Re: Calculus XOR Probability

David R Tribble said:
Brian Chandler said:
Sometimes you seem to agree that (for example) the sequence of pofnats
(0, 1, 2, ...) has no end, and almost always you agree it has no
well-defined end. You say this means it "has no well defined size", yet
you (now) say you advocate "a more numeric approach to infinity", which
appears to mean you insist it must have a "size". It doesn't bother you
that these two claims appear to contradict each other?


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.

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.


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, then we can map each of the hypernaturals to a corresponding real in
[0,1] using a mapping function f(x)=x/Big'un. If y=x/Big'un, then x=y*Big'un,
and that's the inverse of the mapping function. So, over the interval
[0,Big'un] we have floor(Big'un*Big'un-Big'un*0+1)=Big'un^2+1. If we exclude
either Big'un or 0, we have one fewer reals, and exactly Big'un unit intervals
of reals, containing Big'un^2 reals on the entire real line.

So, why would I claim this is the correct solution? Because it agrees with the
normal interval-based geometric interpretation of multiplication. If we have
intervals of n units, and place m of them end to end, we have an overall length
of n*m. So, if we have Big'un reals per unit interval, each occupying a unit
infinitesimal Lil'un-length subinterval, and we have Big'un such intervals, we
would naturally multiply the number of reals/unit times the number of
units/line to get Big'un*Big'un reals/line. So, the Inverse Function Rule, in
combination with the definition of Big'un as both the length of the line in
units and the number of reals per unit, produces results in total agreement
with the most rudimentary use of multiplication of x*y to determine the number
of units contained in a set of x intervals of y units.

I hope you got all that. You miught want to mull it over a bit. :)
--
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, ...
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  • Re: Calculus XOR Probability
    ... if we say that there is a specific infinite number of ... unit intervals on the real line, then we have a cohesive system consistent with ... But this mapping denumerates only some of the reals (0,1] and omits ...
    (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... ...
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  • 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 ...
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