Yeah, if you say determinism means completeness, then "incomplete deterministic" just sounds like "incomplete completeness". Seems like a nosnense term to me. — flannel jesus
seems like you're mixing vocabularies a lot here and generating a lot of unnecessary ambiguity. — flannel jesus
When one definition of determinism is equivalent to "completeness", but then another definition allows you to say "incomplete determinism", and you put pretty close to 0 effort into explaining how that's supposed to make sense, I can't imagine I'm alone in just thinking it's all nonsense from that point on. — flannel jesus
You don't seem interested in trying to make yourself clear, in trying to develop a self-consistent vocabulary for your ideas. You end your post with "Sometimes it still works flawlessly. Sometimes, it doesn't." as if there's nothing at all you could do to clarify your ideas.
Maybe there's not, maybe you can't clarify your ideas. — flannel jesus
The task given to the oracle doesn't make sense. The task given to the oracle is "predict the output of this Thw program, after you feed into the Thw program your prediction for the output of the Thw program."
It's recursive in a way that means the oracle can't even begin. — flannel jesus
Namely, assume toward contradiction that the symbol-printing problem were computably decidable, and fix a method of solving this problem. Using this as a subroutine, consider the algorithm q which on input p, a program, asks whether p on input p would ever print 0 as output. If so, then q will halt immediately without printing 0; but if not, then q prints 0 immediately as output. So q has the opposite behavior on input p with respect to printing 0 as output than p has on input p. Running q on input q will therefore print 0 as output if and only if it will not, a contradiction.
Namely, assume toward contradiction that the symbol-printing problem were computably decidable, and fix a method of solving this problem. Using the oracle as a subroutine, consider the thwarter program which asks to the oracle whether any program p on input p would ever print 0 as output. If the oracle answers that it will print 0, then thwarter itself will not print 0; but if the oracle says that thwarter doesn't print 0, then thwarter does print 0. Running thwarter on itself as input will therefore print 0 as output if and only if the oracle says that thwarter will not, a contradiction.
The sentence, Gödel proves the lack of determinism of deterministic systems, even sounds contradictory. — Tarskian
And who came up with that sentence? — flannel jesus
The sentence, Gödel proves the lack of determinism of deterministic systems, even sounds contradictory. — Tarskian
They... aren't that hard to avoid. You're literally not trying. — flannel jesus
Tarskian, You may be interested in a recent paper by Joel David Hamkins. [...] Terrific, readable paper. Hamkins rocks.
https://arxiv.org/pdf/2407.00680
At bottom, the logic of the argument is like this: if we had a computable way of finding whether existential statements are true, then we could iterate this with negation to also compute ∀∃ assertions, since ∀k∃n ϕ fails just in case there is some k for which the existential statement about it fails. In short, if in general existential statements are decidable, then the whole arithmetic hierarchy collapses.
The effects of diagonalization are important and should be discussed here in PF. It's great that this pops up in several threads and people obviously are understanding it!The environment of the oracle and the thwarter is perfectly deterministic. There is nothing random going on. Still, the oracle cannot ever predict correctly what is going to happen next. The oracle is therefore forced to conclude that the thwarter has free will. — Tarskian
Thanks! Again a fine article, @fishfry, that I have to read. I've been listening to Youtube lectures that Joel David Hamkins gives. They are informative and understandable.You may be interested in a recent paper by Joel David Hamkins. Turing never proved the impossibility of the Halting problem! He actually proved something stronger than the Halting problem; and something else equivalent to it. But he never actually gave this commonly known proof that everyone thinks he did. Terrific, readable paper. Hamkins rocks. — fishfry
Couldn't oracle simply lie to the thwarter. It knows what the thwarter will do. It tells it something else.It is accepted as proof, however, that no oracle can exist that can predict what choices programs will make. — Tarskian
Imagine that you install an app on your phone that can tell you minute by minute what you will be doing at any point in the future along with all possible details?
The existence of this app would prove that you are just an automaton, i.e. a robot. In that case, it would be ridiculous to claim that you have free will. — Tarskian
At what point do you declare my predictive powers eliminate your free will? How many trials must there be and would a single variance re-establish my free will? — Hanover
If I accurately predict the outcome of 50 coin tosses, does that necessarilymake the coin toss outcomes not random? — Hanover
You will never predict correctly what thwarter is going to do. — Tarskian
When you put thwarter in that chaotic system, you suddenly have something freely making decisions while you can impossibly predict what decisions it will make. — Tarskian
Free will is a property of a process making choices. If it impossible to predict what choices this process will make, then it has free will. — Tarskian
you kind of contradict the first half of your post here with the second half. In the first half, you speak as if something being deterministic is basically synonyms with it being predictable, but in the second half you acknowledge that a chaotic system could be deterministic but unpredictable. — flannel jesus
If a chaotic system can be deterministic but unpredictable, then you should be able to imagine software that is chaotic, and thus deterministic and unpredictable, no? — flannel jesus
I think there's a subtly shifting meaning for the word "unpredictable" that's at play there. — flannel jesus
Yes, I think I have lost this debate to Tarskian. — fishfry
You haven't lost any debate, you just made a post with some mistakes. You seem ready to acknowledge them, which is winning in my book. — flannel jesus
The oracle would know how the thwarter would react to its prediction. It could say, "Now that I've told you you will do X, you will do Y, just to thwart me." Which would make the thwarter do X, or Z, or whatever. And the oracle would know every step of the dance. A dance that might go on forever, thwarter never actually doing anything, as oracle endlessly says, "But now that I've said that, you will..." Which oracle would know ahead of time.The thwarter first asks the oracle what it predicts that it will be doing. The oracle then looks at the source code of the thwarter and at the inputs that it would be getting from the environment, and then predicts what the thwarter will be doing. Upon receiving the answer from the oracle, the thwarter does something else instead, because that is exactly how it was programmed. — Tarskian
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