Largest Finite Number

[Bill Dubuque]

Randall Wagoner <lew...@ix.netcom.com> writes:
|
| What is the largest natural number you can think of?

I've appended one of my earlier posts on this topic. All of the
references listed below are expository and should be accessible
to those with a college level education. Rudy Rucker's book
"Infinity and The Mind" is an excellent introduction to most all
aspects of the infinite, accessible to the "educated layperson".

From: w...@zurich.ai.mit.edu (Bill Dubuque)
Subject: Re: Largest natural number with a name?
Date: 1996/01/16
Newsgroups: sci.math

From: bo...@mathworks.com (Bob Silverman)
Date: 15 Jan 1996 11:52:26 -0500

Gary Scott Simon <gars...@nyc.pipeline.com> wrote:
>Michael Glenn writes:
>
>>What is the largest digit with a name? eg: million, ... quadrillion, etc..
>>Can you also include the numerical value eg: 1000 or ... 1 X 10^4 etc...

> Since Mr. Glenn's language suggests that he is looking for the largest
>natural number (rather than the largest "digit") with a name, I'll try to
>be the first to mention a "google-plex", which is 10^(10^100).

Allow me to be the first to mention that the question is not well-posed.
What does the poster mean by "largest digit (sic) with a name"?

Assuming the poster meant "number" rather than digit, the question still
isn't well posed. Largest in the dictionary? Largest that has ever appeared
in a serious math paper? Largest that has ever appeared in print?

There are numbers with names far bigger than googolplex. Skewes number,
for example (e^e^e^e^34). There is also a 'Moser', which is represented
by an iterated form of Ackerman's function then evaluated. These numbers
are too big to even write down.

Perhaps the largest notated integers occur in proof theory and recursion
theory where you'll find notation systems for numbers much larger than
any mentioned above. See the references below, along with my prior
sci.math posts on Godel's incompleteness theorem and natural independence
results for further information (archived in http://www.dejanews.com/ ).

Note that Kreisel has shown that one can extract bounds automatically
from certain proofs and has used the Skewes Prime Number Theorem bounds
as a prototypical example. Another example is the original Ackermanish
bounds on van der Waerdans problem on arithmetic progressions (this
bound has recently been significantly sharpened by Shelah).

-Bill Dubuque

[HFR] Harrington, L.A. et.al. (editors)
Harvey Friedman's Research on the Foundations of Mathematics, Elsevier 1985.

[Kol] Kolata, Gina
Does Godel's theorem matter to mathematics?, Science 218 11/19/1982, 779-780.
reprinted in [HFR]

[Ruc] Rucker, Rudy
Infinity and The Mind, 1995, Princeton Univ. Press.

[Sim] Simpson, Stephen G.
Unprovable theorems and fast-growing functions, Cont. Math. 65 1987, 359-394.

[Smo] Smorynski, Craig (all three papers are reprinted in [HFR])
Some rapidly growing functions, Math. Intelligencer, 2 1980, 149-154.
The Varieties of Arboreal Experience, Math. Intelligencer, 4 1982, 182-188.
"Big" News from Archimedes to Friedman,
Notices Aner. Math. Soc. 30 1983, 251-256.

[Spe] Spencer, Joel
Large numbers and unprovable theorems, Amer. Math. Monthly, Dec 1983, 669-675.

[Peter Davies]

In article <33F2A378...@ix.netcom.com>,

Randall Wagoner <lew...@ix.netcom.com> wrote:
>What is the largest natural number you can think of?
>

>Randall
>
>

As soon as you think of a natural number, you can immediately add one to it
for a larger one. What is the point of your question?

Peter

[Kurt Foster]

Randall Wagoner (lew...@ix.netcom.com) wrote:
: What is the largest natural number you can think of?

:
The largest number YOU can think of -- plus one! ;-)

[Randall Wagoner]

Randall Wagoner wrote:

> What is the largest natural number you can think of?
>

> Randall

Ok, how about this?

What is the highest natural number you can describe in 100 characters or
less?

Randall

[kem...@de.ibm.com]

In <33F2A378...@ix.netcom.com>, Randall Wagoner <lew...@ix.netcom.com> writes:
>What is the largest natural number you can think of?
>
>Randall
>
>

n . Or has it been n+1 ? I've just forgotten it
by counting n! to the n!-th power at n! times.
But remembering the smallest non-natural cardinal w(= small omega) or
aleph_0.
regards

[Randall Wagoner]

[Dilip Andrade]

Randall Wagoner wrote:

> Ok, how about this?
>
> What is the highest natural number you can describe in 100 characters

I don't know, how about

goolglepex^(googleplex^(googleplex^(googleplex^...

stop when you get to 99. I hope that that's big enough, realizing that
there are still an infinite amount of naturals above it.

Dilip Andrade
--------------------------------
My views do not neccessarily reflect thos of my employer.

[John Goodwin]

On Thu, 14 Aug 97 13:26:34 GMT, pet...@interlog.com (Peter Davies)
wrote:

>In article <33F2A378...@ix.netcom.com>,

> Randall Wagoner <lew...@ix.netcom.com> wrote:
>>What is the largest natural number you can think of?
>>
>>Randall
>>
>>
>

>As soon as you think of a natural number, you can immediately add one to it
>for a larger one. What is the point of your question?
>
>Peter

How about just having some FUN

J.G.

[John Goodwin]

On Thu, 14 Aug 1997 11:05:25 -0400, ed...@math.ohio-state.edu (G. A.
Edgar) wrote:

>One of the games analyzed in Berlekamp, Conway, & Guy, WINNING WAYS
>is called "My father makes more than your father." It is played
>as follows: Player A names some positive integer, then player B
>names some positive integer, and the one with the highest number
>wins. Of course there is a simple winning strategy for player B...

I've been thinking about that for all of a quarter of a nanosecond,
and I must admit it's got me completely stumped.

J.G.

[John Goodwin]

On Thu, 14 Aug 1997 11:09:45 -0500, Randall Wagoner
<lew...@ix.netcom.com> wrote:

>Randall Wagoner wrote:
>
>> What is the largest natural number you can think of?
>>
>> Randall
>

>Ok, how about this?
>
>What is the highest natural number you can describe in 100 characters or
>less?
>
>Randall
>

Now this sounds like a LOT of fun.

We would need to set up some rules.

Things like:

Are abbreviations allowed, and to what extent?
[ Perhaps each word would be counted as a symbol]

Do you need to count "^" since it's not necessary in ordinary
mathematical notation.?

Are spaces and new lines free?

Can we leave out punctuation in English?

If we could set up the rules, and get a referee to accept emails until
a certain date. Then a referee to work out the winner.

Then we could spend the next six months arguing about if the referee
had made the correct decision (I don't somehow think that a Casio will
be much help).

[Randall Wagoner]

John Goodwin wrote:

This was meant to be fun.

I'm going to put some limits on it though. What do you think?

The rules are:

Multiplication is the highest operation you can use.
Myriad (M) = 10^4 (10,000) is the highest number that you can use in
any variable. i.e. M*M...100 chars
You can set up as many rules as you like, as long as the total of
non-space characters don't exceed 100. I don't think it will come down
to a few characters. I think it will depend on how well thought out the
rules are.
Functions are allowed in the rules as long as you define everything.

So, if you gave these rules to someone that doesn't know any operations
higher than multiplication, and no number higher then M, he should be
able to determine what you are talking about.

Just so you know what I mean, I'll give you a simple example.

A(1, 1) = M
A(1, j+1) = M*A(k, j)
A(k+1, 1) = A(k, M)
A(k+1, j+1) = M*A(k+1, j)

What is the highest number that can be reached?

Randall

[Arne Dehli Halvorsen]

Randall Wagoner <lew...@ix.netcom.com> wrote in article
<33F2A378...@ix.netcom.com>...

> What is the largest natural number you can think of?

The largest natural number *you* can think of....

SQUARED!!!

(nyah, nyah, nyah)

:-D
Arne

>
> Randall
>
>
>

[G. A. Edgar]

One of the games analyzed in Berlekamp, Conway, & Guy, WINNING WAYS
is called "My father makes more than your father." It is played
as follows: Player A names some positive integer, then player B
names some positive integer, and the one with the highest number
wins. Of course there is a simple winning strategy for player B...

--
Gerald A. Edgar ed...@math.ohio-state.edu

[Dilip Andrade]

Arne Dehli Halvorsen wrote:
>
> Randall Wagoner <lew...@ix.netcom.com> wrote in article
> <33F2A378...@ix.netcom.com>...
> > What is the largest natural number you can think of?
>
> The largest natural number *you* can think of....
>
> SQUARED!!!
>
> (nyah, nyah, nyah)

Just to be a bit of a dink: 1 (nyah, nyah, nyah right back at you.)

Dilip Andrade
--------------------------------------------
Any views I express do not neccessarily reflect the views of my employer

[Robert Israel]

In article <33F39D96...@ix.netcom.com>,

Randall Wagoner <lew...@ix.netcom.com> wrote:
>John Goodwin wrote:
>
>> On Thu, 14 Aug 1997 11:09:45 -0500, Randall Wagoner
>> <lew...@ix.netcom.com> wrote:
>> >
>> >What is the highest natural number you can describe in 100 characters
>> or
>> >less?
>> >

How about:

(the highest number Randall can describe)+1

Robert Israel isr...@math.ubc.ca
Department of Mathematics (604) 822-3629
University of British Columbia fax 822-6074
Vancouver, BC, Canada V6T 1Y4

[Bill Taylor]

In article <33F2A378...@ix.netcom.com>, Randall Wagoner <lew...@ix.netcom.com> writes:

|> What is the largest natural number you can think of?

Unfortunately the margin is too small to contain it...

-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Doctors claim to have discovered a cure for manic depression.

I don't know whether to laugh or cry!
-------------------------------------------------------------------------------

[Jeffrey Lockshin]

It is my opinion that the respondents didn't understand what Randall
was asking. You can always imagine a number larger than any one you
choose - but that's not the point. The point is is that huge numbers
can be entirely ephemeral - a product of the imagination. Very few
people can truly imagine the sheer magnitude of a trillion - it's very
easy to see that you would need more than 31.5 years to count to a
trillion if you counted one number ever second. Yet this is the
magnitude of the national debt.

Take the number 10^100 - a number larger than the number of sub-atomic
particles in the visible universe. This number is _impossible_ to
imagine. Sure, you can say that 10^200 is larger, and that 10^400 is
even larger than that (like one of the respondents said - you can
always take the square). But what Randall was asking should be
rephrased like this:

What is the largest finite number you can think of that has a real,
physical meaning, and is not ephemeral, a product of the imagination?

[Jeffrey Lockshin]

[John Goodwin]

On 15 Aug 1997 12:10:33 -0400, lo...@dialup.ptt.ru (Jeffrey Lockshin)
wrote:

Just for my own amusement I thought up a definition of a very large
number.

You could not work it out very accurately, and it would be something
of a moveable feast.

It was titled the electron possibility index. or EPI (just because I
thought it sounded good).

The definition is.

Take the size of the known universe (or its' maximum projected size if
ever it is determined if it will collapse).

Now take the size of the smallest known subatomic particle.

Now take the time that it takes light to move the distance that is the
diameter of that particle.

Now take the age of the universe (or projected age bang->crunch if
known).

You can work out the calculation for yourself. (I can't be bothered
with only ascii).

Of course you could make it even bigger by calculating every single
possible speed in every possible direction.

Completely pointless of course, and you would need to do quite a lot
of research to calculate it, but it should be a nice big number.

Possibly bigger than a googolplex.

