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From: Jacko on 10 Aug 2010 20:45
Always the same with you unsound logic folks. Semantics over substance.
From: Tim Little on 10 Aug 2010 22:09
On 2010-08-10, Shubee <e.shubee(a)gmail.com> wrote:
> I wonder then if a computable number is the same as my definition of a
> nonrandom sequence. I don't see that my definition requires producing
> an approximation correct to n digits in a finite number of steps, only
> that the nonrandom number can be defined by a rule expressed with a
> finite number of characters.

In that case, no it isn't. Definability isn't the same as
computability in terms of an algorithm. For example, you can easily
define the sequence of Busy Beaver numbers. That sequence is not
computable by any algorithm.

Since you are using definability instead of algorithmic computability,
your notion is exactly Richard's Paradox, arising from ambiguity in
what it means for a number to be "definable". Once the ambiguity is
removed (e.g. by abandoning the unqualified term "definable" for the
more precise "definable in" some system), the paradox disappears.


- Tim
From: Shubee on 10 Aug 2010 22:36
On Aug 10, 9:09 pm, Tim Little <t...(a)little-possums.net> wrote:
> On 2010-08-10, Shubee <e.shu...(a)gmail.com> wrote:
>
> > I wonder then if a computable number is the same as my definition of a
> > nonrandom sequence. I don't see that my definition requires producing
> > an approximation correct to n digits in a finite number of steps, only
> > that the nonrandom number can be defined by a rule expressed with a
> > finite number of characters.
>
> In that case, no it isn't. Definability isn't the same as
> computability in terms of an algorithm. For example, you can easily
> define the sequence of Busy Beaver numbers. That sequence is not
> computable by any algorithm.

The Busy Beaver numbers? You lost me there.

> Since you are using definability instead of algorithmic computability,
> your notion is exactly Richard's Paradox, arising from ambiguity in
> what it means for a number to be "definable". Once the ambiguity is
> removed (e.g. by abandoning the unqualified term "definable" for the
> more precise "definable in" some system), the paradox disappears.

What's so impossible about defining "definable" numbers? I have in
mind something like "A real number is definable if it can be specified
unambiguously by a finite number of characters (either in words,
numbers or mathematical symbols)." I realize that a definition should
be made more rigorous than that, but where's the limit?


From: Tim Little on 11 Aug 2010 00:55
On 2010-08-11, Shubee <e.shubee(a)gmail.com> wrote:
> On Aug 10, 9:09 pm, Tim Little <t...(a)little-possums.net> wrote:
>> For example, you can easily define the sequence of Busy Beaver
>> numbers. That sequence is not computable by any algorithm.
>
> The Busy Beaver numbers? You lost me there.

http://en.wikipedia.org/wiki/Busy_beaver


>> Since you are using definability instead of algorithmic computability,
>> your notion is exactly Richard's Paradox, arising from ambiguity in
>> what it means for a number to be "definable". Once the ambiguity is
>> removed (e.g. by abandoning the unqualified term "definable" for the
>> more precise "definable in" some system), the paradox disappears.
>
> What's so impossible about defining "definable" numbers?

Nothing, but you just have to be quite precise about what counts as a
definition. For example, does "the least natural number not having a
three word definition" count as a definition?


- Tim
From: Shubee on 11 Aug 2010 10:48
On Aug 10, 11:55 pm, Tim Little <t...(a)little-possums.net> wrote:
> On 2010-08-11, Shubee <e.shu...(a)gmail.com> wrote:
>
> > What's so impossible about defining "definable" numbers?
>
> Nothing, but you just have to be quite precise about what counts as a
> definition. For example, does "the least natural number not having a
> three word definition" count as a definition?

That's a good one Tim. Thank you. Isn't the creation of a precise
mathematical language for new axiom sets a standard occurrence in
mathematical logic? Are you saying that this can't be done for the
concept that I'm trying to justify? I realize that something has to
break down somewhere. I simply have no idea where that would be.


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