> On 2 Nov 2018, at 15:02, Philip Thrift <cloudver...@gmail.com> wrote:
> 
> 
> 
> On Friday, November 2, 2018 at 3:45:53 AM UTC-5, Bruno Marchal wrote:
> 
>> On 1 Nov 2018, at 19:43, John Clark <johnk...@gmail.com <javascript:>> wrote:
>> 
>> 
>> On Thu, Nov 1, 2018 at 2:27 PM Philip Thrift <cloud...@gmail.com 
>> <javascript:>> wrote:
>> 
>> > infinite time Turing machines are more powerful than ordinary Turing 
>> > machines
>> 
>> That is true, it is also true that if dragons existed they would be 
>> dangerous and if I had some cream I could have strawberries and cream, if I 
>> had some strawberries.   
>> 
>> > How  "real" you think this is depends on whether you are a Platonist or a 
>> > fictionalist.
>> 
>> No, it depends on if you think logical contradictions can exist, if they can 
>> then there is no point in reading any mathematical proof and logic is no 
>> longer a useful tool for anything.
>> 
> 
> 
> No Turing machine can solve the halting problem. You are right on this. But 
> an oracle can, or a machine with infinite speed can.
> 
> Now, such machine have only be introduced (by Turing) to show that even such 
> “Turing machine with magical power making them able to solve the halting 
> problem” are still limited and cannot solve, for example the totality problem 
> (also an arithmetical). 
> 
> Turing showed that there is a hierarchy of problem in arithmetic, where 
> adding magic (his “oracle”) never make any machine complete. It is a way to 
> show how complex the arithmetical reality is. Adding more and more magical 
> power does not lead to completeness. 
> 
> Post and Kleene have related such hierarchies with the number of alternating 
> quantifiers used in the arithmetical expression. P is a sigma_0 = pi_0 
> formula, without quantifier.
> 
> ExP(x, y). Sigma_1 (negation = AxP(x,y) = Pi_1, more complex than sigma_1, 
> already not computable).
> ExAyP(x, y, z)  = Sigma_2 (beyond today’s math!) (negation = Pi_2).
> Etc. 
> 
> More and more “infinite task” are needed.
> 
> Note that such magic does not change the “theology”. It remains the same 
> variants of the Gödel-Löb-Solovay self-reference logics (G and G*).
> 
> Bruno
> 
> 
> 
> 
> There are other "Turing machine" models other than infinite-time ones people 
> have "invented", e.g. inductive Turing machines:
> 
> Algorithmic complexity as a criterion of unsolvability
> https://pdfs.semanticscholar.org/cd8f/442a9f7667891fff6f276a1bc638dd59b937.pdf
>  :
> 
> Let us take an inductive Turing machine M that given a description of the 
> Turing machine T and first n + 1 words x0, x1, . . . , xn from the list x0, 
> x1, . . . , xn, . . ., produces the (n + 1)th partial output. This output is 
> equal to 1 when the machine T halts for all words x0, x1, . . . , xn given as 
> its input, and is equal to 0 when the machine T does not halt for, at least, 
> one of these words. In such a way, the machine M solves the totality problem 
> for Turing machines.
> 
> ?
> 
> 
> cf.
> https://en.wikipedia.org/wiki/Super-recursive_algorithm#Inductive_Turing_machines
> https://bitrumagora.wordpress.com/about/marl-burgin/
> 
> 
> Nothing is settled in computing.


But this does not clearly violate Church-Thesis. Inference inductive is not the 
same as computing. We know that there are many different Turing machine, which 
are not equivalent for proving or inducting, etc. All humans are like that. We 
are still the same *as* Turing machine (combinators, etc.). Universality is 
with respect to computing, and is false with everything else. Now, if you add 
magical, or actual infinities, or oracles, or infinite speed, then you get 
machine which are no more digital finite machine, and so cannot violate the 
Church-Turing thesis either. 

Bruno





> 
> - pt
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