J.G.

[Ilias Kastanas]

In article <33F39D96...@ix.netcom.com>,
Randall Wagoner <lew...@ix.netcom.com> wrote:
>John Goodwin wrote:
>
>> On Thu, 14 Aug 1997 11:09:45 -0500, Randall Wagoner
>> <lew...@ix.netcom.com> wrote:

>This was meant to be fun.
>
>I'm going to put some limits on it though. What do you think?
>
>The rules are:
>
> Multiplication is the highest operation you can use.
> Myriad (M) = 10^4 (10,000) is the highest number that you can use in
>any variable. i.e. M*M...100 chars
> You can set up as many rules as you like, as long as the total of
>non-space characters don't exceed 100. I don't think it will come down
>to a few characters. I think it will depend on how well thought out the
>rules are.
> Functions are allowed in the rules as long as you define everything.
>
>So, if you gave these rules to someone that doesn't know any operations
>higher than multiplication, and no number higher then M, he should be
>able to determine what you are talking about.

Eh, but then the resulting number cannot be more than M...

>Just so you know what I mean, I'll give you a simple example.
>
> A(1, 1) = M
> A(1, j+1) = M*A(k, j)
> A(k+1, 1) = A(k, M)
> A(k+1, j+1) = M*A(k+1, j)
>
>What is the highest number that can be reached?
>
>Randall

Some suggestions:

M as abbreviation, as well as a bound, is rather artificial. Why
not allow any number formed by 0-9, in base ten as usual.

You don't want "the product of all other entries", "the number
specified in the attached 5 Meg zip file", "the number of permutations of
the elementary particles in the universe"... So, no English (or other
language); just math notation, as follows.

The only predefined functions are + and * ; no "!", exp or
whatever else. Other symbols allowed: , = ) ( > & | ~ ->
(the last ones being "and", "or", "not", "if ... then ...".)

Variables range over N = {0, 1, 2 ... }. Functions are written
as usual, f(x, y) [but y+z, or 4*x]. In a function definition, f(n+1) =
= f(n) +1, "n" is taken as quantified over N ('for all n in {0, 1, 2...}).
Capitalized variables are not quantified, e.g. Y = 4321.

"Terms" are: numbers, variables, functions applied to terms.

The result is specified via a reserved variable Z, by Z = <term>
where <term> evaluates (unambiguously) to a number. It is not required that
an f be defined for arguments not needed in this evaluation.

(This sounds too formal... but it makes clear what is allowed)

So if one wants a sum or product over something (e.g. a factorial),
one must give a recursive definition, g(n+1) = g(n)*(n+1) g(0) = 1.

Now, 100 characters (other than blanks or newlines) are likely to
create practical problems... I would say 50 is preferable.

Ilias

[Jeffrey Lockshin]

[Randall Wagoner]

Ilias Kastanas wrote:

Only varibles can not be greater then M. Not the number itself.

Randall

[Jim P. Ferry]

Man, what an annoying thread!

Hey, how about this: what's the smallest number that's "big"?

Or again: What is "The greatest integer that can be described in less than
100 characters, plus 1"?

You could say intelligent things about these, but they wouldn't involve
surmises like, "Ummmm, 10^100?"

What about: The Ackermann function, A(m,n), whoa, it gets so big that, whoa,
I mean you think its big and then it just blows your mind, like,
hey: you don't know big until you've seen the Ackermann function.

You want really big? There are integers which you could never specify in the
life of the universe, because they have too many (say A(10,10)) digits, and for
for these digits there is no sufficiently concise way to systematize their
description. Well, so what? A(11,11) is still bigger, and I just described it.
A really big integer is one for which you could never specify any integer greater
than it in the life of the universe.

There are numbers like that, and I'm thinking of one. For the sake of brevity,
I won't tell you about it. But not only can I think of it, I can imagine it
physically. Because I say so. Hah!

-Jim Ferry

[Randall Wagoner]

John Goodwin wrote:

> On 15 Aug 1997 02:40:47 GMT, isr...@math.ubc.ca (Robert Israel) wrote:
>
> >In article <33F39D96...@ix.netcom.com>,
> >Randall Wagoner <lew...@ix.netcom.com> wrote:
> >>John Goodwin wrote:
> >>
> >>> On Thu, 14 Aug 1997 11:09:45 -0500, Randall Wagoner
> >>> <lew...@ix.netcom.com> wrote:
> >>> >

> >>> >What is the highest natural number you can describe in 100
> characters
> >>> or
> >>> >less?
> >>> >
> >How about:
> >
> >(the highest number Randall can describe)+1
>

> Yes, you'd need to rule out the answer:
>
> The number described by the runer up + 1
>
> J.G.

Only variables can not be greater then M. Not the number itself.

Randall

[Robert Israel]

In article <so8l1d...@forum.swarthmore.edu>, lo...@dialup.ptt.ru (Jeffrey Lockshin) writes:
|> It is my opinion that the respondents didn't understand what Randall
|> was asking. You can always imagine a number larger than any one you
|> choose - but that's not the point. The point is is that huge numbers
|> can be entirely ephemeral - a product of the imagination. Very few
|> people can truly imagine the sheer magnitude of a trillion - it's very
|> easy to see that you would need more than 31.5 years to count to a
|> trillion if you counted one number ever second. Yet this is the
|> magnitude of the national debt.

What do you mean by ephemeral? This is certainly very real. How well you
can visualise it is another matter. Counting it out one by one is rather
pointless. Well, imagine taking a building 10 metres by 10 metres by 10
metres, and filling it up with grains of sand that are cubes of side 1 mm.
That's a trillion grains of sand.

|> Take the number 10^100 - a number larger than the number of sub-atomic
|> particles in the visible universe. This number is _impossible_ to
|> imagine.

Well, maybe you can't think of it as a count of physical objects. You could
say it's the number of possible strings of 100 base-10 digits. Or slightly
less than the number of permutations of 70 distinct objects.

|> What is the largest finite number you can think of that has a real,
|> physical meaning, and is not ephemeral, a product of the imagination?

Depends on what you mean by a "real, physical meaning".

[Arne Dehli Halvorsen]

How about
googolplex!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!! (stop at 100 chars)

each ! is a factorial sign, so the result should be largish.

Arne

Dilip Andrade <dand...@nortel.ca> wrote in article
<33F366...@nortel.ca>...

> Randall Wagoner wrote:
>
> > Ok, how about this?
> >

> > What is the highest natural number you can describe in 100 characters
>

[John Goodwin]

On 15 Aug 1997 02:40:47 GMT, isr...@math.ubc.ca (Robert Israel) wrote:

>In article <33F39D96...@ix.netcom.com>,
>Randall Wagoner <lew...@ix.netcom.com> wrote:
>>John Goodwin wrote:
>>
>>> On Thu, 14 Aug 1997 11:09:45 -0500, Randall Wagoner
>>> <lew...@ix.netcom.com> wrote:
>>> >

>>> >What is the highest natural number you can describe in 100 characters

[Jeffrey Lockshin]

It is my opinion that the respondents didn't understand what Randall
was asking. You can always imagine a number larger than any one you
choose - but that's not the point. The point is is that huge numbers
can be entirely ephemeral - a product of the imagination. Very few
people can truly imagine the sheer magnitude of a trillion - it's very
easy to see that you would need more than 31.5 years to count to a
trillion if you counted one number ever second. Yet this is the
magnitude of the national debt.

Take the number 10^100 - a number larger than the number of sub-atomic

particles in the visible universe. This number is _impossible_ to

imagine. Sure, you can say that 10^200 is larger, and that 10^400 is
even larger than that (like one of the respondents said - you can
always take the square). But what Randall was asking should be
rephrased like this:

What is the largest finite number you can think of that has a real,

[Anagrams of the Word 'A']

>>What is the largest natural number you can think of?
>

>8

Ooh! What's "8"? That one's new to me.

----------------
| "Here we are now, entertain us!"
| - Nirvana
----------------
Jeff "TechMaster" Pinyan | http://users.bergen.org/~jefpin
I do: HTML!! CGI!! Perl!! JavaScript!! jef...@bergen.org
Got a JavaScript/CGI/Perl question or problem? Let me know!

webXS - the new eZine for WebProgrammers! TechM...@bergen.org
Visit us @ http://users.bergen.org/~jefpin/webXS

[Randall Wagoner]

Randall Wagoner wrote:

> What is the largest natural number you can think of?
>

> Randall

[Jeffrey Lockshin]

Robert Israel wrote:
> What do you mean by ephemeral?

Probably this was an unfortunate choice of words. I meant
"intangible".

>Well, maybe you can't think of it as a count of physical objects.
>You could
>say it's the number of possible strings of 100 base-10 digits. Or
>slightly
>less than the number of permutations of 70 distinct objects.

The first doesn't help you "feel" the weight of these numbers. Again,
it's entirely intellectual - imaginary.

As for the second - give me an example of when you would need this
kind of huge permutation. I know of only one (which I give below), and
as you will see these numbers are still not used.

My example is the calculation of entropy. Now, before everyone jumps
on me like they did in another thread, I want to state write now that
this is NOT, repeat, NOT the definition of entropy. This is merely one
of the ways to theoretically compute it. I am talking about Boltzman's
formula

S=-k*ln(W),

where k is Boltzman's constant and W is the state probability. This
formula is actually derived from the formula

S=S_0 + k*ln(Wth),

where S_0 is some constant (not important here) and Wth is the
thermodynamic probability.

This last is a not-so-good term, since for a system of N molecules, of
which N_i molecules have an energy of u_i, it equals
N!
Wth = ------------------ >> 1.
N_1!*N_2!*...*N_j!

Take 1 mole of something: you have approximately 6*10^23 molecules
already. You can see that Wth is generally _way_ above 10^100. In
fact, for a kilogram or so of gas 10^100 is to Wth as my weight is to
the sun's (this is not a precise calculation, just a simile. However,
it is not far off.) But, as you can see, the sheer magnitude is
reduced to practically nothing, because no one ever needs Wth - they
need ln(Wth). As you can see, the logarithmic function nicely takes
care of these numbers. And no one needs the numbers themselves. So
this permutation of 70 you were talking about is 1) small potatoes and
2) not applicable to any real scientific necessity (as far as I can
see).

>Depends on what you mean by a "real, physical meaning".

Ok. I would say that the largest number that has a real, physical
meaning is the number of elementary particles in the universe. It
seems to me that there can't possibly be any greater number of
anything. Also, as I've said, their possible permutations wouldn't be
counted - all's you need is ln(permutations). This number would be
smaller than the magnitude of the national debt.

You can go the opposite way: what's the smallest number you can think
of? Well, here you hit the fact that you can't keep cutting a stick
into half indefinitely - even if you're doing it mentally. Thing is,
matter is discrete, not continuous. As I recall (this is also not an
exact figure) the smallest unit of length is 10^(-34). I may be wrong,
but it's around there somewhere. Any smaller number, at least for
length, is ridiculous in the light of modern science.

This kind of interpretation is what I think Randall had in mind.

Jeff

[Randall Wagoner]

Ilias Kastanas wrote:

> In article <33F4BCA2...@ix.netcom.com>,
> Randall Wagoner <lew...@ix.netcom.com> wrote:

> >Ilias Kastanas wrote:
> >
> >> In article <33F39D96...@ix.netcom.com>,
> >> Randall Wagoner <lew...@ix.netcom.com> wrote:
> >> >John Goodwin wrote:
> >> >
> >> >> On Thu, 14 Aug 1997 11:09:45 -0500, Randall Wagoner
> >> >> <lew...@ix.netcom.com> wrote:

> ...

