Re: An uncountable countable set

Tony Orlow wrote:

Hmm, it seems to me, Tony, that this post illustrates rather well just
how close to total is your ignorance of what mathematics is, and your
inability to grasp the notion of a formal argument. (So why do I
bother?...)

The theme running through almost all of your comments here is one we
see a lot in JSH arguments. When little children learn arithmetic at
school, it's common to start with positive integers (which form a
semigroup under both addition and multiplication), so lots of things
"can't be done": 3-5, or 11 / 6. Later they learn further concepts,
such as positive rationals, which form a group under multiplication, so
things which previously "couldn't be done" now can. Similarly they
learn about negative numbers, so that now 5-23 can be calculated. In
the context of school arithmetic as a sort of "calculation
engineering", it's reasonable to see this as just learning more and
more powerful ways of "working out the answer". This may well include
what I call "Javascript arithmetic", in which "Infinity" is one
additional value, allowing us to calculate things to do with lenses in
a very useful way.

However, mathematics is the study of formal systems (patterns, if you
like), and of course this includes study of some systems that are
subsets (or embedably isomorphic to subsets) of other systems. When
discussing Fermat's last theorem, a result about the integers, the fact
that there are no solutions to a^3+b^3=c^3 is not contradicted by
writing a=1, b=1, c=cubrt(2).

Anyway,...

imaginatorium@xxxxxxxxxxxxx wrote:
Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx wrote:
Randy Poe wrote:
Tony Orlow wrote:
Virgil wrote:
In article <451bac34@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

If the vase is empty at noon, but not before, how can that not be the
moment that it becomes empty?
Saying that it is empty is quite different from saying anything about a
"last ball". andy does not deny that the vase becomes empty, he just
does not say anything about any "last ball out".
Does that answer the question of **when** this occurs? Of course not.
It does answer the question of "whether" it occurs. "When" is of lesser
importance.
So, you have no answer.
If something doesn't occur, the question "when does it occur"
does not have an answer.
No, but I think the problem is elsewhere, slightly. Is there a formal
definition of what "transition" means? (Not in a nearby pocket "Dict.
of maths." for example)

Seems to me that if you had the graph y = (1 if x<0; 2 if x>=0), and
and associated state transition diagram, then there would be a
"transition" from 1 to 2 "at" x=0. But such terminology does not
capture what the state is "at the point of" the transition, which may
be why it isn't used much. But if a vase has balls in it for values (-1
<= t < 0), I can't see anything actually wrong with saying there is a
transition from empty to non-empty "at" t=-1 and a transition from
non-empty to empty "at" t=0. You need to be careful not to deduce
anything about the _state_ at the two transition values.

After all, consider the graph of y = -x. This is positive for x<0, and
negative for x>0. It's undefined for x=0, but is there not a transition
from positive to negative at x=0?
That all makes very good sense, Brian. I can't see that there isn't a
point of transition, especially when that point is very explicitly
defined to be noon, and that it is at least then, and not until then,
that the vase goes from being non-empty to empty.

Yes, but using this (slightly loose?) "transition" terminology, you
have to be careful that this tells you _nothing_ about the state _at_
noon. Seems to me that all of the following graphs have a "transition"
from positive to negative at x=0, but very different things are true
_at_ 0

"_nothing_"?

Uh, yes, "nothing". What this means is that there is no valid
implication "Some function f has a transition from property P to
property Q at x=0, THEREFORE f(0) = T"

y=-1/x (at x=0 this is undefined)

Or y=1/x. From one side it goes to oo, and from the other, -oo. This is
resolved by application of the number circle concept, where oo=-oo.

Maths isn't about "resolving" things. We are (if you remember) talking
about the unbounded x-y plane, a set including only points (x,y) where
x and y are perfectly ordinary real numbers (porns). oo is not a porn.
[should get lots of hits on this post, eh?] In the field of porns, 1/0
is not defined. Just as cubrt(2) is not defined in the integers.

y = (1 if x<0; -1 if x>=0) : y(0) = -1
Clearly discontinuous - two different formulas

Huh? I said "graphs" above - I might more carefully have said "graphs
of the following functions". The following is a function:

y = (1 if x<0; -1 if x>=0)

Do you disagree? If so would you care to give us your private
definition of "function"? While you're at it, you need to give us your
definition of "formula" too (more shades of Harris...) - of course this
function is discontinuous; who suggested it wasn't? (y=-1/x is
discontinuous too, btw)


f = 1 if x<0; -1 if x>0; purple unicorn if x=0 : well, what can I say?

That's enough. None of those examples pertains to the vase, which is a
very well defined increment of 9 per iteration.

Indeed, I can understand what "increment of 9 per iteration" means in
relation to the vase example. But "increment of 9 per iteration" is not
a function description - for a start, functions don't have
"iterations".


To put it in limit
terms, the "solution" to the vase problem would be equivalent to saying
lim(x->oo: 9x)=0. Ain't the case.

You're trying to jump much faster than you are able. Do you seriously
think there is anyone here who supposes that lim(x->oo: 9x)=0 ?

(I'll carry on with the blue sliver when I have time, which may be in a
day or so)

OK, so the blue sliver. Remind me: do we agree that within an unbounded
x-y plane of porn-pairs, the area between the lobes of the two graphs
y=-1/x and y=-2/x for negative values of x is an unbounded sliver,
going endlessly out to the left and to the top, while getting narrower
and narrower, without ever becoming of zero width? And we agree that if
we shade this sliver with horizontal hatching (all lines y=n for some
constant integer n), there are lots and lots of horizontal line
segments within the sliver, some of which we mentioned explicitly in an
earlier post, but an indefinitely large number of which we could in
principle calculate endpoints for ("I have the time, if you have the
inclination, as Big Ben said to the leaning tower of Pisa").

Can you now consider the following vertical lines in this x-y plane:
x=-5, x=-1.5, x=-7/8, and x=11. I hope it's not too onerous to ask you
how many of the line segments of the horizontal hatching in our sliver
each of these lines intersects with. (I think the answers are pretty
simple, but I'd be grateful to hear them from you.)

Brian Chandler
http://imaginatorium.org

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Relevant Pages

  • Re: An uncountable countable set
    ... Does Han claim that there is any ball put in that is not taken out? ... the vase empties". ... But in order for the vase to transition from not-empty ... If the vase ever became empty, ...
    (sci.math)
  • Re: An uncountable countable set
    ... Does Han claim that there is any ball put in that is not taken out? ... the vase empties". ... But in order for the vase to transition from not-empty ... If the vase ever became empty, ...
    (sci.math)