> "If you have the list of computables, a random real number can be on it to an infinite number
> of digits, and yet not be on the list" True / False / Other

As stated: false. The real either is or is not on the list, but not both.

-------------------------------------------------------------------------------

> "If you have the list of computables, a random real number can
> be on it to an infinite number of digits, and yet not be on
> the list" True / False / Other

True as phrased. (Example: S_3 and 1/3, TX_10 and any q whose
denominator has a prime factor other than 2 or 5)

Herc
--
As our PM for 10 years Mr Howard
What long term goals have you realised?

From:	Will Twentyman
Subject:	Re: WELL WHICH IS IT... ?
Date:	Thu, 20 Jan 2005 22:20:37 -0500

|-|erc wrote:

> > "If you have the list of computables, a random real number can be on it to an infinite number
> > of digits, and yet not be on the list" True / False / Other
>
> As stated: false. The real either is or is not on the list, but not both.
>
> -------------------------------------------------------------------------------
>
> > "If you have the list of computables, a random real number can
> > be on it to an infinite number of digits, and yet not be on
> > the list" True / False / Other
>
> True as phrased. (Example: S_3 and 1/3, TX_10 and any q whose
> denominator has a prime factor other than 2 or 5)

So, apparently Ghost and I interpretted the above differently. I
interpretted it as "Given the list of computables, and a real number X,
X can have all its digits as a single entry on the list, but not be on
the list."

With that reading, it is equivalent to "Given the list of computables,
and a real number X, X can be on the list, but not be on the list,"
which is obviously false.

I'm not sure how Ghost read it.

--
Will Twentyman
email: wtwentyman at copper dot net

From:	\|-\|erc
Subject:	Re: WELL WHICH IS IT... ?
Date:	Fri, 21 Jan 2005 13:34:46 +1000

"Will Twentyman" wrote in >
> > > "If you have the list of computables, a random real number can be on it to an infinite number
> > > of digits, and yet not be on the list" True / False / Other
> >
> > As stated: false. The real either is or is not on the list, but not both.
> >
> > -------------------------------------------------------------------------------
> >
> > > "If you have the list of computables, a random real number can
> > > be on it to an infinite number of digits, and yet not be on
> > > the list" True / False / Other
> >
> > True as phrased. (Example: S_3 and 1/3, TX_10 and any q whose
> > denominator has a prime factor other than 2 or 5)
>
> So, apparently Ghost and I interpretted the above differently. I
> interpretted it as "Given the list of computables, and a real number X,
> X can have all its digits as a single entry on the list, but not be on
> the list."
>
> With that reading, it is equivalent to "Given the list of computables,
> and a real number X, X can be on the list, but not be on the list,"
> which is obviously false.
>
> I'm not sure how Ghost read it.
>

That's because you don't have a predicate in mind for
X is on the list to Y many digits.

Is a simple predicate but you refuse to parse English on the set of terms, instead
you break down each term individually, on, that many digits, in the hope of eliminating
hard questions for Cantorians to answer.

Herc

From:	The Ghost In The Machine
Subject:	Re: WELL WHICH IS IT... ?
Date:	Fri, 21 Jan 2005 05:01:45 GMT

In sci.logic, Will Twentyman

wrote
on Thu, 20 Jan 2005 22:20:37 -0500
<41f0754c$1_1@newsfeed.slurp.net>:
> |-|erc wrote:
>
>> > "If you have the list of computables, a random real number can
>> > be on it to an infinite number of digits, and yet not be on
>> > the list" True / False / Other
>>
>> As stated: false. The real either is or is not on the list,
>> but not both.
>>
>> ---------------------------------------
>>
>> > "If you have the list of computables, a random real number can
>> > be on it to an infinite number of digits, and yet not be on
>> > the list" True / False / Other
>>
>> True as phrased. (Example: S_3 and 1/3, TX_10 and any q whose
>> denominator has a prime factor other than 2 or 5)
>
> So, apparently Ghost and I interpretted the above differently. I
> interpretted it as "Given the list of computables, and a real number X,
> X can have all its digits as a single entry on the list, but not be on
> the list."
>
> With that reading, it is equivalent to "Given the list of computables,
> and a real number X, X can be on the list, but not be on the list,"
> which is obviously false.
>
> I'm not sure how Ghost read it.
>

Hmm....well, if one takes my favorite class TX_10, then it
is clear that all of pi's finite prefixes are within TX_10,
but pi, being irrational, is not.