> >> > Multiplication is the highest operation you can use.
> >> > Myriad (M) = 10^4 (10,000) is the highest number that you can
> use in
> >> >any variable. i.e. M*M...100 chars

> ...

> >> >What is the highest number that can be reached?
> >> >
> >> >Randall
>
> >> Some suggestions:
> >>
> >> M as abbreviation, as well as a bound, is rather
> artificial.
> >> Why not allow any number formed by 0-9, in base ten as usual.
> >>
> >> You don't want "the product of all other entries", "the
> number

> >> specified in the attached 5 Meg zip file", "the number of
> permutations of

> >Only varibles can not be greater then M. Not the number itself.
> >
> >Randall
>
> I was joking about the number and the someone who understands
> only
> < M ... because _I_ don't understand the purpose of limiting the
> explicit
> numbers to < 10000. Or the abbreviation; if one wants 10000, it
> costs five
> characters; let him decide. The huge numbers the contest will
> generate will
> not be due to "lots of digits" anyway! I think it's simpler to have
> no
> abbrev., and permit any base-10 digit string.
>
> And 100 chars asks for trouble. Even 50 may be too much.
>
> Ilias

Ok then, no M. I was just trying to make it simple.

Randall

[Randall Wagoner]

Benjamin J. Tilly wrote:

> In article <33F646B1...@ix.netcom.com>
> Randall Wagoner <lew...@ix.netcom.com> writes:
>
> > Benjamin J. Tilly wrote:
> >
> > > In article <33F32DC9...@ix.netcom.com>

> > > Randall Wagoner <lew...@ix.netcom.com> writes:
> > >
> > > > Randall Wagoner wrote:
> > > >

> > > > > What is the largest natural number you can think of?
> > > > >
> > > > > Randall
> > > >
> > > > Ok, how about this?
> > > >

> > > > What is the highest natural number you can describe in 100
> > > characters or
> > > > less?
> > >

> > > The answer to this depends on whom I am describing it to. But if
> you
> > > know what a Turing Machine is, then in plain English the following
>
> > > describes one heck of a big number...
> > >
> > > b(n) = the longest that it can take to halt from a blank tape on
> an
> > > n-state Turing machine. b(b(99))
> > >
> > > (OK, so BB(n) is the usual notation for b(n).) Now just how big is
>
> > > this
> > > number? Well let me indicate it as follows. Imagine a computer
> > > program,
> > > any computer program, which you could write on any computer in the
>
> > > world that you can think of. Imagine that this program was
> designed to
> > >
> > > let you take some description in a particular format and generate
> a
> > > number in base 10 from it. Imagine running it on a computer with
> > > infinite capacity, and feeding it an input file where the numbers
> were
> > >
> > > the size of numbers on a microfiche and the input file filled the
> > > observable universe. Feed it the input file that you think will
> get
> > > this computer to produce the largest possible number.
> > >
> > > b(b(99)) is larger than that.
> > >
> > > In fact it is a LOT larger than that.
> > >
> > > So do I win some sort of pissing contest with this? :-P
> > >
> > > Ben Tilly
> > >
> > > PS This number has some interesting properties. One of them is
> that no
> > >
> > > consistent axiom system stateable within the space of the known
> > > universe is capable of proving that any particular number written
> in
> > > base 10 is an upper bound for b(b(99)). No, this is not a
> > > contradictory
> > > statement, and understanding what it means emphasizes why it is
> that
> > > some people lean towards constructivism. :-)
> >
> > I'm confused. Are you discribing a number or are you discribing a
> > machine that will describe a number?
>
> I am describing an explicit (though not computable from the given
> definition) number. I am giving you a sense of how big that number is
> by comparing it with any number specified by a computation which might
>
> be explicitly specified (although not acutally done) in our universe,
> and noting that the one that I gave is larger.
>
> Ben Tilly

Ok. Now describe a number using 100 characters or less. But you have to
give the full description. You can't compare it to something else. In
other words, you can't say, "The number of seconds it would take to walk
to the end of the universe.". Nobody would know by that description what
the number is.

I gave a simple example of a description of a number. I also didn't use
any operation higher then multiplication. Here it is if you missed that
post.

M = 10000

A(1, 1) = M
A(1, j+1) = M*A(k, j)
A(k+1, 1) = A(k, M)
A(k+1, j+1) = M*A(k+1, j)

What is the highest number that can be reached from these rules.

Like I said, this is just an example.

Randall

[Randall Wagoner]

Randall Wagoner wrote:

> What is the largest natural number you can think of?
>
> Randall

You all are missing my point!

Any idiot can take someone else's number and add one to it. That takes
no thought. I want you to come up with your own numbers.

I suggest you think about what I am really asking and come up with a
number that you can be proud of.

Also, I asked what was the highest number that my rules describe, but
NOBODY gave me an answer. I assume that my simple example is the largest
so far.

Randall

[John Goodwin]

On 15 Aug 1997 22:07:36 GMT, j...@hydra.cfm.brown.edu (Jim P. Ferry)
wrote:

>Man, what an annoying thread!

Then why are you reading it, and prolonging it?

snip

>
>What about: The Ackermann function, A(m,n), whoa, it gets so big that, whoa,
> I mean you think its big and then it just blows your mind, like,
> hey: you don't know big until you've seen the Ackermann function.
>
>You want really big? There are integers which you could never specify in the
>life of the universe, because they have too many (say A(10,10)) digits, and for
>for these digits there is no sufficiently concise way to systematize their
>description. Well, so what? A(11,11) is still bigger, and I just described it.

snip

But we were thinking of using some intelligence, so rather than just
going to from A(10,10) to A(11,11), try

A(A(9,9),A(9,9))

By the way, if there are any children reading this, DONT TRY THIS AT
HOME.

There is a possibility that you could get a floating point overflow
error!

(I can't remember Ackerman's function, but I think it does something
like this itself. I programmed it once (in Algol!), as a demonstartion
of how recursion can use up memory at an alarming rate.)

J.G.

[Ilias Kastanas]

[Benjamin J. Tilly]

In article <33F32DC9...@ix.netcom.com>
Randall Wagoner <lew...@ix.netcom.com> writes:

> Randall Wagoner wrote:
>
> > What is the largest natural number you can think of?
> >
> > Randall
>

[Randall Wagoner]

Benjamin J. Tilly wrote:

I'm confused. Are you discribing a number or are you discribing a

machine that will describe a number?

Randall

[KRamsay]

In article <nuo3ck...@forum.swarthmore.edu>, lo...@dialup.ptt.ru
(Jeffrey Lockshin) writes:

|Very few
|people can truly imagine the sheer magnitude of a trillion - it's very
|easy to see that you would need more than 31.5 years to count to a
|trillion if you counted one number ever second.

Thousand years, you mean.

|Yet this is the

|magnitude of the national debt.

Think of things like the national debt on a per-capita basis.

Keith Ramsay There is nothing on this earth, and little beyond it,
kra...@aol.com that nobody ever denounces. -- Matt McIrvin

[Benjamin J. Tilly]

In article <33F646B1...@ix.netcom.com>
Randall Wagoner <lew...@ix.netcom.com> writes:

I am describing an explicit (though not computable from the given

[Benjamin J. Tilly]

In article <33F671E5...@ix.netcom.com>
Randall Wagoner <lew...@ix.netcom.com> writes:

> Ok. Now describe a number using 100 characters or less. But you have to
> give the full description. You can't compare it to something else. In
> other words, you can't say, "The number of seconds it would take to walk
> to the end of the universe.". Nobody would know by that description what
> the number is.
>

In case you missed it, here was my description.

b(n) = the longest that it can take to halt from a
blank tape on an n-state Turing machine. b(b(99))

That takes 100 characters, and if you know what a Turing machine is
then it is well defined. I counted spaces as key-strokes and it is
written in simple English.

If you don't know what a Turing machine is, then the description of one
would take longer than 100 characters. However it can be done in a
paragraph. And it really does define a unique number. (We cannot
compute that number though!)

> I gave a simple example of a description of a number. I also didn't use
> any operation higher then multiplication. Here it is if you missed that
> post.
>

My description needed no operations at all. :-)

> M = 10000
>
> A(1, 1) = M
> A(1, j+1) = M*A(k, j)
> A(k+1, 1) = A(k, M)
> A(k+1, j+1) = M*A(k+1, j)
>
> What is the highest number that can be reached from these rules.
>
> Like I said, this is just an example.

If you disallow the use of the word "Turing machine", or any term which
depends on one, then I can virtually guarentee that it is nowhere near
what I called b(b(99)). (OK, so there are some other things out there
which are also large for similar reasons. However if you do not accept
computers, then statements in terms of the minimal number of steps
needed to prove the statement of length at most n whose minimal proof
is of maximal size are unlikely to fit in your definition...)

Ben Tilly

[Ariel Scolnicov]

J...@opticon.demon.co.uk (John Goodwin) writes:

> On 15 Aug 1997 22:07:36 GMT, j...@hydra.cfm.brown.edu (Jim P. Ferry)
> wrote:
>
> >Man, what an annoying thread!
>
> Then why are you reading it, and prolonging it?
>
> snip
> >
> >What about: The Ackermann function, A(m,n), whoa, it gets so big that, whoa,
> > I mean you think its big and then it just blows your mind, like,
> > hey: you don't know big until you've seen the Ackermann function.
> >
> >You want really big? There are integers which you could never specify in the
> >life of the universe, because they have too many (say A(10,10)) digits, and for
> >for these digits there is no sufficiently concise way to systematize their
> >description. Well, so what? A(11,11) is still bigger, and I just described it.
> snip
>
> But we were thinking of using some intelligence, so rather than just
> going to from A(10,10) to A(11,11), try
>
> A(A(9,9),A(9,9))
>
> By the way, if there are any children reading this, DONT TRY THIS AT
> HOME.
>
> There is a possibility that you could get a floating point overflow
> error!

Nah, no way. You're defining the function entirely in terms of
recursion and adding 1. If you're working in floating point, you'll
never overflow when adding one, you'll just lose the one.

The message to kids around the world is this: DON'T TRY THIS WITH
FLOATING POINT ARITHMETIC! You'll get stuck in an infinite loop!

The reason not to try this at home is that...

CALCULATING NUMBERS BEYOND A(5,5) WILL CAUSE STACK OVERFLOW AND VOID
ANY WARRANTY.

[AJJMAT]

The Penquin Book of Interestesting & Curious Numbers
lists numbers with a "real, physical meaning" in ascending order.

It is a very good book and worth buying at around £7.00.

Andy Jones

[Randall Wagoner]

AJJMAT wrote:

Question: What is the first number that is NOT interesting or curious?

Randall

[Larry Taylor]

Randall Wagoner wrote:
> <snip>
> My simple example is Ackerman's function. I'll define it again.

>
> M = 10000
>
> A(1, 1) = M
> A(1, j+1) = M*A(k, j)

?? You mean A(k,j+1) on the left OR A(1,j) on the right

> A(k+1, 1) = A(k, M)
> A(k+1, j+1) = M*A(k+1, j)
>

> What would be the size of this number?
>
> I have a much greater function, but I'm waiting for someone to come up
> with the number from this one.
>
> Randall

The above is NOT Ackermann's function and certainly doesn't grow as fast
as Ackermann's function. The Ackermann property depends essentially on
using the function value as an input to the function in the recursive
definiton as in the 3rd equation below. One variation:

A(1,j) = 2^j j >=2
A(i,1) = A(i-1,2) i >=2
A(i,j) = A(i-1,A(i,j-1)) i,j >= 2

So,
A(1,j) = 2^j
A(2,j) = 2^... ^2 (the exponent has j terms in it
A(3,j) = 2^ ...^2 (the exponent has the same number of terms as the
value of A(3,j-1) !!

