On Sunday, March 3, 2019 at 11:29:32 AM UTC-6, John Clark wrote:
>
>
>
> On Sun, Mar 3, 2019 at 9:26 AM Bruno Marchal <mar...@ulb.ac.be 
> <javascript:>> wrote:
>
>>
>> >> The 8000th Busy Beaver Number can be named but not calculated even 
>>> theoretically,
>>
>>
>> *> The busy beaver function is not computable, but on each individual n, 
>> it is computable theoretically, *
>>
>
> No it is not, not if n= 7918, to compute that the program would have to 
> solve the Halting Problem. The first 4 Busy beaver numbers have been 
> computed and Scott Aaronson proved that the 7918th Busy Beaver Number is 
> not computable, most people think n=5 is not computable either but that has 
> not been proved.
>
> The 7918th Busy Beaver Number 
> <https://www.scottaaronson.com/busybeaver.pdf>
>
> *> The 8000h BB number is well defined, *
>>
>
> Yes.
>  
>
>> *> so it is a (finite) number, *
>>
>
> Yes,
>
> *> and so you there exist a finite program computing it*
>>
>
> No. The 8000th Busy Beaver Number is the largest number of FINITE 
> operations a 8000 state Turing Machine will make before it halts. Some 
> programs we can observe halting and with others it's easy to prove will 
> never halt, that's why we know the first 4 Busy Beaver Numbers, but Turing 
> Proved you can't do that in general and  Aaronson proved you can't do that 
> for the 7918th; and you probably can't even do it for the 5th.
>
> It is entirely possible that the 5th Busy Beaver number is  47,176,870 
> because a 5 state Turing Machine has been found that halts after 47,176,870 
> operations,  the problem is there are still 5 different 5 state turing 
> machines that are well past 47,176,870 and they have not halted. If none of 
> those 5 machines ever halts then 47,176,870 really and truly is the 5th 
> Busy Beaver Number, but if that is the case we will never know that is 
> the case because we'll never know that none of those 5 machines ever halts. 
>
> John K Clark
>



The original issue is what real numbers can be *described* or *defined.*

      https://twitter.com/JDHamkins/status/1100090709527408640 
<https://www.google.com/url?q=https%3A%2F%2Ftwitter.com%2FJDHamkins%2Fstatus%2F1100090709527408640&sa=D&sntz=1&usg=AFQjCNGtkxSJwylKQNS1lOTcYCkL9y9Fvg>

Must there be numbers we cannot describe or define? Definability in 
mathematics and the Math Tea argument 



If a program "represents" a real number (e.g. in the spigot sense), then 
that could be said to "define" it.

But what noes it mean for a real number to be "defined"?

- pt



 

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