Re: An uncountable countable set

Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx wrote:
Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx wrote:
Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx wrote:
Tony Orlow wrote:
David Marcus wrote:

<snip>

Suppose I define the following function, referring to sliver-1, which
is the area between y=-2/x and y=-1/x for x<0. ("sleight" stands for
'sliver height', not 'sleight of hand'...)

sleight(x) = -2/x +1/x for x<0; 0 elsewhere

Uh huh. For x<0 as opposed to x>=0. No declared point of discontinuity
there....

OK. Let's see if it's possible to understand what, if anything, you
mean by "function".

Do you agree that the graph of y=-1/x for x < 0 is one lobe of a
hyperbola?

Do you agree that the graph of y=-2/x for x < 0 is one lobe of another
hyperbola?

Do you agree that in the unbounded x-y plane, these two lobes define a
"sliver", a boomerang-shaped area, extending indefinitely 'left' and
also extending indefinitely 'upward' (using these directional terms in
the sense of looking at a conventional graph)?

Do you agree that for any simply-connected area (think that's the right
term) within the x-y plane we could consider the function that maps x
to the vertical measure* of the area at the particular x value?

( * a term I've made up. If you don't understand ask; if anyone knows a
proper word, please tell me)

By way of a different example, consider the circle radius 1, centre (0,
0), and find its 'height()' function. For any value of x outside the
range (-1, +1), the vertical measure is zero, because, obviously, the
circle only extends horizontally from -1 to +1. Within that range, the
vertical measure is equal to the height of an ellipse centred on the
origin, of width 2 and height 4, so (if I calculate correctly) the full
function is given by:

height(x) = 2 * sqrt(1-x^2) for -1 < x < 1
height(x) = 0 otherwise

Please tell me: is this a function? Is it a continuous function? If so,
does it have a "declared discontinuity"?

You might like to do the same for the function height() of a rectangle
diagonal from (0,0) to (3, 57).

If you somehow claim that there _is_ no function representing the
height of the sliver at a particular value of x, you really need to
give us your definition of "function".

If you agree there is such a function, why not try to write it down?

You may or may not agree that this function is discontinuous - in any
event, please explain whether my description above of the hyperbola
lobes and the "sliver" has already included a "declared discontinuity".
If not, does that mean there might be different ways of writing the
same function, possibly some including a "declared discontinuity",
others not.

<snip>

On the contrary, the process *at noon* is completely well-defined.

Then how come no one can say what happens "at noon", which doesn't
happen "before noon"? You're stretching to the point of breaking.

No, I'm slightly lost. Don't understand the relation between the
comma-separated clauses of the first sentence.

A
ball is inserted in the vase or removed from the vase only at a time
that is -1/n for some pofnat n. There is no pofnat m such that -1/m =
0. Therefore no ball is either inserted or removed at noon. (This
really is elementary, you know.)

Well then, nothing can change at noon that was true at every time before
noon, when there is a growing positive number of balls in the vase. What
changed at noon?

Every time *before* noon was a time at which a ball was still to be
removed. Give me a (real, genuine, numerical) value of a time before
noon, and I will give you the number of a ball that has yet to be
removed from the vase. *At* the time noon, there is no ball that has
yet to be removed. (In normal logic, this means the vase is empty.)

Nothing, since nothing happened "at" noon.

No ball movement, no.

So, the
number of balls in the vase at noon is growing is growing,

No, the number of balls at any time, however small, *before* noon is
growing. And the smaller the time before noon, the crazier it is
growing.


... since the distinction between
iterations disappears. How do you know there are countably many
iterations, and not some uncountably number? You don't.

Of course I do.

Ah, Zeno told you....

Try being less obnoxious. In the end you might make yourself look less
silly.

The problem explicitly says balls with natural numbers
(pofnats) on them.

And this set ends where? Nowhere. Well, actually, at noon. Isn't that a
tad artificial, and somewhat contradictory?