Since A(3,1) = 2^2^2, A(3,2) = 2^...^2 (16 terms in exponent)
But A(3,3) has A(3,2) terms in the exponent....

and A(4,j)....???

Larry Taylor

[Randall Wagoner]

Larry Taylor wrote:

Actually Archimedes defined what I wrote. I was saving the Ackermann
Generalized Exponential function until someone came close to that one.
It's a double iteration process.

G(1, k, j) = j*k
G(n+1, 1, j) = j
G(n+1, k+1, j) = G(n, G(n+1, k, j), j)

G(2, k, j) = j^k
G(3, k, j) = j^(j^(j^(j^....k times

Lets just take M = 10000.
G(M, M, M) = ???

Also, you can not use operations higher than multiplication. You are
using "^".

Randall

[John Goodwin]

On 17 Aug 1997 15:57:19 +0300, Ariel Scolnicov
<ari...@mangal.cs.huji.ac.il> wrote:

You missed this, somewhat pertinant part from my post!

:(I can't remember Ackerman's function, but I think it does something

:like this itself. I programmed it once (in Algol!), as a demonstartion
:of how recursion can use up memory at an alarming rate.)

Are you saying that there is no variable in Ackerman's function that
gets very big.

That is what I remembered from many years ago, but since the person to
whom I was replying seemed to think it would generate a large number,
I assumed I must have been mistaken.

I take it A(5,5) is just an example. Would the actual parameters not
depend on the amount of memory available for the stack?

Can someone PLEASE post the definition of Ackerman's function.

J.G.

[Randall Wagoner]

John Goodwin wrote:

My simple example is Ackerman's function. I'll define it again.

M = 10000

A(1, 1) = M
A(1, j+1) = M*A(k, j)

[Randall Wagoner]

Benjamin J. Tilly wrote:

> > M = 10000
> >
> > A(1, 1) = M
> > A(1, j+1) = M*A(k, j)
> > A(k+1, 1) = A(k, M)
> > A(k+1, j+1) = M*A(k+1, j)
> >

> > What is the highest number that can be reached from these rules.
> >
> > Like I said, this is just an example.
>
> If you disallow the use of the word "Turing machine", or any term
> which
> depends on one, then I can virtually guarentee that it is nowhere near
>
> what I called b(b(99)). (OK, so there are some other things out there
> which are also large for similar reasons. However if you do not accept
>
> computers, then statements in terms of the minimal number of steps
> needed to prove the statement of length at most n whose minimal proof
> is of maximal size are unlikely to fit in your definition...)
>
> Ben Tilly

You have to define Turing Machine. You can not assume everyone knows
what that is.

I could say, the number A(M, M), where A() is Ackermann's exponential
function. If you don't know what that is, you have know idea the size of
that number.

Randall

[Bill Taylor]

Randall Wagoner <lew...@ix.netcom.com> writes:

|> Question: What is the first number that is NOT interesting or curious?

-------------------------------------------------------------------------------
The smallest uninteresting natural number is *not* thereby interesting;
...but it *is* very meta-interesting !
-------------------------------------------------------------------------------

-------------------------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
Trollin' Trollin' Trollin'
Keep that flamebait rollin'....
-------------------------------------------------------------------------------

[Richard White]

Randall Wagoner <lew...@ix.netcom.com> writes:

> Question: What is the first number that is NOT interesting or curious?

Due to the phenomenal success of my first mathematics humor book "The
Hithchiker's Guide to Calculus 2", I have been asked to do a spoof of
number theory. Maybe I will. Or maybe I'll die of inductive poisoning
first. Who knows?

Anyway, this is one of the questions that I have considered. By "first",
I assume you mean "least". Certainly the uninteresting numbers are
well-ordered, no?

At the risk of boring everyone, I'd like to take a shot at derailing the
standard reasoning, which goes like this. Since the first number is
interesting by virtue of its being the first number that's uninteresting,
that would make the second number that's uninteresting the first number
that's uninteresting. Lather, rinse, repeat.

However - and I haven't seen anyone make this objection (but then I
haven't looked either) - the second number's claim to fame is exactly the
same as the first number's claim. Ergo, the second uninteresting number
truly is uninteresting. Maybe everyone else has a different favorite
uninteresting number.

Happy Times Upon You, Richard.

[Gene W. Smith]

In article <edgar-ya02408000...@news.math.ohio-state.edu>,
G. A. Edgar <ed...@math.ohio-state.edu> wrote:

>One of the games analyzed in Berlekamp, Conway, & Guy, WINNING WAYS
>is called "My father makes more than your father."

My brother and I used to play a version of this game. The rules were
each side must define a positive integer which did not reference the
other side's number, and write it down on a regulation piece of
paper. It started to go down hill when I introduced the Ackermann
function, and really went into the tank when I introduced logic. It
became too difficult to prove who had won the game.

[Randall Wagoner]

Ilias Kastanas wrote:

> In article <33F671E5...@ix.netcom.com>,

> Randall Wagoner <lew...@ix.netcom.com> wrote:
> >Benjamin J. Tilly wrote:
> >

> >> In article <33F646B1...@ix.netcom.com>
> >> Randall Wagoner <lew...@ix.netcom.com> writes:
> >>

> >> > Benjamin J. Tilly wrote:
> >> >
> >> > > In article <33F32DC9...@ix.netcom.com>
> >> > > Randall Wagoner <lew...@ix.netcom.com> writes:
> >> > >
> >> > > > Randall Wagoner wrote:
> >> > > >
> >> > > > > What is the largest natural number you can think of?
> >> > > > >
> >> > > > > Randall
> >> > > >
> >> > > > Ok, how about this?
> >> > > >
> >> > > > What is the highest natural number you can describe in 100
> >> > > characters or
> >> > > > less?
> >> > >
> >> > > The answer to this depends on whom I am describing it to. But

> if
> >> you

> >> > > know what a Turing Machine is, then in plain English the
> following
> >>
> >> > > describes one heck of a big number...
> >> > >

> >> > > b(n) = the longest that it can take to halt from a blank tape
> on
> >> an
> >> > > n-state Turing machine. b(b(99))
> >> > >

> >Ok. Now describe a number using 100 characters or less. But you have
> to
> >give the full description. You can't compare it to something else. In
>
> >other words, you can't say, "The number of seconds it would take to
> walk
> >to the end of the universe.". Nobody would know by that description
> what
> >the number is.
> >

> >I gave a simple example of a description of a number. I also didn't
> use
> >any operation higher then multiplication. Here it is if you missed
> that
> >post.
> >

> > M = 10000
> >
> > A(1, 1) = M
> > A(1, j+1) = M*A(k, j)
> > A(k+1, 1) = A(k, M)
> > A(k+1, j+1) = M*A(k+1, j)
> >
> >What is the highest number that can be reached from these rules.
> >
> >Like I said, this is just an example.
>

> So maybe now you appreciate why I wrote up "rules of the game"
> the
> way I did! I excluded English; in English you can refer to "Turing
> Machi-
> nes"... or whatever. I gave an explicit framework of what is allowed
> --
> which includes your example.
>
> Also, your example defines A. It doesn't make much sense to
> limit
> "explicit" numbers to being <= M... Why "just" A(M, M)... when it is
> always
> possible to have A(A(M, M), A(M, M)), and so on? Which is why I did
> not
> impose any such bound.
>
> Ilias

If someone thinks A(A(M, M), A(M, M)) is greater then coming up with a
new set of rules, then let them. I'm not limiting explicit numbers per
se. I'm just truing to make it less complicated by only using M to be
the only *constant*. I maybe should have said that instead of
*variable*.

My only concern is everything has to be defined, and in under 100
characters.

Randall

[Patrick E. White]

Benjamin J. Tilly wrote:

> (OK, so BB(n) is the usual notation for b(n).) Now just how big is this
> number? Well let me indicate it as follows. Imagine a computer program,
> any computer program, which you could write on any computer in the
> world that you can think of. Imagine that this program was designed to
> let you take some description in a particular format and generate a
> number in base 10 from it. Imagine running it on a computer with
> infinite capacity, and feeding it an input file where the numbers were
> the size of numbers on a microfiche and the input file filled the
> observable universe. Feed it the input file that you think will get
> this computer to produce the largest possible number.

Don't you need to worry about whether this computer can be described in
less than b(99) states? ... Well, I guess about any computer can...

> PS This number has some interesting properties. One of them is that no
> consistent axiom system stateable within the space of the known
> universe is capable of proving that any particular number written in
> base 10 is an upper bound for b(b(99)). No, this is not a contradictory
> statement, and understanding what it means emphasizes why it is that
> some people lean towards constructivism. :-)

Now this is interesting. What about b(b(2))? It probably doesn't share
this "unprovable" property? What can you say about b(n) in general with
respect to this property? Anything stronger than "except for finitely
few n, ..."

[Ben Soares]

Randall Wagoner wrote:
>
> John Goodwin wrote:
>
> > On Thu, 14 Aug 1997 11:09:45 -0500, Randall Wagoner
> > <lew...@ix.netcom.com> wrote:
> >

> > >Randall Wagoner wrote:
> > >
> > >> What is the largest natural number you can think of?
> > >>
> > >> Randall

The largest natural number I can think of using less than 100 characters
is:

"The largest natural number defined by less than 100 characters"

I only used 62 characters.

Ben
*(:-)

[John Goodwin]

On Mon, 18 Aug 1997 17:59:04 +0100, Ben Soares <b...@dcs.st-and.ac.uk>
wrote:

This is illogical.

"The largest natural number defined by less than 100 characters ^ 10"

67 characters, and much bigger (except that it doesn't really make
sense).

Comments Please.

J.G.

[Randall Wagoner]

John Goodwin wrote:

Doesn't look like anyone wants to play

Randall

[John Goodwin]

X-Humour: On

On Sun, 17 Aug 1997 12:20:06 -0500, Randall Wagoner
<lew...@ix.netcom.com> wrote:

>AJJMAT wrote:
>
>> The Penquin Book of Interestesting & Curious Numbers
>> lists numbers with a "real, physical meaning" in ascending order.
>>
>> It is a very good book and worth buying at around £7.00.
>>
>> Andy Jones
>
>

>Question: What is the first number that is NOT interesting or curious?
>

>Randall
>

31,427,638

(Although some have suggested that it is interesting because it is the
first number that is not interesting).

J.G.