Or, one can take a constant such as Chaitin's Omega.
This constant is a perfectly good real number, but is
not computable, therefore cannot be on any list of computables.
However, all of its prefixes easily can be (since one way
of "computing" a finite sequence is simply to recite it).

That was more or less my thinking; apologies if that wasn't
horribly clear.

Of course |-|erc isn't being horribly clear here, either.
Having a real number be on a list of numbers "to an infinite
number of digits" isn't all that well-defined an operation.

--
#191, ewill3@earthlink.net
It's still legal to go .sigless.

From:	stephen at nomail.com
Subject:	Re: WELL WHICH IS IT... ?
Date:	21 Jan 2005 06:48:16 GMT

In sci.math The Ghost In The Machine wrote:
: In sci.logic, Will Twentyman
:
: wrote
: on Thu, 20 Jan 2005 22:20:37 -0500
:> So, apparently Ghost and I interpretted the above differently. I
:> interpretted it as "Given the list of computables, and a real number X,
:> X can have all its digits as a single entry on the list, but not be on
:> the list."
:>
:> With that reading, it is equivalent to "Given the list of computables,
:> and a real number X, X can be on the list, but not be on the list,"
:> which is obviously false.
:>
:> I'm not sure how Ghost read it.
:>

: Hmm....well, if one takes my favorite class TX_10, then it
: is clear that all of pi's finite prefixes are within TX_10,
: but pi, being irrational, is not.

: Or, one can take a constant such as Chaitin's Omega.
: This constant is a perfectly good real number, but is
: not computable, therefore cannot be on any list of computables.
: However, all of its prefixes easily can be (since one way
: of "computing" a finite sequence is simply to recite it).

Of course, even though all of the prefixes of Chaitin's Omega
will appear on a list of computables we cannot actually determine
that they are prefixes of Chaitin's Omega.

Stephen

From:	The Ghost In The Machine
Subject:	Re: WELL WHICH IS IT... ?
Date:	Sat, 22 Jan 2005 00:01:39 GMT

In sci.logic, stephen@nomail.com

wrote
on 21 Jan 2005 06:48:16 GMT
:
> In sci.math The Ghost In The Machine wrote:
>: In sci.logic, Will Twentyman
>:
>: wrote
>: on Thu, 20 Jan 2005 22:20:37 -0500
>:> So, apparently Ghost and I interpretted the above differently. I
>:> interpretted it as "Given the list of computables, and a real number X,
>:> X can have all its digits as a single entry on the list, but not be on
>:> the list."
>:>
>:> With that reading, it is equivalent to "Given the list of computables,
>:> and a real number X, X can be on the list, but not be on the list,"
>:> which is obviously false.
>:>
>:> I'm not sure how Ghost read it.
>:>
>
>: Hmm....well, if one takes my favorite class TX_10, then it
>: is clear that all of pi's finite prefixes are within TX_10,
>: but pi, being irrational, is not.
>
>: Or, one can take a constant such as Chaitin's Omega.
>: This constant is a perfectly good real number, but is
>: not computable, therefore cannot be on any list of computables.
>: However, all of its prefixes easily can be (since one way
>: of "computing" a finite sequence is simply to recite it).
>
> Of course, even though all of the prefixes of Chaitin's Omega
> will appear on a list of computables we cannot actually determine
> that they are prefixes of Chaitin's Omega.