Does the infinite series (gosh, it's amazing, but I believe there are
tiny fragments of mathematics you have actually managed to grasp) 1 +
1/2 + 1/4 + 1/8 + 1/16 + .... "end" anywhere. No it doesn't. But if you
represent this as an unending sequence of blocks laid in a row, the
blocks do not extend indefinitely, ever though there is no last one of
them.



The sequence either of insertions or of removals is
immediately mapped onto the pofnats.

The sequence of both intertwined amounts to a linearly increasing sum,
when kept in their stated order.

There can - by definition - only
be a countable sequence of pofnats. (Actually, in non-mumbo-jumbo,
"uncountable sequence" is a contradiction in terms.)

That definition is bunk, when discussing sequences where there are
clearly infinitely-indexed iterations, such as occur at noon.

Oh, dear, oh dear. "Occur at noon"? Means what? There are no
"infinitely-indexed iterations", because no ball is ever put in the
vase unless it has a pofnat written on it.

DO YOU UNDERSTAND THAT?


Then absolutely nothing happens at noon, since that would require
infinite n, to change the fact that, at every moment from 11:59 until
then, there were a nonzero number o balls in the vase.

Sorry, I'm lost. Nothing happens at one time to change the fact that at
another time something was true? No of course it doesn't.



When only the
pofnats are involved, so only are finite times before noon, during which
this fact holds true. Do you have a refutation of that argument,
specifically?

No, of course not, because it's true. For any time *before* noon, there
are balls yet to be removed. Yes, of course the number of balls is
increasing like crazy. Yes, of course the limit to the number of balls
in the vase as t -> noon diverges (increases without limit). But no,
there are no balls in the vase at noon. You admit (I think) that there
are no balls with actual genuine pofnats written on them (because of
course, even you have to agree that they have been removed). So we are
left with some ethereal "unidentified, and unidentifiable-in-principle
pseudo-pofnats" supposed to leap in just to make the function behave in
conformity with your high-school-limited intuition. Hmmph - you then
attempt snide remarks about "phantoms" to refer to things your
high-school intuition hasn't prepared you for.

<snipabit>

Your point escapes me here. Is maths restricted to the activity of
"having an equation". (I can see that school maths might look that
way...)

Mathematical evaluation involves deriving a value, generally from other
values, generally formulaically.

It does?

You throw all the naturals in a bag and pretend you have some
specific number of them,

No I do not. It is only _you_ who talks of "specific infinities" and
various other nonsense in which you pretend "Big'un" (or whatever it is
at the moment) is part of finite arithmetic.

Oh. What was Aleph_0 again?

You have, I'm sure been told dozens, if not hundreds of times - Aleph_0
is the name for the cardinality one might explain to children as "you
can count, and reach any of them, but the counting never stops".

Aleph_0 is very definitely not "part of finite arithmetic".

Brian Chandler
http://imaginatorium.org

.

Relevant Pages

  • Re: An uncountable countable set
    ... discontinuity" to which different rules apply? ... There is a discontinuity in the number of ball s in the vase every time ... the differing numbers of balls in the vase at different times to make ... If by "graph" you mean the sliver, ...
    (sci.math)
  • Re: infinity
    ... >>> the vase keeps growing as you approach noon. ... the algorithm which describes the filling of the vase with balls ... Start with an empty vase. ... we try to connect some logical reasoning (about putting ...
    (sci.math)
  • Re: infinity
    ... the set of balls in the vase at state E ... >> consists of a finite number of sets or an infinite number of sets. ... The sum of an infinite series does depend on the number ...
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  • Re: infinity
    ... by definition the vase is empty at state E. ... the set of balls in the vase at state E ... When you say "There is no change whether or not there are an infinite number of sets I_n", ... given by the sum of an infinite series. ...
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  • Re: An uncountable countable set
    ... -1/n, where n is a natural number, there are balls in the vase. ... Let S be the set of naturals on balls removed before noon. ...
    (sci.math)
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