Re: What is the largest integer you can write in 5 seconds?

[John Clark]

On Mon, Mar 11, 2019 at 3:41 AM Liz R  wrote:

*> I have a simpler answer!*
> *"the largest integer you can write in 5 seconds"*
> *...can be written in 5 seconds.*
>

I can beat that and it would take even less time to write:

"the largest integer you can write in 5 YEARS"

 John K Clark


>

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Re: What is the largest integer you can write in 5 seconds?

[John Clark]

On Mon, Mar 11, 2019 at 3:39 AM Liz R  wrote:

*> Graham's number tetrated Graham's number times? That took about 5
> seconds, does it come close?*


Tetration is computable and the Busy Beaver Function grows faster than ANY
computable function. We don't know what BB(7) is but we do know its
larger, probably a LOT larger, than 10^((10^10)^(10^10)^7) ; and we're
talking about BB(8000).

John K Clark

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Re: What is the largest integer you can write in 5 seconds?

[Liz R]

I have a simpler answer!

"the largest integer you can write in 5 seconds"

...can be written in 5 seconds.

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Re: What is the largest integer you can write in 5 seconds?

[Liz R]

Graham's number tetrated Graham's number times? That took about 5 seconds, 
does it come close?

On Wednesday, 6 March 2019 07:06:24 UTC+13, John Clark wrote:
>
> It's easy to prove that the Busy Beaver Function grows faster than *ANY* 
> computable function because if there were such a faster growing function 
> you could use it to solve the Halting Problem. So if you're ever in a 
> contest to see who can name the largest integer in less than 5 seconds just 
> write BB(9000) and you'll probably win.
>
> John K Clark
>
>

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Re: What is the largest integer you can write in 5 seconds?

[Bruno Marchal]

> On 8 Mar 2019, at 22:00, John Clark  wrote:
> 
> On Fri, Mar 8, 2019 at 5:05 AM Bruno Marchal  > wrote:
> 
>  > BB(8000) is stil an infinitesimal (so to speak) compared to 
> f_epsilon_0(BB(8000)).
> 
> I don't know what "f_epsilon_0" is but if its computable then BB[BB(8000)]

Good point. But now, using epsilon_0, which is the constructive ordinal 

 omega^(omega^(omega^(omega^…, or equivalently omega tetrated to omega 
(tetration is the iteration of exponentiation, like exponentiation is the 
iteration of multiplication, and multiplication is the iteration of addition).

You can iterate the application of BB epsilon_0 times, and actually, alpha 
times for alpha any constructive ordinal.

If the problem is to name a large number, or just to point to a large number, 
in all case the winner is the one who will be able to use the constructive 
ordinal. 

Now, as you don’t ask for a name (definite description, programs to build that 
name) the question raised if you could not iterate BB on non constructive 
ordinals, but that will make the pointing even more fuzzy and not well 
definite. Continuing this process will lead to inconsistency, at to some 
undecidable threshold.

Bruno




> would be a larger number than that because BB grows faster than any 
> computable function.
> 
> John K Clark
> 
> 
> 
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Re: What is the largest integer you can write in 5 seconds?

[Bruno Marchal]

> On 8 Mar 2019, at 15:49, John Clark  wrote:
> 
> On Fri, Mar 8, 2019 at 5:05 AM Bruno Marchal  > wrote:
> 
> >> Assuming you're just using 2 symbols (like 0 and 1) there are (16001)^8000 
> >>  different 8000 state Turing Machines. And that is a very large number but 
> >> a finite one. And one of those machines makes the largest number of FINITE 
> >> operations before halting. And that number of operations is BB(8000).  
> >> Even theoretically, much less practically,  you can never compute that 
> >> number but I have given a unique description of it, no other 8000 state 
> >> Turing Machine has that propertie.
>  
> > Yes, sure BB(8000) is a precise well defined finite number. But it is no 
> > what logician and philosopher call a “name”, where the number should be 
> > computable in principle. My point is just a vocabulary point,
> 
> I agree it's just a question of vocabulary but to avoid confusion if logician 
> and philosophers want to use commonly used words then their technical meaning 
> should have some relationship to their common meaning. Parents can give a 
> precise definition to their child (he's the only kid in the crib) so they can 
> "name" him even though they can't calculate him.
>  
> > and a way to remind a nice problem which I have used to illustrate some 
> > less known application of Cantor’s diagonal.
> 
> It always seemed to me that if Cantor had taken just one more small step he 
> could have proven the existence of non computable numbers more than 40 years 
> before Turing did.


Similarly, if you read Plotinus’ Ennead “On the Number”, you can see that 
Plotinus was foreseeing the Difficulty that Cantor was confronted with the 
notion of set, notably by trying to get a number of the numbers, which was a 
natural idea for a platonician, but one of those ideas which leads to 
conceptual difficulties, and theological one too, as Cantor saw by having an 
heavy correspondence with the catholic clergy. 
I think that if we would not have been obliged, by violence and terror, to 
separate science and theology, and to not have mixed it with the State(s), the 
whole “Church-Turing-Gödel” revolution could have appeared 500 hundred years 
before. I take the discovery of the Universal Machine, made by Babbage, Post, 
Church Turing is the biggest discovery made by the humans ever. It changes 
literally everything including the conception we can have on everything.

Bruno






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> 
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Re: What is the largest integer you can write in 5 seconds?

[John Clark]

On Fri, Mar 8, 2019 at 5:05 AM Bruno Marchal  wrote:

 > *BB(8000) is stil an infinitesimal (so to speak) compared to
> f_epsilon_0(BB(8000)).*


I don't know what "f_epsilon_0" is but if its computable then BB[BB(8000)]
would be a larger number than that because BB grows faster than any
computable function.