Actually we can -- once we have Chaitin's Omega. I'll admit I'm
not sure, but I suspect we can get a good chunk of Omega's digits,
just not *all* of it (the proof that we can't involves constructing
a proper set of counterexamples, but one can still apply heuristics).

But this is probably getting close to hairsplitting. The fact is
that any list of numbers derived from a computing machine
mechanism of some sort means that that list is denumerable,
and therefore a proper subset of all reals. Therefore, it is
possible to construct a real that, even though all of its prefixes
are somewhere on the list, the actual real is not.

It's a bit like |-|erc's favorite bete noir: 1/3 in {.3, .33, .333, ...}
= S_3 = {(10^k - 1)/ (3 * 10^k): k >= 0, k in J}.

Or, to be crystal clear on the matter, let's list some
possible questions:

- Is N = {1, 2, 3, ...} countable? Yes. (By definition.)

- Is J = N union {0} union {-k: k in N} countable? Yes.
(Order the elements 0, 1, -1, 2, -2, ... .)

- Is Q = {a/b: a, b in J, b != 0} countable? Yes.
(Order the elements 0/1, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1,
1/3, -1/3, 2/2, -2/2, 3/1, -3/1, 1/4, -1/4, 2/3, -2/3,
3/2, -3/2, 4/1, -4/1, 1/5, -1/5, ..., and discard duplicates.)

- Is TX_10 = {k/10^n: n >= 0, k, n in J} countable? Yes.

- Is R, the set of all real numbers, countable? No.

- Is C = {a + bi: a, b in R} countable? No.

- Is S_3 = {(10^k - 1)/ (3 * 10^k): k >= 0, k in J} countable? Yes.

- Are all of 1/3's base-10 prefixes
(where prefix10(x,n) = floor(x * 10^n)/10^n, x > 0)
in S_3? Yes.

- Is 1/3 in S_3? No. (The proof involves induction and
divisibility of powers of 10 by 3, and is left to the interested
reader.)

- Are all r in R such that all of r's prefixes are in TX_10? Yes.

- Is any r in R guaranteed to be in TX_10? No. (r = pi)

- Are all q in Q such that all of q's prefixes are in TX_10? Yes.

- Is any q in Q guaranteed to be in TX_10? No. (q = 1/3)

I could go into issues such as a set being dense around a point,
what points a set is dense around, and such, but that would
probably take me a bit far afield.

If one wants to expand things a bit, let A be the set of all
roots of all irreducible polynomial equations

c_n * x^n + c_{n-1} * x^{n-1} + ... + c_1 * x + c_0 = 0

where c_i are integers (the _algebraic numbers_),
c_n != 0, and c_i != 0 except for the special case x + 0 = 0.
Furthermore, define functions Re(a+bi) = a, Im(a+bi) = b.
It is provable (though again beyond my scope here) that all
such roots are in C, and therefore A is a subset of C --
a proper subset, as it turns out.

- Is A countable? Yes. (Group the equations by the sum of
the absolute value of the coefficients, plus the highest
exponent, and then pick an order for each "subcluster".)

- Are all x in A such that all Re(x)'s prefixes are in TX_10? Yes.

- Are all x in A such that all Im(x)'s prefixes are in TX_10? Yes.

- Is that Re(x) guaranteed to be in TX_10? No.

- Is that Im(x) guaranteed to be in TX_10? No.

- Are all x in A such that all Re(x)'s prefixes are in Q? Yes.

- Are all x in A such that all Im(x)'s prefixes are in Q? Yes.

- Is that Re(x) guaranteed to be in Q? No.

- Is that Im(x) guaranteed to be in Q? No.

- Is Q a proper subset of A? Yes. (x = sqrt(2); q = a/b <=> ax - b = 0.)