[Ilias Kastanas]

In article <33F671E5...@ix.netcom.com>,
Randall Wagoner <lew...@ix.netcom.com> wrote:
>Benjamin J. Tilly wrote:
>
>> In article <33F646B1...@ix.netcom.com>
>> Randall Wagoner <lew...@ix.netcom.com> writes:
>>
>> > Benjamin J. Tilly wrote:
>> >
>> > > In article <33F32DC9...@ix.netcom.com>
>> > > Randall Wagoner <lew...@ix.netcom.com> writes:
>> > >

>> > > > Randall Wagoner wrote:
>> > > >
>> > > > > What is the largest natural number you can think of?
>> > > > >
>> > > > > Randall
>> > > >

>> > > > Ok, how about this?
>> > > >
>> > > > What is the highest natural number you can describe in 100
>> > > characters or
>> > > > less?
>> > >
>> > > The answer to this depends on whom I am describing it to. But if
>> you
>> > > know what a Turing Machine is, then in plain English the following
>>
>> > > describes one heck of a big number...
>> > >
>> > > b(n) = the longest that it can take to halt from a blank tape on
>> an
>> > > n-state Turing machine. b(b(99))
>> > >

>> > > (OK, so BB(n) is the usual notation for b(n).) Now just how big is
>>
>> > > this
>> > > number? Well let me indicate it as follows. Imagine a computer
>> > > program,
>> > > any computer program, which you could write on any computer in the
>>
>> > > world that you can think of. Imagine that this program was
>> designed to
>> > >
>> > > let you take some description in a particular format and generate
>> a
>> > > number in base 10 from it. Imagine running it on a computer with
>> > > infinite capacity, and feeding it an input file where the numbers
>> were
>> > >
>> > > the size of numbers on a microfiche and the input file filled the
>> > > observable universe. Feed it the input file that you think will
>> get
>> > > this computer to produce the largest possible number.
>> > >

>> > > b(b(99)) is larger than that.
>> > >
>> > > In fact it is a LOT larger than that.
>> > >
>> > > So do I win some sort of pissing contest with this? :-P
>> > >
>> > > Ben Tilly
>> > >

>> > > PS This number has some interesting properties. One of them is
>> that no
>> > >
>> > > consistent axiom system stateable within the space of the known
>> > > universe is capable of proving that any particular number written
>> in
>> > > base 10 is an upper bound for b(b(99)). No, this is not a
>> > > contradictory
>> > > statement, and understanding what it means emphasizes why it is
>> that
>> > > some people lean towards constructivism. :-)
>> >

[Bruce Cox]

>>>>> "JG" == John Goodwin <J...@opticon.demon.co.uk> writes:
JG> Can someone PLEASE post the definition of Ackerman's function.

ack(0, j) = j + 1 (j >= 0)
ack(i, 0) = ack(i - 1, 1) (i > 0)
ack(i, j) = ack(i - 1, ack(i, j - 1)) (i > 0, j > 0)

Cheers,
bruce

--
Bruce Cox br...@maths.usyd.edu.au
School of Mathematics and Statistics F07 +61 2 9351 3814
University of Sydney 2006
AUSTRALIA

[Jim Sarbeck]

: In article <33f8c610....@news.demon.co.uk>, J...@opticon.demon.co.uk

: (John Goodwin) wrote:
:
: : On Mon, 18 Aug 1997 17:59:04 +0100, Ben Soares <b...@dcs.st-and.ac.uk>
: : wrote:
: :
: : >Randall Wagoner wrote:
: : >>
: : >> John Goodwin wrote:
: : >>

: : >> > On Thu, 14 Aug 1997 11:09:45 -0500, Randall Wagoner
: : >> > <lew...@ix.netcom.com> wrote:
: : >> >

: : >> > >Randall Wagoner wrote:
: : >> > >
: : >> > >> What is the largest natural number you can think of?
: : >> > >>
: : >> > >> Randall

: : >
: : >The largest natural number I can think of using less than 100 characters

: : >is:
: : >
: : > "The largest natural number defined by less than 100 characters"
: : >
: : >I only used 62 characters.
: : >
: :
: : This is illogical.
: :
: : "The largest natural number defined by less than 100 characters ^ 10"
: :
: : 67 characters, and much bigger (except that it doesn't really make
: : sense).
: :
: : Comments Please.

:
: Well, can you 'Think of it'? How about
:
:
9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9
:
: Does that make sense? It's Really Big!
:
: Perhaps
:
:
9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999!
:
: is worth thinking about for a while.
:
: or
:
: Let g=googolplex:
:
g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g
:
: I wonder which is largest.

Or even

9!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Wowie!

Regards,
Jim Sarbeck
*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*

[Randall Wagoner]

John Goodwin wrote:

> On Thu, 14 Aug 1997 01:19:36 -0500, Randall Wagoner
> <lew...@ix.netcom.com> wrote:
>
> >What is the largest natural number you can think of?
> >

> def F( f, m, n )
> n=1: m f m
> n!=1:F(f, m, n-1 )
> end
>
> h=F( ^, 9, 9 )
> b=F( ^, h, h )
> v=F( ^, b, b )
> d=F( ^, v, v )
>
> Explanation
>
> The function F takes three parameters, An arithmetic operator f, and
> two numbers m and n.
>
> It is defined such that if n is one, the result is n f n, otherwise
> F( f, m, n-1 )
>
> eg F( ^, 2, 2)
> is ( 2^2 ) ^ (2^2)
>
> Then we define a hoogleplex (h) as F( ^, 9, 9 )
>
> and use it again in our function F to define a boogleplex (b), a
> voogleplex (v), and finally, we use out voogleplex in F to get us a
> doogleplex.
>
> I suspect that is the largest number yet submitted to this thread,
> and, ignoring spaces and newlines, it takes up 74 characters.
>
> You could, of course get in to more Xplexes by doing another 2
> "passes"
>
> There are, of course ways to get it even bigger, by defining the most
> concise notation for the definition, and nesting the F functions.
>
> The only problem is that the numbers involved are so awesome that it
> would be difficult to compare them with similarly awesome numbers
> generated by a different process.
>
> I hope that this is what Randall had in mind when he originally posed
> the question.
>
> J.G.

The problem with yours is, it takes an operator i. e. "^". Didn't I say
none higher than multiplication? Mine takes a number to be the operator.
So I can get operators as high as my constant, M.

By the way, it can be proven which set of rules grows faster, but I'm
not sure how to do it. Does anyone know how?

Randall

[Randall Wagoner]

John Goodwin wrote:

Here is a better explination of Ackermann's Generalized Exponential
Function.

It's a double iteration process.

G(1, k, j) = j*k
G(n+1, 1, j) = j
G(n+1, k+1, j) = G(n, G(n+1, k, j), j)

G(2, k, j) = j^k
G(3, k, j) = j^(j^(j^(j^....k times

To explain G(4, k, j), I have to show what "Tetration" means. It's
the next higher operation after exponentiation.

2 tetra 2 = 2 ^ 2
2 tetra 3 = 2 ^ (2 ^ 2) = 2 ^ 4
2 tetra 4 = 2 ^ (2 ^ (2 ^ 2)) = 2 ^ (2 ^ 4) = 2 ^ 16
2 tetra 5 = 2 ^ (2 ^ (2 ^ (2 ^ 2))) = 2 ^ (2 ^ 16) = 2 ^ 64,536

3 tetra 3 = 3 ^ (3 ^ 3) = 3 ^ 27
3 tetra 4 = 3 ^ (3 ^ (3 ^ 3)) = 3 ^ (3 ^ 27)

10 tetra 3 = 10 ^ (10 ^ 10) = 10 ^ 10,000,000,000

So G(4, k, j) = j tetra k

G(5, k, j) = j tetrated k times
G(5, 2, 3) = 2 tetra (2 tetra 2) = 2 tetra 4 = 2 ^ 16
G(5, 3, 3) = 3 tetra (3 tetra 3) = 3 tetra (3 ^ 27)

G(5, 10, 3) = 10 tetra (10 tetra 10) = Very freaking large!

As you can see, G(n, k, j) as n grows is going to be Very very freaking
large!

Now, lets just take M = 10000.

G(M, M, M) = ???

Randall

[John Goodwin]

On Tue, 19 Aug 1997 10:10:30 GMT, J...@opticon.demon.co.uk (John
Goodwin) wrote:

Please ignore the last post I made in this thread. I had not yet
received the definition of Ackerman's function, which seems very
similar.

In any case, I cut and pasted an intermediate version, and it's not at
all what was intended, in fact,it's virtually gibberish!

As you can see, I am answering the original question, before the
stipulation about not using ^.

It is obvious that the way to generate a large number is by some sort
of advanced recursive function, but it is the compactness that is
important, and I posted something I was working on along the way.

This is the corrected version:

>On Thu, 14 Aug 1997 01:19:36 -0500, Randall Wagoner
><lew...@ix.netcom.com> wrote:
>
>What is the largest natural number you can think of?
>

def F( m, n )
n=1: m ^ m
n!=1:F( F( m, m-1 ), F( m, m-1))
end

h=F( 9, 9 )
b=F( h, h )
v=F( b, b )
d=F( v, v )

Explanation

The function F takes two parameters m and n.

It is defined such that if n is one, the result is m ^ m, otherwise
F(F( m, m-1 ), F( m, m-1))

eg F( 2,2 )

is ( 2^2 ) ^ (2^2)

which is 256

but F( 2, 3 )
is ( 256 ^ 256 ) ^ ( 256 ^ 256 )

Then we define a hoogleplex (h) as F( 9, 9 )

and use it again in our function F to define a boogleplex (b), a
voogleplex (v), and finally, we use out voogleplex in F to get us a
doogleplex.

I suspect that is the largest number yet submitted to this thread,

and, ignoring spaces and newlines, it takes up 76 characters.

You could, of course get in to more Xplexes by doing another 2

"passes", and start out with F( 999, 999 ).

[John Goodwin]

On Mon, 18 Aug 1997 07:06:10 -0400, Richard White <wh...@csee.usf.edu>
wrote:

>Randall Wagoner <lew...@ix.netcom.com> writes:
>
>> Question: What is the first number that is NOT interesting or curious?
>

>Due to the phenomenal success of my first mathematics humor book "The
>Hithchiker's Guide to Calculus 2", I have been asked to do a spoof of
>number theory. Maybe I will. Or maybe I'll die of inductive poisoning
>first. Who knows?
>
>Anyway, this is one of the questions that I have considered. By "first",
>I assume you mean "least". Certainly the uninteresting numbers are
>well-ordered, no?
>
>At the risk of boring everyone, I'd like to take a shot at derailing the
>standard reasoning, which goes like this. Since the first number is
>interesting by virtue of its being the first number that's uninteresting,
>that would make the second number that's uninteresting the first number
>that's uninteresting. Lather, rinse, repeat.
>
>However - and I haven't seen anyone make this objection (but then I
>haven't looked either) - the second number's claim to fame is exactly the
>same as the first number's claim. Ergo, the second uninteresting number
>truly is uninteresting. Maybe everyone else has a different favorite
>uninteresting number.

This argument (the one you are debunking), seems similar to a paradox
that I have never suceeded in nailing. It goes like this:

You sit alone in a room, with only a clock for company. I tell you
that I am going to come into the room every hour, exactly on the hour,
and kill you the twelth time I enter. You, however have a gun with one
bullet in it. You are a good shot, and if the bullet hits me in the
chest I'm history, and you can escape. However, I will be wearing
invisible body armour on each occasion I come into the room bar one.

I'm sure that by this point you will see the similarity (others may
not).

I have reworded the paradox from the way I originaly heard it to make
it a little more focused.

Any thoughts?

J.G.

[Danny Kaye]

In article <33F902A3...@ix.netcom.com>,
lew...@ix.netcom.com (Randall Wagoner) wrote:

> Doesn't look like anyone wants to play
>
> Randall
>

Why are you surprised? I thought this was sci.math not "my
numbers bigger than yours" :)

danny

email d...@maths.ntu.ac.uk
www http://euler.ntu.ac.uk/dk/dk.html

[John Goodwin]

On Tue, 19 Aug 1997 18:25:52 GMT, dann...@cix.compulink.co.uk
("Danny Kaye") wrote:

>In article <33F902A3...@ix.netcom.com>,
>lew...@ix.netcom.com (Randall Wagoner) wrote:
>
>> Doesn't look like anyone wants to play
>>
>> Randall
>>
>
>Why are you surprised? I thought this was sci.math not "my
>numbers bigger than yours" :)

There is, in fact, a lot of very interesting theory behind the most
compact definition of very large numbers.