John K Clark

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Re: What is the largest integer you can write in 5 seconds?

[John Clark]

On Fri, Mar 8, 2019 at 5:05 AM Bruno Marchal  wrote:

>> Assuming you're just using 2 symbols (like 0 and 1) there are
>> (16001)^8000  different 8000 state Turing Machines. And that is a very
>> large number but a finite one. And one of those machines makes the largest
>> number of FINITE operations before halting. And that number of operations
>> is BB(8000).  Even theoretically, much less practically,  you can never
>> compute that number but I have given a unique description of it, no other
>> 8000 state Turing Machine has that propertie.
>
>

*> Yes, sure BB(8000) is a precise well defined finite number. But it is no
> what logician and philosopher call a “name”, where the number should be
> computable in principle. My point is just a vocabulary point,*
>

I agree it's just a question of vocabulary but to avoid confusion if
logician and philosophers want to use commonly used words then their
technical meaning should have some relationship to their common meaning.
Parents can give a precise definition to their child (he's the only kid in
the crib) so they can "name" him even though they can't calculate him.


> > *and a way to remind a nice problem which I have used to illustrate
> some less known application of Cantor’s diagonal.*
>

It always seemed to me that if Cantor had taken just one more small step he
could have proven the existence of non computable numbers more than 40
years before Turing did.

 John K Clark

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Re: What is the largest integer you can write in 5 seconds?

[Bruno Marchal]

> On 7 Mar 2019, at 15:04, John Clark  wrote:
> 
> On Thu, Mar 7, 2019 at 8:18 AM Bruno Marchal  > wrote:
> 
> > Usually, when asked to name a big number, we mean to provide a number that 
> > e can compute in a finite time (no matter how long). BB(8000) will be 
> > rejected, because it is not a definite description, or name, because BB is 
> > not computable.
> 
> Assuming you're just using 2 symbols (like 0 and 1) there are (16001)^8000  
> different 8000 state Turing Machines. And that is a very large number but a 
> finite one. And one of those machines makes the largest number of FINITE 
> operations before halting. And that number of operations is BB(8000).  Even 
> theoretically, much less practically,  you can never compute that number but 
> I have given a unique description of it, no other 8000 state Turing Machine 
> has that propertie.


Yes, sure BB(8000) is a precise well defined finite number. But it is no what 
logician and philosopher call a “name”, where the number should be computable 
in principle. My point is just a vocabulary point, and a way to remind a nice 
problem which I have used to illustrate some less known application of Cantor’s 
diagonal. A constructive one, in this case. Then even using BB, you should 
still use the diagonalization, as BB(8000) is stil an infinitesimal (so to 
speak) compared to f_epsilon_0(BB(8000)).

Bruno





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Re: What is the largest integer you can write in 5 seconds?

[John Clark]

On Thu, Mar 7, 2019 at 8:18 AM Bruno Marchal  wrote:

*> Usually, when asked to name a big number, we mean to provide a number
> that e can compute in a finite time (no matter how long). BB(8000) will be
> rejected, because it is not a definite description, or name, because BB is
> not computable.*
>

Assuming you're just using 2 symbols (like 0 and 1) there are (16001)^8000
different 8000 state Turing Machines. And that is a very large number but a
finite one. And one of those machines makes the largest number of FINITE
operations before halting. And that number of operations is BB(8000).  Even
theoretically, much less practically,  you can never compute that number
but I have given a unique description of it, no other 8000 state Turing
Machine has that propertie.

 John K Clark

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Re: What is the largest integer you can write in 5 seconds?

[Bruno Marchal]

> On 5 Mar 2019, at 19:05, John Clark  wrote:
> 
> It's easy to prove that the Busy Beaver Function grows faster than ANY 
> computable function because if there were such a faster growing function you 
> could use it to solve the Halting Problem. So if you're ever in a contest to 
> see who can name the largest integer in less than 5 seconds just write 
> BB(9000) and you'll probably win.





Usually, when asked to name a big number, we mean to provide a number that e 
can compute in a finite time (no matter how long). BB(8000) will be rejected, 
because it is not a definite description, or name, because BB is not computable.

How to name a big number? You can start with a sequence of growing function, 
like addition, multiplication, expoenntation, iteration, quintation, hexaxion, 
etc. (each one is just the iteration of the preceding one).

Then, calling those functions F_0, F_1, F_2, … F_i, … you can get a “limit” by 
diagonalising them, which gives a growing total function, as all F_i are total, 
and it grow more quickly:

G_0(n) = F_n(n) + 1,

And then a full new sequence G_0, G_1, G_2, …

And you can diagonalise on that sequence too, and again and again. This will 
work on the whole range of the constructive (aka recursive) ordinals.

So, if you need to write the description of a big number, the usual method will 
be to name a big constructive infinite ordinal, like epsilon_zer, for example, 
although there are much bigger one … and write:

F_epsilon-0 (999).

See my old post on this published here a (long) time ago, for a more detailed 
account, going far above epsilon_zero.

It is a cute problem we can ask six years old children. Usually some write 
9+9+9+9+… +9, on the whole paper. Some write 9*9*9*9*9¨… *9. Later they get 
better ideas, and can discuss this all along their secondary school. To get the 
transfinite constructive original, you need the second recursion theorem of 
Kleene, which is con course much more advanced.

O course, if a non computable number number is asked, and still want to win, 
you can do the same statring from BB(8000), like 

F_epsilon-0(BB(8000)). But again, that is usually not accepted, because the 
“name” is not constructive, and usually a name must be a constructive definite 
description.

Bruno





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