Re: Calculus XOR Probability

David R Tribble said:
Tony Orlow wrote:
It is an axiomatic statement in my system that there is a unit infinity N
which is the length of the real line, and therefore the number of unit
intervals, or naturals, on the real line, as well as being the number of
nipotent infinitesimal reals in the unit interval.


Virgil said:
A "number" is nilpotent only if it is not zero but some natural power of
it is zero (i.e., x*x*...*x is zero for some mathematically finite
number of multiplications).


Tony Orlow wrote:
perhaps "nilpotent" isn't exactly the right word, but it's not far from
correct. If you divide the unit interval into N subintervals, the endpoints of
those subintervals are infinitesimally close, so between them is only one real
number.

Well, you can divide [0,1] into N open intervals, e.g.:
I(1) = (1/2, 1)
I(2) = (1/3, 1/2)
I(3) = (1/4, 1/3)
...
I(n) = (1/(n+1), 1/n)

Between each pair of adjacent intervals I(n) and I(n+1) is a single
real point 1/(n+1).

But this is not to say that there are only N possible intervals in
[0,1]. It's obvious that each interval I(n) above is "wider" than one
point, and therefore contains an infinite number of real points.
There are, in fact, c possible intervals (which is quite a lot more
than N).

Here's another way to look at it. Consider the following list of
reals:
x(0) = 0.0
x(1) = 0.01
x(2) = 0.012
x(3) = 0.0123
...
x(12) = 0.0123456789101112
...
x(n) = x(n-1) + n/10^(floor(log10(n))), (or something like that).

I've just defined N real points in the interval [0.0, 0.01234...), and
even though I've used up all of the naturals in N, I'm nowhere near
exhausting all the reals in [0,1]. I haven't even used up all the
reals between 0.0 and 0.01, such as 0.005.

Yet another way to look at it is to realize that even though we
can map all the naturals to all the rational numbers in [0,1],
our mapping still omits an infinitude of irrational numbers;
between any two rationals in our list, there are an infinite
number of irrationals that are not included in our list.

In fact, you can't find any mapping between the naturals and the
reals in [0,1] (or any other interval) that includes all of the reals
in the interval. We therefore conclude that there must be more
reals than naturals, even the reals in a small interval like
[0, 0.01234...].

Your theorem states just the opposite, that there is a one-to-one
mapping of naturals in N to all the reals in [0,1]. Perhaps you
could show us a mapping that proves your theorem?

Doing so, by the way, would also provide a well-ordering for
the reals, as a nice side effect.



Okay, David, but the well ordering that results is just Ross's contiguous
degenerate intervals and nilpotent infinitesimals. Remember, first, that the
"N" above is the number of hypernaturals, finite and infinite, and the length
of the real line, not the set of finite naturals, okay? It's an "uncountable"
number, which I have now taken to calling BigOne, to avoid such confusion. So,
given BigOne naturals from 0 to BigOne-1, we simply apply the formula f(n)
=n/BigOne, and each natural n maps directly to the nth infinitesimal interval
within [0,1). That may seem simplistic, but there is no other way to map them
than with a function involving infinity, besides the H-riffics, that I can
think of, and it works perfectly with the Inverse Fucntion Rule.

--
Smiles,

Tony
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, ... number of reals on the line. ... to the subsequent logic that claims such a set cannot have infinite values. ... standard orderings, since sets in general don't come with little tags ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, ... number of reals on the line. ... to the subsequent logic that claims such a set cannot have infinite values. ... don't have a definition for an arbitrary set of its "standard ordering" ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... you had said that the existence ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... The equality between element count and value in the naturals holds in the infinite case, showing clearly that the set is only actually infinite when it contains infinite values. ...
    (sci.math)
  • 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: Uncountable sets in CZF?
    ... naturals bijectively to the reals. ... relatively modern response to the vagaries of infinitesimals of Newton ... from the naturals to the reals was the definition of the Natural/Unit ... properties similar to EF in monotonically mapping. ...
    (sci.math)
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