Scott Mayers wrote: ↑Tue Mar 03, 2020 8:58 pm
No. It is incomplete with respect to a finite time.
That's not really in your control, is it? It depends entirely on the time-complexity of the problem and the size of your inputs.
Scott Mayers wrote: ↑Tue Mar 03, 2020 8:58 pm
If you allow time as a static dimension, then technically this could solve the problem but would require a rule that uses contradiction to SEPARATE into two different universes and what I use for my theory.
"Separating into two different universes" is nothing more than a non-deterministic Turing machine cloning itself. Concurrency/Parallelism. You are arriving at the doorstep of P vs NP.
Scott Mayers wrote: ↑Tue Mar 03, 2020 8:58 pm
Because Totality would contain absolutely all, the kind of machine that would permit completion is itself Totality which is never "complete" but you can still close it in using limits and Calculus.
Naturally. Numerical methods (sheer brute force) is the only way to solve some problems. Whether you will be successful or not at this still depends on the computational resources at your disposal. Space, Time, Entropy.
Scott Mayers wrote: ↑Tue Mar 03, 2020 11:09 am
I don't know what you are thinking on recursion. But as I just mentioned, with respect to Totality, it is 'complete' by label and yet never complete in 'times'. But given all possibilities in Totality takes in all the continuum of infintes, including time, physics itself would be 'complete' with respect to it but not necessarily provable by expected standards. It doesn't exclude any possibility.
You are confusing your perspectives here. A priori vs a posteriori.
A posteriori any function which has halted (past tense) is "Total". This is an empirical assertion and needs not be "proven".
A priori some functions can be proven to be Total. E.g we can prove that they WILL halt (given enough time).
Neither of those two functions address the issue of totality with respect to recursive functions. There are some recursive functions that may not terminate. EVER. Even if they had infinite time/memory - they will continue to run for for eternity. One trivial example of such a function is an infinite loop.
Scott Mayers wrote: ↑Tue Mar 03, 2020 11:09 am
?? You sound schizoid here. What are you meaning when you say that "I insist that I am permitted to express a contradiction!" if you are already in agreement to the Incompleteness Theorem you are challenging Pete about?
I am in agreement with the incompleteness theorem
AND the completeness theorem.
I agree in as much as I understand the implications of each theorem. There exists a trade-off (read: choice) between consistency, completeness and unrestricted recursion.
Choose any two.
Pete has chosen Consistency + Completeness.. Therefore he has given up the expressive power of recursion for
Walther recursion which is provably terminating (read: finite) form of recursion.
That may be an acceptable loss to Pete, but it's not an acceptable loss for me.