Watch and learn!

J.G.

[Jim P. Ferry]

> What is the highest natural number you can describe in 100 characters or
> less?

100 characters or 1 character makes no difference: x

Of course I have to tell you what "x" means, which may take many, many
characters. To say 9!!!!!!!!...! or to nest Ackermann functions (or pre-nested
Ackermann functions), or even to write 9999...9 assumes one understands the
context, and to place no limits on the context makes the problem silly. Hence
x.

The best response I've seen in this thread is Ben Tilly's:

b(n) = the longest that it can take to halt from a
blank tape on an n-state Turing machine. b(b(99))

because it specifies a fairly well-defined context (the lore of mathematicians
and the comprehension of English), and doesn't attempt to cheat by making
definitions that don't count toward the character count, or by weaseling with
the character set (e.g., "my set of characters shall be the 2^16 ordered pairs
of ascii characters").

I'm surprised that people have bothered monkeying with nesting Ackermann
functions rather than "Tilly functions". E.g.:

b(n,k) = the longest that it can take to halt from a
blank tape on an b(n,k-1)-state Turing machine.
b(n,0) = n.

I also wonder why saying "Your number plus 1" is considered asinine, whereas
saying "Your system, nested once more" isn't. It can't be because the latter
gives numbers that are "much bigger", since surely readers of sci.math realize
that notion of relatively bigger is as relative as the notion of big. Just as
multiplying by a million ceases to significantly increase numbers the size of
10^(10^(10^10)), so passing from B(n,n) to B(B(n),B(n)) doesn't really matter
when B(n) is huge enough.

This discussion of trying to come up with large integers reminds me of the
discussion of trying to "Godelize" in "Godel,Escher,Bach". Or of trying to
express large ordinal numbers.

-Jim Ferry

[Randall Wagoner]

Jim P. Ferry wrote:

Ben Tilly's response is not a definition, because Turing machine has no
meaning. The Ackermann function does, because it is based on
multiplication. If you can define Turing machine in under 100
characters, then go for it.

So far, my generalized exponential function is by far the fastest
growing yet to be posted.

I should have asked, "Who can come up with the fastest growing
function?". Maybe we would have less misunderstandings.

Randall

[Randall Wagoner]

John Goodwin wrote:

I find it fascinating!

Randall

[Norman D. Megill]

In article <5td6ki$6...@cocoa.brown.edu>,
Jim P. Ferry <j...@hydra.cfm.brown.edu> wrote:
[pretty good post]

>
>This discussion of trying to come up with large integers reminds me of the
>discussion of trying to "Godelize" in "Godel,Escher,Bach". Or of trying to
>express large ordinal numbers.

Perhaps a parallel game could have set theorists trying to outdo each
other by naming ever larger cardinals.

Here's what I could find, in increasing order of size (someone check me).
For me the second on the list already hurts my brain.

aleph-0
aleph-1
aleph-omega
inaccessible cardinals
hyper-inaccessible cardinals
super-hyper-inaccesible cardinals
Mahlo cardinals
hyper-Mahlo cardinals
indescribable cardinals
weakly compact cardinals
ineffable cardinals
partition cardinals
Ramsey cardinals
measurable cardinals
strongly compact cardinals
supercompact cardinals
extendible cardinals

--Norm

[Jim Sarbeck]

In article <33f8c610....@news.demon.co.uk>, J...@opticon.demon.co.uk
(John Goodwin) wrote:

: On Mon, 18 Aug 1997 17:59:04 +0100, Ben Soares <b...@dcs.st-and.ac.uk>
: wrote:
:
: >Randall Wagoner wrote:
: >>
: >> John Goodwin wrote:
: >>

: >> > On Thu, 14 Aug 1997 11:09:45 -0500, Randall Wagoner
: >> > <lew...@ix.netcom.com> wrote:
: >> >

: >> > >Randall Wagoner wrote:
: >> > >
: >> > >> What is the largest natural number you can think of?

: >> > >>
: >> > >> Randall
: >
: >The largest natural number I can think of using less than 100 characters
: >is:
: >
: > "The largest natural number defined by less than 100 characters"
: >
: >I only used 62 characters.
: >
:
: This is illogical.
:
: "The largest natural number defined by less than 100 characters ^ 10"
:
: 67 characters, and much bigger (except that it doesn't really make
: sense).
:
: Comments Please.

Well, can you 'Think of it'? How about

9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9^9

Does that make sense? It's Really Big!

Perhaps

9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999!

is worth thinking about for a while.

or

Let g=googolplex:
g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g^g

I wonder which is largest.

Regards,
Jim Sarbeck
*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*

[John Goodwin]

On Thu, 14 Aug 1997 01:19:36 -0500, Randall Wagoner
<lew...@ix.netcom.com> wrote:

>What is the largest natural number you can think of?
>

def F( f, m, n )
n=1: m f m
n!=1:F(f, m, n-1 )
end

h=F( ^, 9, 9 )
b=F( ^, h, h )
v=F( ^, b, b )

d=F( ^, v, v )

Explanation

The function F takes three parameters, An arithmetic operator f, and

two numbers m and n.

It is defined such that if n is one, the result is n f n, otherwise
F( f, m, n-1 )

eg F( ^, 2, 2)

is ( 2^2 ) ^ (2^2)

Then we define a hoogleplex (h) as F( ^, 9, 9 )

and use it again in our function F to define a boogleplex (b), a
voogleplex (v), and finally, we use out voogleplex in F to get us a
doogleplex.

I suspect that is the largest number yet submitted to this thread,

and, ignoring spaces and newlines, it takes up 74 characters.

You could, of course get in to more Xplexes by doing another 2
"passes"

There are, of course ways to get it even bigger, by defining the most

[John Goodwin]

On Tue, 19 Aug 1997 17:34:16 GMT, J...@opticon.demon.co.uk (John
Goodwin) wrote:

Third time lucky !!

>>What is the largest natural number you can think of?
>>

>def F( m, n )
>n=1: m ^ m

>n!=1:F( F( m, n-1 ), F( m, n-1))

>end
>
>h=F( 9, 9 )
>b=F( h, h )
>v=F( b, b )

>d=F( v, v )
>
>Explanation
>

>The function F takes two parameters m and n.
>
>It is defined such that if n is one, the result is m ^ m, otherwise

>F(F( m, n-1 ), F( m, n-1))

>
>eg F( 2,2 )
>is ( 2^2 ) ^ (2^2)
>
>which is 256
>
>but F( 2, 3 )
>is ( 256 ^ 256 ) ^ ( 256 ^ 256 )
>
>

>Then we define a hoogleplex (h) as F( 9, 9 )

>
>and use it again in our function F to define a boogleplex (b), a
>voogleplex (v), and finally, we use out voogleplex in F to get us a
>doogleplex.
>
>I suspect that is the largest number yet submitted to this thread,

>and, ignoring spaces and newlines, it takes up 76 characters.

>
>You could, of course get in to more Xplexes by doing another 2

>"passes", and start out with F( 999, 999 ).
>

[John Goodwin]

Those cardinals with the nice red cloak thingies who all meet up
whenever the guy with the white cloak thingie dies.

J.G.

[John Goodwin]

On Tue, 19 Aug 1997 17:54:01 -0500, Randall Wagoner
<lew...@ix.netcom.com> wrote:

snip

>
>Ben Tilly's response is not a definition, because Turing machine has no
>meaning. The Ackermann function does, because it is based on
>multiplication. If you can define Turing machine in under 100
>characters, then go for it.
>
>So far, my generalized exponential function is by far the fastest
>growing yet to be posted.
>
>I should have asked, "Who can come up with the fastest growing
>function?". Maybe we would have less misunderstandings.
>

I have changed my mind about this enterprise.

I had thought it would be a lot of fun, but after some analysis, I
think that it all comes down to the most compact definition of a
suitable recursive function, and the most efficient form of nesting it
in 100 characters.

And I still believe that it would just end up in squabbles over:

a) The rules
b) Who got the biggest number anyway

If we take the Ackerman function (G below), which gives the biggest
number?

Let F(n) = G(n,n,n)
F(F(9))

or, as Randall has posited

Let M=10000
G(M,M,M)

It's the first example, without a doubt, but it requires 4 extra
characters for it's definition. You can hardly come up with a ratio of
the number obtained divided by the number of characters used to define
it, well, at least not one that most people will understand!

The trick is, to always let the recursion do the work, and as far as I
can see, the only way to do that is to keep nesting.

In the example above, making the number of characters the same:

Let M=100000000
G(M,M,M)

is still nowhere near as effective as the nested example.

I suspect that if you tried to define a *more* recursive function than
Ackerman, the extra characters used in the definition would lose more
from the reduction of nesting than they would gain in the more
powerful definition.

I could well be wrong!

Then take the definition of Ackerman. As given by Randall it requires
54 characters.

G(1, k, j) = j*k
G(n+1, 1, j) = j
G(n+1, k+1, j) = G(n, G(n+1, k, j), j)

What about

X( j, 1 ) = j * j
X( j, n+1 ) = X( j, n ) * j

This requires 17 characters, and will grow at a much slower rate, BUT,
with the 37 characters saved you could get another 12 levels of
nesting.

Do the extra 37 characters required for Ackerman yield an improvement
on an extra 12 levels of nesting on the X function described above? I
have a very strong suspicion that they don't.

Without nesting, it is an integer power function.

At the first level of nesting, the second parameter has the same
function as the first in G above, after that you're in uncharted
waters.
Or, you could ban nesting, and end up coming up with an extended, 4,
5, or 6 parameter version of Ackerman.

In the end, it all comes down to the most efficient method of
specifying recursion.

J.G.

[John Goodwin]

On 19 Aug 1997 22:27:29 GMT, j...@hydra.cfm.brown.edu (Jim P. Ferry)
wrote:

>> What is the highest natural number you can describe in 100 characters or
>> less?
>
>100 characters or 1 character makes no difference: x
>
>Of course I have to tell you what "x" means, which may take many, many
>characters. To say 9!!!!!!!!...! or to nest Ackermann functions (or pre-nested
>Ackermann functions), or even to write 9999...9 assumes one understands the
>context, and to place no limits on the context makes the problem silly. Hence
>x.
>
>The best response I've seen in this thread is Ben Tilly's:
>
> b(n) = the longest that it can take to halt from a
> blank tape on an n-state Turing machine. b(b(99))
>

The problem with this approach is that the definition of the number
would not be self sufficient. You need to define a turing machine.

>I also wonder why saying "Your number plus 1" is considered asinine,

Because if you actually had a competition, and two people used this
kind of trick, determining the outcome would involve the resolution of
an infinitly recursive formula.

>whereas
>saying "Your system, nested once more" isn't.

It is, for the same reason.

>It can't be because the latter
>gives numbers that are "much bigger", since surely readers of sci.math realize
>that notion of relatively bigger is as relative as the notion of big. Just as
>multiplying by a million ceases to significantly increase numbers the size of
>10^(10^(10^10)), so passing from B(n,n) to B(B(n),B(n)) doesn't really matter
>when B(n) is huge enough.

?

>This discussion of trying to come up with large integers reminds me of the
>discussion of trying to "Godelize" in "Godel,Escher,Bach". Or of trying to
>express large ordinal numbers.

You seem not to have noticed that the whole thing is intended to be
just a piece of fun.

There is no need for these huge definitions, it's just a sort of
puzzle. That's why Randall put certain constraints on the answer, so
that more people could have a go. Including non-professional
mathematicians.

