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From: Sylvia Else on 5 Jun 2010 04:54
On 5/06/2010 7:55 AM, |-|ercules wrote:
> According to sci.math, the number line consists mostly of "dark numbers".
>
> You see, 99.999999.... nearly % of real numbers are not computable,
> there is
> no program that represents them.
>
> A computer can calculate ANY digit sequence up to INFINITE length.
>
> Sci.math will make a minor correction there, a computer can only calculate
> all digit sequences up to ALL finite lengths.
> This is like saying a computer can only calculate all digit sequences up
> to INFINITE finite lengths.

A computer is a finite state machine, so although it may continue to
compute forever, its outputs must start to repeat. Any number whose
sequence of digits does not appear in that recurring output will never
be output by that computer no matter how long you wait.

If you don't constrain the computer's storage space, letting it be
"inifinite" you still have the problem that the set of storage locations
will be countably infinite, and thus the set of all of its states will
also be countably infinite, as will the set of all of its possible outputs.

Since the reals are not countable, the computer will still omit most of
them from its output. That is, it will not be able to compute them.

Of course, you can hypothesise a computer that is not subject to those
constraints, but that's really just assuming the answer.

Sylvia.
From: Sylvia Else on 5 Jun 2010 08:22
On 5/06/2010 7:54 PM, |-|ercules wrote:
> "Sylvia Else" <sylvia(a)not.here.invalid> wrote
>> On 5/06/2010 7:55 AM, |-|ercules wrote:
>>> According to sci.math, the number line consists mostly of "dark
>>> numbers".
>>>
>>> You see, 99.999999.... nearly % of real numbers are not computable,
>>> there is
>>> no program that represents them.
>>>
>>> A computer can calculate ANY digit sequence up to INFINITE length.
>>>
>>> Sci.math will make a minor correction there, a computer can only
>>> calculate
>>> all digit sequences up to ALL finite lengths.
>>> This is like saying a computer can only calculate all digit sequences up
>>> to INFINITE finite lengths.
>>
>> A computer is a finite state machine, so although it may continue to
>> compute forever, its outputs must start to repeat. Any number whose
>> sequence of digits does not appear in that recurring output will never
>> be output by that computer no matter how long you wait.
>>
>> If you don't constrain the computer's storage space, letting it be
>> "inifinite" you still have the problem that the set of storage
>> locations will be countably infinite, and thus the set of all of its
>> states will also be countably infinite, as will the set of all of its
>> possible outputs.
>>
>> Since the reals are not countable, the computer will still omit most
>> of them from its output. That is, it will not be able to compute them.
>>
>> Of course, you can hypothesise a computer that is not subject to those
>> constraints, but that's really just assuming the answer.
>>
>> Sylvia.
>
> Your argument for uncountable infinity is based on the fact that reals are
> not countable?
>
> Perhaps you should freshen up with a nice poll.
>>> The proof of higher infinities than 1,2,3...oo infinity relies on the
>>> fact that there
>>> is no box that contains all and only all the label numbers of the
>>> boxes that
>>> don't contain their own label number.
>
> True or False?

False.

Sylvia.
From: David Bernier on 8 Jun 2010 00:44
|-|ercules wrote:
> "David Bernier" <david250(a)videotron.ca> wrote
>>>
>>> NO BOX EVER CONTAINING THE NUMBERS OF BOXES NOT CONTAINING THEIR OWN
>>> NUMBER
>>> MEANS HIGHER INFINITIES EXIST.
>>>
>>> TRUE OR FALSE?
>>>
>>>
>>> BTW, dozens of famous mathematicians dispute Cantor's proof, so maybe
>>> having a 95iq
>>> is a basis for not following it, but not "following" it is not a basis
>>> for a 95iq.
>>
>> How should one go about defining existence?
>
>
> As in higher infinities exist?
>
> That's very deep and profound, the type of answer that might pop into my
> head in a few
> days time.
>
> Would you rather say "higher infinities" are useful descriptions for
> certain operations?
[...]

I think a lot of the time, in the history of mathematics,
new fields of study arise from trying to answer "popular"
mathematical problems/questions of the day; other times,
it can be through trying to solve applied problems in
science or engineering (or even medicine or genetics).

According to the book
"Georg Cantor: his mathematics and philosophy of the infinite",
by Joseph Warren Dauben, Cantor at first was concerned
with convergence of trigonometric sums, i.e.
Fourier series. That led him to study sets of points
where a given Fourier series didn't converge (or
vice versa).

Cf.:
< http://books.google.ca/books?id=n3t4b6GUlhAC > .

As is well-known, there was Russell's paradox from
around 1900 and other developments ( well-ordered sets,
the use of the Axiom of Choice) which gave rise to
vigorous debates in the early 1900's .

These days, I think mathematicians in general are less
concerned with logic and foundations questions.

One of the most active forums for this is
the FOM (Foundations of Mathematics) mailing
list. The FOM list info page, with a link to
its archives, can be found here:
< http://www.cs.nyu.edu/mailman/listinfo/fom > .

David Bernier
From: Nam Nguyen on 8 Jun 2010 01:07
David Bernier wrote:

> I think a lot of the time, in the history of mathematics,
> new fields of study arise from trying to answer "popular"
> mathematical problems/questions of the day; other times,
> it can be through trying to solve applied problems in
> science or engineering (or even medicine or genetics).

I agree: as well as biology and AI.

> These days, I think mathematicians in general are less
> concerned with logic and foundations questions.

Well, perhaps in general. But in particular, certain new
fields might very much concern with foundations of
mathematical reasoning. Let me give 2 examples.

Suppose we're FOL formalizing biological processes
(mitosis, evolution, etc...). How would we define
"the standard model" of mathematical abstract life,
knowing abstract cells could be defined to be finite
or infinite?

Another example would be suppose we define AI formal
system as one in which some of its theorems would be
syntactically isomorphic to a formal system in which
in turn there's a formula than can't be model theoretically
truth definable. The clause "a formula than can't be
model theoretically truth definable" would require
a close inspection of our current knowledge about
the foundation of reasoning via FOL.


From: Nam Nguyen on 8 Jun 2010 01:25
Nam Nguyen wrote:
> David Bernier wrote:
>
>> I think a lot of the time, in the history of mathematics,
>> new fields of study arise from trying to answer "popular"
>> mathematical problems/questions of the day; other times,
>> it can be through trying to solve applied problems in
>> science or engineering (or even medicine or genetics).
>
> I agree: as well as biology and AI.
>
>> These days, I think mathematicians in general are less
>> concerned with logic and foundations questions.
>
> Well, perhaps in general. But in particular, certain new
> fields might very much concern with foundations of
> mathematical reasoning. Let me give 2 examples.
>
> Suppose we're FOL formalizing biological processes
> (mitosis, evolution, etc...). How would we define
> "the standard model" of mathematical abstract life,
> knowing abstract cells could be defined to be finite
> or infinite?
>
> Another example would be suppose we define AI formal
> system as one in which some of its theorems would be
> syntactically isomorphic to a formal system in which
> in turn there's a formula than can't be model theoretically
> truth definable.

> The clause "a formula than can't be
> model theoretically truth definable" would require
> a close inspection of our current knowledge about
> the foundation of reasoning via FOL.

For adding some clarity to the issue, such formula could be
named "transcendental formula". In English, with some
glossing over, such AI definition would be something like:

An FOL AI formal system is one of which an instance (model)
would discuss about transcendental formulas.


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