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I see your pathological proof of existence and raise you an asymptotic probability of one: “The halting problem for Turing machines is perhaps the canonical un
by Xcelerate 3y ago
I see your pathological proof of existence and raise you an asymptotic probability of one:
“The halting problem for Turing machines is perhaps the canonical undecidable set. Nevertheless, we prove that there is an algorithm deciding almost all instances of it.”
https://arxiv.org/pdf/math/0504351.pdf https://arxiv.org/pdf/math/0504351.pdf
- riku_iki 3y ago"almost all" is very speculative term. What is "almost all" of natural numbers for example?..
- drdeca 3y agoA subset with asymptotic density of 1?
- riku_iki 3y agoin common math/geometry, all natural numbers altogether I think will have density 0..
- drdeca 3y agoThe asymptotic density of a subset of the natural numbers, is defined as the limit of (number of elements of the set which are less than N)/N , as N goes to infinity.
- riku_iki 3y agoOk, by this definition I believe if you consider set of natural number without say n^2 numbers, then density will be 1, or "almost all", while class we ignored and similar classes are quite large and important.
- drdeca 3y agoYes, the asymptotic density of “natural numbers which are not perfect squares” is 1. I.e. “almost all natural numbers are not perfect squares”. And similarly for many other important classes. Almost all natural numbers are not prime. Etc. etc.