Liste des Groupes | Revenir à c theory |
On 28/03/2024 16:53, olcott wrote:I disagree yet prior to my infinitesimal number system there was noThat is a property of the numbers 0.9, 0.99, 0.999 and soYes.Yet it seems that wij is correct that 0.999... would seem toThat /cannot/ be correct in the "real" numbers, in which there
be infinitesimally < 1.0.
are no infinitesimals [basic axiom of the reals]. In other systems of
numbers, it could be correct,but that will depend on what is meant byApproaching yet never reaching 1.0.
"0.999..",
on arranged as a sequence [and of many other sequences], but is not
/yet/ a value. Not until you explain what you mean. In conventional
mathematics, it is usually taken to mean the limit of that sequence
expressed as a real number, where "limit" has a precise meaning as
discussed and formalised in the 19thC. That limit is 1.
Not a tinyYes we all agree that 0.999... never gets to 1.0.
bit less than one, not some new sort of object, but 1, exactly. You
and Wij may find that surprising, or even nonsensical, but it is what
the mathematics tells us from the axioms of the real numbers and from
the definition of "limit". If you want the answer to be different,
then that must follow from different axioms and definitions. Until
you and/or Wij tell us what those are, there is nothing further useful
to be said.
Been there done that.Again, there are no infinite or infinitesimal real numbers, soand note that if you appeal to something that mentions limits
to define this, then you have to explain how infinite and infinitesimal
numbers are handled in the definition.
if you want an infinitesimal in your answer, it is incumbent on you to
explain what you are using /other than/ conventional maths.
That is inconsistent. They are exactly the same points up untilI didn't ask what "geometric point" and "number line" are, butOf course by geometric point I must mean a box of chocolates and byOne geometric point on the number line.Until you describe the axioms of what you mean by "geometric
[0.0, 1.0) < [0.0, 1.0] by one geometric point.
point" and "number line", this is meaningless verbiage. Give your
number line I mean a pretty pink bow. No one would ever suspect that
these terms have their conventional meanings.
what axioms you think they have. In conventional mathematics, those two
intervals have /exactly/ the same measure even though they are not
exactly the same sets of points.
If you get a different answer [andI am just showing EXACTLY where the conventional notions lead.
have not simply made a mistake], it /must/ be because you are using
different axioms. What are they?
I just proved otherwise. [0.0, 1.0] has all of the same pointsBut "one geometric point" has measure zero. Not "never actuallyaxioms, and it becomes possible to discuss this. Until then, we areYet it is never actually zero because it is possible to specify a
entitled to assume that you and Wij are talking about the "traditional"
"real" numbers [as used in engineering, etc.] in which there are no
infinitesimals, and so the only interpretation we can make of the size
of "one geometric point" is the usual "measure", which is zero.
line segment that is exactly one geometric point longer than another.
[0.0, 1.0] - [0.0, 1.0) = one geometric point.
zero", but actually and really zero. Unless, that is, you are using someConventional interval notion proves otherwise.
different and as yet unexplained axioms/definitions. Which are ...?
Les messages affichés proviennent d'usenet.