far.
The similarity lies in that you can't know with certainty whether the provable second-order theorems are a tiny first-order subset or not.
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branchandroot
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Date: 2013-07-03 07:58 am (UTC)The BB variant goes: given enough time, we can proceed to find ever more complex busy beavers. A Turing machine can't, therefore we're doing hypercomputation.
The counter goes: but at any given point, there's a pre-prepared TM that can solve all BBs up to the point you've discovered. So there's no reason to presume that you can keep on solving ever more complex BBs; perhaps you're just a very complex TM and haven't reached your limit yet.
(At least that's how I understand it.) So the Busy Beaver variant can only be inductive: it can say something to the effect of "we've surpassed ourselves this many times and we keep doing so, so the evidence that there's some hidden limit is starting to look awfully specific whereas the alternate explanation is much simpler". It can't deductively prove anything by puzzle-solving alone since the bar can always be set higher than whatever has been solved so far.
The similarity lies in that you can't know with certainty whether the provable second-order theorems are a tiny first-order subset or not.