Re: math : the problem with points.

Mariano wrote:

On Apr 5, 12:40 pm, Julio Di Egidio
<ju...@xxxxxxxxxxxxx> wrote:
G. Rodrigues wrote:
On Sat, 05 Apr 2008 07:08:47 EDT, amy666
<tommy1...@xxxxxxxxxxx> fed
this fish to the penguins:

Tim Little wrote :

(and fishfry about the same )

On 2008-04-04, amy666 <tommy1...@xxxxxxxxxxx>
wrote:
So, what do you actually mean by "beginning"?

the first positive real number > 0.

Does not exist.

Of course this is not a problem specific of the
continuum, since the
rational numbers already have this "defect":
there is
no first
rational number > 0.

If we get into the domain of computable numbers
(actual numbers), and take closed interval arithmetic
as reference, that lower limit (actually) exists and
it is the floating-point limit to underflow,
intrinsic to any specific implementation. It is, in
fact, under all respects, a rational number.

You are using a very, very restricted definition of
what computable
is...

I'd rather call it "sharp"; I'll try to show how:

In general, and IMVH-Though-Frank-Opinion, we won't get this unified Theory until we don't take _machines_ into proper account. A _computable number_ is a h-algorithm, as well as it is a wf-formula, and as it is a c-set. In many respects I guess we could simply call it "a rational number", although we must not forget the foundational role of the "extended containment constraint", along with the dis-paradoxical "empty-set-prevalence", about "where all failing (excepting, unclosed) machines dis-belong" [*].

With some notational license:

Q = the "computables", always constructible (and computable) with respect to a specific theory (and machine);

-Q = the "non-computables", never constructible (nor computable) with respect to any specific theory (nor machine);

-Q = (Entire \ Q) <= (Empty \ Q) <= Empty; thus, -Q = Empty (by containment, zero)
Q = (Entire \ -Q) = (Entire \ Empty) = Empty (by substitution, broad)
Q = (Entire \ Entire \ Q) = (Entire \ Entire) = Entire (by prevalence, sharp)

That is, it seems: -Q = Empty and Q = Entire, satisfying containment and inductively congruent.

(Digression, tentative, with interval vocabulary: the so called "irrationals" in this model rather become "non-degenerate rationals", definition itself congruent, AFAI-Can-Tell, to how we actually construct them. OTOH, Infinity too is _actual_ (constructible and computable in the given sense), and so is the Entire and the Empty intervals, along with Overflow mirroring Underflow. However, endpoints at Infinity, Overflow and Underflow (even at Zero?), though constructible, follow by a costruction which is of a different nature than that of the finite rationals (degenerate or non degenerate intervals with finite-non-special endpoints). Further distinct nature for the endpoints of the Empty interval.)

I've as well tried to show how this concept of _computable number_ can close the domain of _tractability_, and indeed none-of-that that-there-is out-of-that _domain_, apart from the _empty set_ (which, ultimately, happens to be "us", the _actual non-existent_), _actually exists_, so that, for instance, _intelligence_ is out of this reach as in so far it is within "us".

Indeed, that means something quite interesting is missing from our so defined "domain of tractability", though *that* _something_, up to here, cannot be talked of formally (Socrates-in-Ludwig docent). The enclosure brought by a "full" definition -I can only imagine- revolves around an even more fundamental "stipulation".

Julio

[*] Cfr. TOA: http://mathforum.org/kb/message.jspa?messageID=6128704

P.S. Sorry, an OT maybe, but it keeps buzzing in my head... it's not QED, it's QDE, and it stands for Quod Demonstrandum Est (What To-Prove There-Is): straight BNF.
.

Relevant Pages

  • Re: math : the problem with points.
    ... this fish to the penguins: ... continuum, since the ... rational numbers already have this "defect": ...
    (sci.math)
  • Re: math : the problem with points.
    ... this fish to the penguins: ... (and fishfry about the same) ... that lower limit exists ... Those who let computers do their thinking deserve what they get from ...
    (sci.math)
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