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shminux comments on Pluralistic Existence in Many Many-Worlds - Less Wrong Discussion
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Sure, I can imagine a magic box which accepts any algorithm and tells you whether it halts. Therefore a universe with such a box has a right to exist.
Doesn't that just mean your imagination is self-contradictory?
In what sense? It does not make me go mad or anything, it's just one of the many programs my brain runs.
I have very high confidence that the following question will not make your imagination explode. However, if you are at all worried about that, please stop reading.
What do you imagine the magic box will do if you feed it the algorithm: "Feed this algorithm to the magic box. If it says it halts, then go into an infinite loop. If it says it doesn't halt, then halt."
If you presuppose that the universe is not "mathematically possible", you can't really prove that it is possible for anyone to ask it that question. For that matter, it might just say "it halts" and be right. (It's a mathematically impossible world, and you're using math both when you're deciding what the algorithm will do and what the box should answer.)
By the way, the usual statement for the halting problem is that you can't make an algorithm that solves it, and by algorithm it usually means something a Turing machine, i.e. everything the algorithm does can be done by a Turing machine. In this case, assuming it makes sense to use math to reason about it, “Feed this algorithm to the magic box” is not actually something a Turing machine can do (it only has heads and tapes, no magic boxes). If you also give it the magic box it's no longer just a Turing machine, it's a [Turing machine + Turing oracle], which is something a bit different.
Imagine the magic box accepts algorithms expressed in Lisp (which theoretically allows unlimited memory). How do you express "feed something to the magic box" in Lisp?
By the way, does anyone know if it's proven impossible (or even if discussed) to build a limited halting-problem solving algorithm that works for all algorithms except those that contain (complete or limited) halting solver algorithms as subroutines, in which case they also halt but say something like "don't be an ass"?
And your point is?
Well, either your magic box can't cope with algorithms that talk about the magic box itself, or there's a contradiction going on.
And what's so bad about that?
Nothing's bad about it, but I don't think you can actually imagine the thing you said you could!
Maybe we have different definitions for the term imagine. As far as I'm concerned, by describing your question you imagined it. If you are worried about it being logically inconsistent in this particular universe, imagine a universe where an algorithm's halting behavior changes after it's been fed through the magic box in question. My universe - my rules. Or lack thereof.
Okay, at this point I think we have different definitions for "universe". The one you're describing can't be consistently described.
Absolutely. It generates numbers at random and in one universe, it happens to always be right.
I don't see the relevance, though.
The relevance is that since our imagination runs on the Turing machine of our brains, whatever we can imagine is as likely to exist as any construct based on mathematical axioms, like Tegmark level 4.
Why are you jumping from some symbols being rearranged on a Turing machine to assuming unknown and arbitrarily complex instantations loosely resembling said symbols? Of course a brain 'imagining' something exists on level 4, but why credit any particular form of imagination as being coherent and also greater than mathematics? If you imagine a square triangle, how is that a refutation of Tegmark level 4, rather than, say, evidence that a brain can emit two words in succession which don't mean anything?
Why privilege TL4 over your imagination?
So basically, that's all that your point boils down to? "never mind the failure of millennia of imagination-based reasoning and the striking success of mathematical reasoning in those millennia, I'm just going to make imagination the arbiter of metaphysical possibility even if that means embracing contradictions and other such nonsense"? That's pretty lame.
So you refused to understand my original point and resorted to misrepresenting, strawmanning and eventually insults? Nice. Tapping out.