In fact, I now tend to agree that it's pointless, because it all comes
down to the most efficient method of defining recursion!

It would need someone to come up with a technique more powerful than
recursion to do any better, and such a technique seems unlikely (if it
is to be defined and used in 100 characters)!

J.G.

[Randall Wagoner]

John Goodwin wrote:

> In the end, it all comes down to the most efficient method of
> specifying recursion.
>
> J.G.

It can be proven that a function taking x into G(x, x, x) (doubly nested
function) is by-end-pieces greater than any function that can be defined
using single nesting. By-end-pieces I mean after a certain finite number
x, it grows bigger and stays bigger.

There are ways to create functions with more than two nestings. But I
don't think you can create a function that takes two numbers as
arguments and an argument that represents the number of nestings the
function will be. At least I haven't found one. If you understand that,
you will realize that that would be an ungodly fast growing
function.

Does anyone know of a way to prove which functions grows faster, or know
of a function that takes the number of nestings as an arguments? If so,
you will be king.

Randall

[Alan Morgan]

In article <5tdktr$e...@northshore.shore.net>,

Norman D. Megill <n...@shore.net> wrote:

>Perhaps a parallel game could have set theorists trying to outdo each
>other by naming ever larger cardinals.
>
>Here's what I could find, in increasing order of size (someone check me).
>For me the second on the list already hurts my brain.
>
>aleph-0
>aleph-1
>aleph-omega
>inaccessible cardinals
>hyper-inaccessible cardinals
>super-hyper-inaccesible cardinals

Are these serious????

>Mahlo cardinals
>hyper-Mahlo cardinals
>indescribable cardinals
>weakly compact cardinals
>ineffable cardinals

What the f are ineffable cardinals?

>partition cardinals
>Ramsey cardinals
>measurable cardinals
>strongly compact cardinals
>supercompact cardinals
>extendible cardinals

Alan

[Norman D. Megill]

In article <5tesqf$bil$1...@news.interlog.com>,

Peter Davies <pet...@interlog.com> wrote:
>>On 19 Aug 1997 22:31:23 -0400, n...@shore.net (Norman D. Megill) wrote:
>>

>>>Here's what I could find, in increasing order of size (someone check me).
>>>For me the second on the list already hurts my brain.
>>>
>>>aleph-0
>>>aleph-1
>>>aleph-omega
>>>inaccessible cardinals
>>>hyper-inaccessible cardinals
>>>super-hyper-inaccesible cardinals

>>>Mahlo cardinals
>>>hyper-Mahlo cardinals
>>>indescribable cardinals
>>>weakly compact cardinals
>>>ineffable cardinals

>>>partition cardinals
>>>Ramsey cardinals
>>>measurable cardinals
>>>strongly compact cardinals
>>>supercompact cardinals
>>>extendible cardinals
>

>None of the cardinals on your list, from inaccessible on, are known to exist.
>In fact, if you could prove the existence of even an inaccessible then you
>could prove the consistency of ZFC within ZFC. This, of course, is not
>possible. In summary, inaccessible cardinals MIGHT exist but we will never
>know. On the other hand, if they don't exist it would be possible to prove
>they don't. All that would be needed is to prove some inconsistent result
>follows from their existence.
>
>I have always found theorems which depend on the existence of inaccesible (or
>larger) cardinals very unsatisfying. However, this has not stopped me from
>proving such theorems!

Of course, you are assuming that "known to exist" means having
existence provable in ZFC. ZFC is "known" by general agreement
(with no known inconsistency). But even aleph-0 is "known" to exist
only because we assume it exists. It is just an axiom. It is not
mandated by God (or is it? :). ZFC - INF + ~INF is also a set theory
not known to be inconsistent; aleph-0 would then not exist. I,
for example, am philosophically reluctant to affirm the "existence"
(whatever that means) of even aleph-0 but keep such feelings private
and prove things with it anyway.

All we are doing is adding additional such axioms to ZFC. Like ZFC
itself, all are ultimately speculative (necessarily so by Godel). As
soon as any such axiom is found to be inconsistent with the previous we
of course would banish it from the list.

Perhaps a cleaner list might be limited to those cardinals whose
existence is not provable from ZFC - INF + the previous ones on the
list. In which case we omit aleph-1 and aleph-omega.

Or, if we think ZFC is sacred, have yet another game for wimps, "the
largest cardinal less than inaccessible that can be described with
less than 100 characters".

--Norm
(added sci.logic to followup)

[Ilias Kastanas]

In article <5t5ubr$utc$1...@dartvax.dartmouth.edu>,
Benjamin J. Tilly <Benjamin...@dartmouth.edu> wrote:
>In article <33F671E5...@ix.netcom.com>

>Randall Wagoner <lew...@ix.netcom.com> writes:
>
>> Ok. Now describe a number using 100 characters or less. But you have to
>> give the full description. You can't compare it to something else. In
>> other words, you can't say, "The number of seconds it would take to walk
>> to the end of the universe.". Nobody would know by that description what
>> the number is.
>>

>In case you missed it, here was my description.

>
> b(n) = the longest that it can take to halt from a
> blank tape on an n-state Turing machine. b(b(99))
>

>That takes 100 characters, and if you know what a Turing machine is
>then it is well defined. I counted spaces as key-strokes and it is
>written in simple English.
>
>If you don't know what a Turing machine is, then the description of one
>would take longer than 100 characters. However it can be done in a
>paragraph. And it really does define a unique number. (We cannot
>compute that number though!)

>
>> I gave a simple example of a description of a number. I also didn't use
>> any operation higher then multiplication. Here it is if you missed that
>> post.
>>

>My description needed no operations at all. :-)

>
>> M = 10000
>>
>> A(1, 1) = M
>> A(1, j+1) = M*A(k, j)
>> A(k+1, 1) = A(k, M)
>> A(k+1, j+1) = M*A(k+1, j)
>>
>> What is the highest number that can be reached from these rules.
>>
>> Like I said, this is just an example.
>

>If you disallow the use of the word "Turing machine", or any term which
>depends on one, then I can virtually guarentee that it is nowhere near
>what I called b(b(99)). (OK, so there are some other things out there
>which are also large for similar reasons. However if you do not accept
>computers, then statements in terms of the minimal number of steps
>needed to prove the statement of length at most n whose minimal proof
>is of maximal size are unlikely to fit in your definition...)

I've proposed a set of rules for this... one thing is, English
clearly has to go. A kind of equational language is to be used.

Now this does not outlaw computers (TMs). The language, similar
to one by Kleene, can define any partial recursive function; so you can
have universal ones etc. Of course it takes space to define them!

On the other hand, Busy_Beaver and variants are not computable
(recursive). You might want to set up some recursive approximation...

By the way, some other posters find the problem "meaningless"...
_With_ the rules, this is not so. There is a maximum number attainable.

Some say "it's just recursion". Note that it accounts for _all_
recursive functions, i.e. all computable ones (Church's thesis). Recursion
is not just "one thing"... and surely is not limited to producing the
Ackermann function!

Also, the problem is not identical to "rapidly growing functions".
A g might dwarf f for n > some K way out; but we have a space bound, and
it could be that within what we can use f does better.

Numbers will easily be impossible, in any practical sense, to
actually evaluate... especially if my suggestion to lower the bound to
50 characters is not adopted.

Ilias

[Gene W. Smith]

In article <33faaf79...@news.demon.co.uk>,
John Goodwin <J...@opticon.demon.co.uk> wrote:

>In fact, I now tend to agree that it's pointless, because it all comes
>down to the most efficient method of defining recursion!

I don't see why. What violates the rules in this:

"f(n) is the number of steps it takes to get to zero by writing n base
2, subtracting one, interpreting that in base 3, and repeating with
ever increasing base."

Now pick f(n) for a large starting value. I don't know what you mean
exactly by "the most efficient method of defining recursion", but I
doubt this qualifies.

[Hauke Reddmann]

Norman D. Megill (n...@shore.net) wrote:
:
: Perhaps a parallel game could have set theorists trying to outdo each
: other by naming ever larger cardinals.

...and you didn't invent this game. (M.Gardner in SciAm?
D.Hofstadter, ditto? Dunno where I read this before.)

Now, can you give us definitions of the beasties below?
Please use a countable amount of words. :-)
:
:
: aleph-0

: aleph-1
: aleph-omega
: inaccessible cardinals
: hyper-inaccessible cardinals
: super-hyper-inaccesible cardinals
: Mahlo cardinals
: hyper-Mahlo cardinals
: indescribable cardinals
: weakly compact cardinals
: ineffable cardinals
: partition cardinals
: Ramsey cardinals
: measurable cardinals
: strongly compact cardinals
: supercompact cardinals
: extendible cardinals

:
--
Hauke Reddmann <:-EX8
fc3...@math.uni-hamburg.de PRIVATE EMAIL
fc3...@rzaixsrv1.rrz.uni-hamburg.de BACKUP
redd...@chemie.uni-hamburg.de SCIENCE ONLY

[Peter Davies]

In article <33fab04e...@news.demon.co.uk>,

J...@opticon.demon.co.uk (John Goodwin) wrote:
>On 19 Aug 1997 22:31:23 -0400, n...@shore.net (Norman D. Megill) wrote:
>

>>In article <5td6ki$6...@cocoa.brown.edu>,

>>Jim P. Ferry <j...@hydra.cfm.brown.edu> wrote:

>>Here's what I could find, in increasing order of size (someone check me).
>>For me the second on the list already hurts my brain.
>>

>>aleph-0
>>aleph-1
>>aleph-omega
>>inaccessible cardinals
>>hyper-inaccessible cardinals
>>super-hyper-inaccesible cardinals
>>Mahlo cardinals
>>hyper-Mahlo cardinals
>>indescribable cardinals
>>weakly compact cardinals
>>ineffable cardinals
>>partition cardinals
>>Ramsey cardinals
>>measurable cardinals
>>strongly compact cardinals
>>supercompact cardinals
>>extendible cardinals
>

>Those cardinals with the nice red cloak thingies who all meet up
>whenever the guy with the white cloak thingie dies.
>
>J.G.
>

None of the cardinals on your list, from inaccessible on, are known to exist.

In fact, if you could prove the existence of even an inaccessible then you
could prove the consistency of ZFC within ZFC. This, of course, is not
possible. In summary, inaccessible cardinals MIGHT exist but we will never
know. On the other hand, if they don't exist it would be possible to prove
they don't. All that would be needed is to prove some inconsistent result
follows from their existence.

I have always found theorems which depend on the existence of inaccesible (or
larger) cardinals very unsatisfying. However, this has not stopped me from
proving such theorems!

Peter Davies

[KRamsay]

In article <33FA2409...@ix.netcom.com>, Randall Wagoner

<lew...@ix.netcom.com> writes:
|So far, my generalized exponential function is by far the fastest
|growing yet to be posted.

The busy-beaver function BB(n) or b(n) grows much faster; I think
you mean your function is the fastest one which was defined with
certain limited means, not assuming the reader knows what a
Turing machine is, perhaps.

Keith Ramsay There is nothing on this earth, and little beyond it,
kra...@aol.com that nobody ever denounces. -- Matt McIrvin

[Bill Taylor]

|> >aleph-0
|> >aleph-1
|> >aleph-omega

Hey, what happened to 2^aleph-0, you continuum hypothesizer, you!

|> >inaccessible cardinals
|> >hyper-inaccessible cardinals
|> > ... ...

|> Those cardinals with the nice red cloak thingies who all meet up
|> whenever the guy with the white cloak thingie dies.

