Re: Calculus XOR Probability

David R Tribble said:
Tony Orlow wrote:
It doesn't bother me that they appear to YOU to contradict each other. I have
repeatedly said that the unboundedness of the finites poses problems for
measuring the set of finite naturals, and that no real size can be attributed
to this set. However, if we say that there is a specific infinite number of
reals in each unit interval, and that there is an equally infinite number of
unit intervals on the real line, then we have a cohesive system consistent with
rules governing finite sets of such values, namely, the Inverse Function Rule.


David R Tribble said:
If your "Inverse Function Rule" means that for every "unit interval on
the real line" corresponding to some natural k that there is an
corresponding "inverse" real x in each "unit interval" such as (0,1],
then this is simply the mapping:
x = 1/k for all k=1,2,3,...
k = 1/x for all x in (0,1].

But this mapping denumerates only some of the reals (0,1] and omits
a much larger number of them completely (e.g., x=2/3). So you can't
use this mapping to say that the number of reals in a unit interval is
the same as the number of unit naturals on the real line.


Tony Orlow wrote:
No, David, you misunderstand the Inverse Function Rule. Here it is again.

Given a quantitative set S mapped from the naturals using f(n) for n in !N, and
given g(x) s.t. f(g(x))=g(f(x))=x (g is the inverse of the mapping function f)
that the size of the set S between values A and B is floor(g(A)-g(B)+1) (I
think I remember that correctly heh). This works for all finite sets mapped
from some finite set of naturals, as long as N and S are order isomorphic. The
rule is not itself a mapping, but a statement regarding the relationship
between the size of the mapped set and the mapping function.

Now, the equality between the number of reals in the unit interval and the
number of unit intervals on the real line does not directly follow from this
rule, but it's consistent with it. If we declare axiomatically that the number
of reals in the unit interval is Big'un, each occupying 1/Big'un=Lil'un space
within that interval, and the number of unit intervals on the real line is
Big'un, ...

By doing that you've defined something you call "the real number line
with unit intervals" that is entirely different from the standard real
number line of mathematics. This is because the number of reals
in a "unit interval" (e.g., [0,1]) is c, but the number of "unit
intervals on the real line", in other words, the integers, is Aleph_0,
and c > Aleph_0.

Not really. The number of hypernaturals is as uncountable as the reals in
[0,1], even according to nonstandard accepted hyperreal systems.



... then we can map each of the hypernaturals to a corresponding real
in [0,1] ...

You are attempting to define a "real number line" that contains
infinite reals, and then declare that it is the same as the real number
line of standard arithmetic. You're also attempting to mix the
nonstandard hyperreals in with the standard reals, and then declare
this mix is the real number line. Both of these attempts are wrong.

Wrong, morally? Wrong, how? The standard system only concerns itself with
finite n. So what? Is the real number line finite? If so, how long is it? I am
attempting to create a system that doesn't make me wretch, so if I end up
mixing different concepts that don't traditionally play together, well, whaddya
expect?


If you're going to invent terms like Bu and Lu and employ the
hyperreals to do it, then you have to state quiet plainly and obviously
that you're not talking about the standard real number line, but about
something else.

I think I've been pretty transparent about that, although I wasn't aware that
the size of the continuum was consider countable. Where is this axiomatically
stated that the real line has no two points infinitely distant from each other?
Let me get my crosshairs on it...



I hope you got all that. You miught want to mull it over a bit. :)

I understand quite well what you're attempting to define. But it is
not the real number line of standard mathematics. It is probably
not compatible with the nonstandard set of hyperreals, either,
because they don't act the way you think they do.


I rather think they will mesh well with Robinson's hyperreals, but I'll have to
research that. We'll discuss that eventually.

--
Smiles,

Tony
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... But this mapping denumerates only some of the reals (0,1] and omits ... What set of naturals is TO using as his domain for this function. ... and the number of unit intervals on the real line is Big'un, ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... rules governing finite sets of such values, namely, the Inverse Function Rule. ... But this mapping denumerates only some of the reals (0,1] and omits ... number of unit intervals on the real line does not directly follow from this ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... Tony Orlow wrote: ... Show us your definition of "infinite length". ... last of a set of contiguous unit intervals with a 1-1 correspondence to ... as the unit intervals are in 1-1 correspondence with the reals in ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... For instance, if there are Big'un reals in [0,1), there are 2*Big'un reals ... How many rationals are there in [0,1)? ... Divided by Big'un unit intervals overall, ... the concept of cardinality can do for you... ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... For instance, if there are Big'un reals in [0,1), there are 2*Big'un reals ... How many rationals are there in [0,1)? ... Divided by Big'un unit intervals overall, ... the same as the number of rational expressions per unit ...
    (sci.math)
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