Now let M_h be the set of all numbers written out by a
class of halting Turing machines (alphabet [0-9]),
where the decimal point is assumed to be in front
of the first digit, and let I=[0,1) be a subset of R
and T_10 = TX_10 intersection [0,1)
= {k/10^n: n >= 0, 0 <= k < 10^n, n, k in J}.

For simplicity we define an empty tape to be 0 and
ignore issues with blanks interspersed with the digits.

We don't particularly care where the head ends up (although
we could modify the definition so that the head position
defines the decimal point; the details are left to the
interested reader).

- Is M_h countable? Yes. (Turing machines can be encoded
into N; the mapping is 1-1 but not onto.)

- Are all m in M_h such that all m's prefixes are in T_10? Yes.

- Is m in T_10? Yes, since m must be finite.

- Is M_h = T_10? Yes.

Lucky |-|erc. But if we let M_i be the set of all numbers
written out by a class of *non* halting Turing machines
(with certain restrictions so that we don't get into
a machine that just moves its head back and forth; I'm not
entirely sure how to specify such properly at this time):

- Is M_i countable? Yes.

- Are all m in M_i such that all m's prefixes are in T_10? Yes.

- Is m guaranteed to be in T_10? No.

And of course, if we define M = M_h union M_i:

- are all i in I such that i is in M? No.

- Is M = I? No. (This is just a rephrasing of the preceding question.)

I could get into the realm of coin flips but things there
get a little dicey becaues of the ...0111...(2) = ...1000...(2)
issue.

>
> Stephen

--
#191, ewill3@earthlink.net
It's still legal to go .sigless.

From:	rupertmccallum at yahoo.com
Subject:	Re: WELL WHICH IS IT... ?
Date:	22 Jan 2005 15:22:23 -0800

|-|erc wrote:
> wrote in
> >
> > |-|erc wrote:
> > > > "If you have the list of computables, a random real number can
be
> > on it to an infinite number
> > > > of digits, and yet not be on the list" True / False / Other
> > >
> >
> > It's possible that for each k, the real agrees with some member of
the
> > list to k digits, but that the real is not on the list.
> >
>
> .......therefore true?
>
> ........therefore false?
>
> BE ASSERTIVE MAN!
>
> Its not newspeak I wish to clarify, its using English for proper
comprehension that
> your syntax seems unable to grasp.
>
> Herc

Well, it depends what you mean by "be on it to an infinite number of
digits". I gave you one interpretation on which it's true.

From:	rupertmccallum at yahoo.com
Subject:	Re: WELL WHICH IS IT... ?
Date:	20 Jan 2005 19:06:44 -0800

|-|erc wrote:
> > "If you have the list of computables, a random real number can be
on it to an infinite number
> > of digits, and yet not be on the list" True / False / Other
>

It's possible that for each k, the real agrees with some member of the
list to k digits, but that the real is not on the list.

> As stated: false. The real either is or is not on the list, but
not both.
>
>
-------------------------------------------------------------------------------
>
> > "If you have the list of computables, a random real number can
> > be on it to an infinite number of digits, and yet not be on
> > the list" True / False / Other
>
> True as phrased. (Example: S_3 and 1/3, TX_10 and any q whose
> denominator has a prime factor other than 2 or 5)
>
>
> Herc
> --
> As our PM for 10 years Mr Howard
> What long term goals have you realised?

From:	\|-\|erc
Subject:	Re: WELL WHICH IS IT... ?
Date:	Fri, 21 Jan 2005 13:12:58 +1000

wrote in
>
> |-|erc wrote:
> > > "If you have the list of computables, a random real number can be
> on it to an infinite number
> > > of digits, and yet not be on the list" True / False / Other
> >
>
> It's possible that for each k, the real agrees with some member of the
> list to k digits, but that the real is not on the list.
>

........therefore true?

.........therefore false?

BE ASSERTIVE MAN!

Its not newspeak I wish to clarify, its using English for proper comprehension that
your syntax seems unable to grasp.

Herc