Re: Calculus XOR Probability

Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:

It's increasingly hard to imagine how you expect any mathematical
argument to convince you of something about which you have already
decided "it is simply true".

Suppose I have a set X.

Suppose I define a limit on sequences of X such that lim n->oo {x_n} is
well-defined for all sequences of elements in X.

Suppose I define a function f : X -> R (the real numbers).

Now, suppose as a /result/ of these /definitions/, I can explicitly
provide a sequence {x_n}, such that I can provide a proof that f(x_n) =
1 for all n, but I can also prove that f(lim n->oo {x_n}) = 0.

I'm not claiming at the moment I have neccesarily done this; I'm just
asking you to /suppose/ that I had, for the sake of argument.

I would then conclude that whatever notion of limit you are using is inadequate
for measuring whatever f(x_n) is supposed to be measuring.


f(x_n) isn't "measuring" something; it's a function. Functions simply
yield values in their range, for each element in their domain.

I don;t understand what you mean by "measure". What does the function
f: R->R defined by f(x) = sin(x) + cos(x) "measure"?


Which, if any, of the following would you then claim?

(1) Therefore, "infinite induction" does not hold in this case.

Not when there is another obvious explanation for the error.


(2) Therefore, lim n->oo {x_n} is actually not a member of X, it is
something else.

No, I wouldn't say that.


Then why, in this very post alone, do you say the following things? For
the purposes of the following, I will use "X" to mean "the set of all
sets of pairs in R^2", and "{s_n}" to mean "the sequence of staircases
as sets of pairs in R^2", and "D" to mean "the set of pairs in R^2
defined by {(a,b): b = 1 - a, a,b >= 0}":

* This works well for measuring the diagonal, but not for summing
infinitesimal segments, such as the staircase in the limit, without
some notion of infinitesmals.

Here you claim that lim {s_n} is not in X, because lim {s_n} contains
"infintesimal segments", which elements of X do not; they are solely
sets of pairs in R^2.

* [...] why do you question that result, and not the exact equivalence
between that curve and the diagonal?

Here you claim that lim {s_n} cannot be equal to D in X, solely on the
basis of equivalence by set membership; because D is clearly in X, this
implies that lim {s_n} is something other than a set of pairs; i.e., is
not in X.

* No, it implies that the diagonal is different from the fractal
diagonal.

Here you claim that lim {s_n} is not equal to D in X, because D is a
"diagonal", and lim {s_n} is a "fractal diagonal", i.e., cannot be
expressed soley as set of pairs (thus is not in X).

* No, the usual metric isn't wrong, but you're not really using it on
the staircase, when you ignore the infinitesimal right angles in it.

Here you claim that lim {s_n} is not in X, because lim {s_n} contains
"infinitesimal right angles", and thus is not in X, which contains only
sets of pairs in R^2.

* It's not the "usual metric" that fails. I misspoke. It's the false
equivalence between the diagonal and the staircase in the limit.

Again you claim that lim {s_n} cannot be equal to D in X, because lim
{s_n} is something other than a set of pairs; i.e., is not in X.

* It was never my premise that the staircase in the limit is exactly
the diagonal in every sense.

Here you claim that lim {s_n} cannot be equal to D in X even though
they contain the same elements (pairs); although that is not consistent
with the definition of "A = B" with regards to two sets; therefore lim
{s_n} is not in X.

* You have your pairs in your limit, but not your pairs of pairs which
are required by the dist() metric. It's those pairs of pairs which are
distinguishable from those in the diagonal by always having a
corresponding element equal from one pair to the other.

Here you claim that lim {s_n} cannot be equal to D in X even though
they contain the same elements (pairs); because lim {s_n} is not
actually a set of pairs in R^2, it is a set of pairs of pairs in R^2;
and thus not in X.

* [...] my immediate thought is that the pairs that define the points
in the curve can be replaced by a sequence of pairs that denote the x
and y changes of a series of segments, which in the limit approach the
curve.

Here you declare that lim n->oo {s_n} is equivalent to some other limit
which has a range different from X; which is saying that the limit is
not in X.


(3) Therefore, f(x_n) is actually not 1 for some natural n.

I would say, given that it's inductively provable that it is, then it is.


(4) Therefore, lim n->oo {x_n} is not "the infinite case".

I would say there is an issue between your limit and f(x).


(5) {Insert other possible objection here}.

You are currently claiming "(2)", in our non-hypothetical example of
the staircases.


<snip>

It's not the "usual metric" that fails. I misspoke. It's the false equivalence
between the diagonal and the staircase in the limit.

We are now talking about /one/ thing: the /set/ D = {(a,b) : b = 1 - a,
a,b >= 0} = lim n->oo {C_n}, where C_n is the nth staircase.

In what way is it a "false equivalence" to say that

{(a,b) : b = 1 - a, a,b >= 0} = {(a,b) : b = 1 - a, a,b >= 0}?

Cheers - Chas

.

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