You sig-stealer, you! ;-)

==============================================================================
Bill Taylor W.Ta...@math.canterbury.ac.nz
------------------------------------------------------------------------------
Some say the pope is the greatest cardinal.
But others insist this cannot be so, as every pope has a successor.
------------------------------------------------------------------------------

[Benjamin J. Tilly]

In article <33F883...@mitre.org>
"Patrick E. White" <wh...@mitre.org> writes:

> Benjamin J. Tilly wrote:
>
> > (OK, so BB(n) is the usual notation for b(n).) Now just how big is this
> > number? Well let me indicate it as follows. Imagine a computer program,
> > any computer program, which you could write on any computer in the
> > world that you can think of. Imagine that this program was designed to
> > let you take some description in a particular format and generate a
> > number in base 10 from it. Imagine running it on a computer with
> > infinite capacity, and feeding it an input file where the numbers were
> > the size of numbers on a microfiche and the input file filled the
> > observable universe. Feed it the input file that you think will get
> > this computer to produce the largest possible number.
>
> Don't you need to worry about whether this computer can be described in
> less than b(99) states? ... Well, I guess about any computer can...
>
Both the file and the computer together can I believe. b(n) truly grows
at an astounding rate! (I have vague memories saying that b(6) is
something like 10^12. And it does NOT slow down after that.)
>
> > PS This number has some interesting properties. One of them is that no
> > consistent axiom system stateable within the space of the known
> > universe is capable of proving that any particular number written in
> > base 10 is an upper bound for b(b(99)). No, this is not a contradictory
> > statement, and understanding what it means emphasizes why it is that
> > some people lean towards constructivism. :-)
>
> Now this is interesting. What about b(b(2))? It probably doesn't share
> this "unprovable" property? What can you say about b(n) in general with
> respect to this property? Anything stronger than "except for finitely
> few n, ..."

Consider an axiom system capable of encoding arithmetic. We can design
a Turing machine which provably enumerates all possible proofs and
disproofs in that axiom system, and will halt if it comes across a
proof of a contradiction.

Then for the axiom system to prove that this program does not halt
would by Goedel's theorem prove that it is inconsistent. (And hence the
program does halt!:-)

However the axiom system certainly can prove that the program takes
longer than N steps to run where N is any specified integer. (Just run
a simulation!) And since that is insufficent to prove that it does not
halt, then N must not be something which that axiom system proves is an
upper bound.

If you push this argument slightly, then the answer turns out to be
that there is no upper bound for b(n) provable within a given
consistent axiom system that can describe arithmetic if n is more than
a fixed amount larger than the number of bits required to specify the
axiom system.

Ben Tilly

[John Goodwin]

On 20 Aug 1997 17:24:22 -0400, gws...@river.gwi.net (Gene W. Smith)
wrote:

>In article <33faaf79...@news.demon.co.uk>,
>John Goodwin <J...@opticon.demon.co.uk> wrote:
>
>>In fact, I now tend to agree that it's pointless, because it all comes
>>down to the most efficient method of defining recursion!
>
>I don't see why. What violates the rules in this:
>
>"f(n) is the number of steps it takes to get to zero by writing n base
>2, subtracting one, interpreting that in base 3, and repeating with
>ever increasing base."

It depends which rules you are referring to. If you use Randall's
rules:

1) It uses English
2) It is significantly greater than 100 characters
3) It does not terminate (assuming I am interpreting it correctly)

e.g 4 to base 2 is 100
100 - 1 is 11
11 to base 3 is 4 which to base 2 is 100
100 - 1 is 11
11 to base 5 is 6
110 - 1 is 101
101 to base 6 is 37 etc.

It does seem to me, however, that Randall's rules were specifically
set up to favour the Ackerman function.

>Now pick f(n) for a large starting value. I don't know what you mean
>exactly by "the most efficient method of defining recursion", but I
>doubt this qualifies.

It doesn't, because for a recursive function to be of any use, it must
terminate.

J.G.

[Randall Wagoner]

John Goodwin wrote:

I don't understand the example?

> It does seem to me, however, that Randall's rules were specifically
> set up to favour the Ackerman function.
>

Actually, I had no rules in the beginning. I made 100 characters the
limit to keep things somewhat simple. The Ackermann function has well
below 100 chars, only 54. I just picked 100 as a nice even number.

[-Mammel,L.H.]

In article <01bca94d$3d842e60$36528fc2@ntmaskin>,
Arne Dehli Halvorsen <a...@computas.no> wrote:
>How about
>googolplex!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
>!!!!!!!!!!!!!!!!!!!! (stop at 100 chars)
>
>each ! is a factorial sign, so the result should be largish.
>
>Arne

This and the exponential tower of 100 googolplexes are both
less than an exponential tower of 103 10's.

Consider (10^10^100)^(10^10^100) = 10^(10^100*10^10^100) =
10^10^(100+10^100) ~ 10^10^10^100.

Now if you take 10^10^100 to THAT power, the extra exponentiation
in the base will have even less effect. In the end the tower of
100 googolplexes is only enhanced over a tower of 100 10's by the
extra length at the top : 10^10^10^2+ replacing 10.

Now consider the googolplex factorial(100) . N! < N^N, so an
upper bound to each factorial iteration is formed by raising
a tower of tens to itself. Following a procedure similar
to the above, we see that each time we "squish them together",
(tower of N)^(tower of N) = 10^( tower of N-1 * tower of N ) and
the taller tower accommodates the shorter ( by 1 ) tower with just
the tiniest increment to the top exponent, and the top 2+ will never
get much bigger. The whole thing stays less than a tower of
103 10's.

I think the "power of the tower" is amazing, and it's almost
frightening to contemplate that this is just ONE step past
exponentiation in the generalized Ackermann sequence of operations.

Lew Mammel, Jr.

[John Goodwin]

I may have misinterpreted the method, I think the definition may be a
little ambiguous. Which bit do you not understand?

I was using n=4.

Written in base 2 that is 100
Subtract 1 (still in base 2) gives 11
Interpret that as 11 to base 3 gives 4

and so on....

>> It does seem to me, however, that Randall's rules were specifically
>> set up to favour the Ackerman function.
>>
>
>Actually, I had no rules in the beginning. I made 100 characters the
>limit to keep things somewhat simple. The Ackermann function has well
>below 100 chars, only 54. I just picked 100 as a nice even number.

I meant that you specified no arithmetic operator greater than *,
which, as it happens, allows Ackerman's function, but would not allow
an almost identical function which used ^ in place of *, and would
thus be much bigger for any given set of parameters.

J.G.

[Chris Menzel]

Norman D. Megill (n...@shore.net) wrote:

: Of course, you are assuming that "known to exist" means having

: existence provable in ZFC. ZFC is "known" by general agreement
: (with no known inconsistency). But even aleph-0 is "known" to exist
: only because we assume it exists. It is just an axiom.

True enough, but the natural numbers provide our postulation of
aleph_0 with a certain intuitive warrant that is lacking in the case
of, say, the proposition that there is an inaccessible, where we don't
have a similarly intuitive structure that exhibits the cardinality in
question. Putting it as you have above sorta makes it sound like the
two axioms are on an epistemological par. To the contrary,
justification for large cardinal axioms has to proceed on much dicier
grounds, e.g., the intuition (?) that the set theoretic universe is
*so* rich that any conceivable way of extending it has got to be
realized in fact.

==================================================================
Christopher Menzel | web: philebus.tamu.edu/~cmenzel
Philosophy, Texas A&M University | net: chris....@tamu.edu
College Station, TX 77843-4237 | vox: (409) 845-8764
==================================================================

[Benjamin J. Tilly]

In article <33FA2409...@ix.netcom.com>
Randall Wagoner <lew...@ix.netcom.com> writes:

[...]

> Ben Tilly's response is not a definition, because Turing machine has no
> meaning. The Ackermann function does, because it is based on
> multiplication. If you can define Turing machine in under 100
> characters, then go for it.

[...]

Can you define multiplication in under 100 characters from the Peano
Axioms?

It is all a question of what you assume. (As I said in my original
post.)

Ben Tilly

[John Goodwin]

On 22 Aug 1997 02:24:28 GMT, Benjamin...@dartmouth.edu (Benjamin
J. Tilly) wrote:

In the context of this thread that is not necessary.

In general, in a group such as sci.math, it is quite likely that most
of the people who use the group will have some familiarity with
multiplication.

The same cannot be said for the definition if a turing machine.

J.G.

[Randall Wagoner]

-Mammel,L.H. wrote:

> In article <01bca94d$3d842e60$36528fc2@ntmaskin>,
> Arne Dehli Halvorsen <a...@computas.no> wrote:
> >How about
> >googolple

In other words, G(4, k, j)

[Randall Wagoner]

-Mammel,L.H. wrote:

> In article <33F785...@csubak.edu>,
> Larry Taylor <lta...@csubak.edu> wrote:
> >Randall Wagoner wrote:
> >> <snip>
> >> My simple example is Ackerman's function. I'll define it again.

> >>
> >> M = 10000
> >>
> >> A(1, 1) = M
> >> A(1, j+1) = M*A(k, j)
> >

> > ?? You mean A(k,j+1) on the left OR A(1,j) on the right
>
> He's copying this stuff out of Chapter 3 of INFINITY AND THE MIND.
> This accounts for the Ackermann/Archimedes confusion as well
> as the cited error, although M is supposed to be a myriad-myriad
> = 10^8.
>
> Lew Mammel, Jr.

He is right. I did get it out of that book. It is the best book that
I've read on the subject to date.

Any comments on it?

Randall

[Randall Wagoner]

John Goodwin wrote:

Actually, if you put "^" in place "*" in the rules I gave, it wouldn't
make much difference. If you add one to the first argument, you would
have the same function with "^" has the beginning operation. The first
argument is what specifies the operation. So, what is one more?

Randall

[Randall Wagoner]

Benjamin J. Tilly wrote:

> In article <33FA2409...@ix.netcom.com>
> Randall Wagoner <lew...@ix.netcom.com> writes:
>
> [...]
> > Ben Tilly's response is not a definition, because Turing machine has
> no
> > meaning. The Ackermann function does, because it is based on
> > multiplication. If you can define Turing machine in under 100
> > characters, then go for it.
> [...]
>
> Can you define multiplication in under 100 characters from the Peano
> Axioms?
>

Nope! I don't even want to try.

But, I said multiplication is the highest operation that can be used.

> It is all a question of what you assume. (As I said in my original
> post.)

Multiplication is a lot more common then Turing machine. that's why I
picked it to be among the operations to be used.

Now, forgive me for my ignorance, but could you define the Turing
machine for me?

Thanks in advance.

Randall

[Richard White]

On Fri, 22 Aug 1997, Randall Wagoner wrote:

> -Mammel,L.H. wrote:
>
> [snip, RW]

>
> > He's copying this stuff out of Chapter 3 of INFINITY AND THE MIND.
> > This accounts for the Ackermann/Archimedes confusion as well
> > as the cited error, although M is supposed to be a myriad-myriad
> > = 10^8.
>

> He is right. I did get it out of that book. It is the best book that
> I've read on the subject to date.
>
> Any comments on it?

Well, lets just say that there are other authors whose writing style I
like better than Rudy Rucker's. If, say, Douglas R. Hofstadter had
written that book, I think it would have been much more readable.

Happy Times Upon You, Richard.

[Peter Davies]

In article <5ti42a$gr6$1...@news.tamu.edu>,

Thanks for that! Norman's response was to my post and I was about to respond
but you have said more or less what I wanted to say, only much more
eloquently!

Peter Davies