Re: Calculus XOR Probability

In article <MPG.1ebaa6a691b21d1c98ac72@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

David R Tribble said:

the Inverse Function Rule.

That hoary impossibility again? That is a stupid as TO's perpetual
appeal to his largest finite natural argument.

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

What is a "quantitative set" other than just a set? Until this is
properly defined, the rest is nonsense.

mapped from the naturals

What set of naturals is TO using as his domain for this function. In
defining functions which are to be invertedone must be very careful with
domains and codomains.

using f(n) for n
in !N,

What is "!N"?

and given g(x) s.t. f(g(x))=g(f(x))=x

This is nonsense unless functions f and g both have the same domain and
the same codomain, which is patently not the case here.

(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)

Since A and B are undefined, the above is nonsense.

(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.

What "N" is that? and if it and S are order isomorphic, there is no need
for distinct sets, why not take S = N?



The rule is not itself a
mapping, but a statement regarding the relationship between the size
of the mapped set and the mapping function.

It is not only not a mapping, nor a function, it is not mathematically
recognizable as anything at all.



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.

How is something consistent with a monumental inconsistency?


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.

And it immediately and logically follows that 2 = 1.



So, why would I claim this is the correct solution?

Overweening ego?




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

Not hardly!
.

Relevant Pages

  • 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)
  • Re: Zenkins paper on Cantor
    ... > as least as many reals not in that list as are in the list to start with. ... A bijection between the naturals and reals is not accepted by ... My perception _was_ that you could toss out antidiagonals all day and after ... the mapping must be that way. ...
    (comp.theory)
  • Re: Zenkins paper on Cantor
    ... > as least as many reals not in that list as are in the list to start with. ... A bijection between the naturals and reals is not accepted by ... My perception _was_ that you could toss out antidiagonals all day and after ... the mapping must be that way. ...
    (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: induction vs Cantor
    ... > that creates a new mapping for each natural number. ... > between the naturals and the reals. ... It's a pity that you don't understand what a Proof By Contradiction means. ...
    (sci.logic)