logical induction can’t be sure if its operating in a nonstandard context or not.
The question specified “all my variables are implicitly natural numbers”. Why can’t there be traders that specialize on questions specifically about standard numbers and ignore others? (I assume that the natural numbers are standard numbers, correct?) Also, what’s the connection between nonstandard numbers and your Godel-like proof?
If you have any procedure that can perfectly distinguish between true and false statements within the structure of PA, then you can make a “this statement is false” and get a contradiction.
I think it would be fun to find a concrete example… here’s my attempt...
P=”the logical inductor will assign probability <0.5 to P in the limit of infinitely many steps”
ϕ(n) = “the logical inductor will assign probability <0.5 to P after n inference steps”
so P=∀nϕ(n).
Then maybe any individual ϕ(n) will be true, but the algorithm will never assign >0.5 probability to P=∀nϕ(n).
Did I get that right? I’m very out of practice with my self-referential math so I have low confidence. :-P
Your self referential maths is nearly right. Technically the limit is defined as ∃N∀n>N:ϕ(n) as your ∀n:ϕ(n) could fail because ϕ(1) is false, and logical induction only behaves well in the limit.
The paper says that Pn=" the logical inductor will assign probability <0.5 to Pn after n inference steps”
tend to 0.5 as n→∞. But I havn’t yet worked out the infinite case.
The question specified “all my variables are implicitly natural numbers”. Why can’t there be traders that specialize on questions specifically about standard numbers and ignore others? (I assume that the natural numbers are standard numbers, correct?)
You can’t do that because non-standard numbers look really similar to standard numbers from the inside. There is no formula ϕ(x) that is true on all standard numbers, and false on nonstandard numbers.
Suppose you had a logical induction procedure that had the property that after updating on ϕ(1),ϕ(2)... it believed ∀x:ϕ(x).
So suppose you had p1,p2...∈[0,1] with each pn depending only on ϕ(1)..ϕ(n). and pn→1if∀x:ϕ(x)else0 as n→∞. (Produced by your alternative to logical induction, these represent the probability assigned to ∀x:ϕ(x))
Suppose you prove all that in PA. As PA has nonstandard models, the proof also holds in those. But we can pick a ϕ such that ∀x:ϕ(x) is true in the standard model, but false for some nonstandard x. There must exist nonstandard n such that 1−pn is infinitesimal.
Otherwise the formula ∃m:1−pm<1/k would distinguish standard k from nonstandard k. Assigning infinitesimal probability to something that actually happend is a form of bad behaviour that I think can be transported back to the standard domain.
I think that the resulting behaviour is consistent, but results in poor behaviour.
Sorry if these are stupid questions...
The question specified “all my variables are implicitly natural numbers”. Why can’t there be traders that specialize on questions specifically about standard numbers and ignore others? (I assume that the natural numbers are standard numbers, correct?) Also, what’s the connection between nonstandard numbers and your Godel-like proof?
I think it would be fun to find a concrete example… here’s my attempt...
P=”the logical inductor will assign probability <0.5 to P in the limit of infinitely many steps”
ϕ(n) = “the logical inductor will assign probability <0.5 to P after n inference steps”
so P=∀nϕ(n).
Then maybe any individual ϕ(n) will be true, but the algorithm will never assign >0.5 probability to P=∀nϕ(n).
Did I get that right? I’m very out of practice with my self-referential math so I have low confidence. :-P
Your self referential maths is nearly right. Technically the limit is defined as ∃N∀n>N:ϕ(n) as your ∀n:ϕ(n) could fail because ϕ(1) is false, and logical induction only behaves well in the limit.
The paper says that Pn=" the logical inductor will assign probability <0.5 to Pn after n inference steps”
tend to 0.5 as n→∞. But I havn’t yet worked out the infinite case.
You can’t do that because non-standard numbers look really similar to standard numbers from the inside. There is no formula ϕ(x) that is true on all standard numbers, and false on nonstandard numbers.
Suppose you had a logical induction procedure that had the property that after updating on ϕ(1),ϕ(2)... it believed ∀x:ϕ(x).
So suppose you had p1,p2...∈[0,1] with each pn depending only on ϕ(1)..ϕ(n). and pn→1 if ∀x:ϕ(x) else 0 as n→∞. (Produced by your alternative to logical induction, these represent the probability assigned to ∀x:ϕ(x))
Suppose you prove all that in PA. As PA has nonstandard models, the proof also holds in those. But we can pick a ϕ such that ∀x:ϕ(x) is true in the standard model, but false for some nonstandard x. There must exist nonstandard n such that 1−pn is infinitesimal.
Otherwise the formula ∃m:1−pm<1/k would distinguish standard k from nonstandard k. Assigning infinitesimal probability to something that actually happend is a form of bad behaviour that I think can be transported back to the standard domain.
I think that the resulting behaviour is consistent, but results in poor behaviour.
Will post more as I think of it.