Re: An uncountable countable set

stephen@xxxxxxxxxx wrote:
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Randy Poe <poespam-trap@xxxxxxxxx> wrote:

<snip>

What is the number of the ball which, when removed,
makes the vase empty?
I know the kind of nonsense you will spout in answer to
those questions, but the answers within our axiom system
are: (1) there is no t<noon which is the moment just
before noon. For any t<noon, there is t < t' < noon.
(2) There is no such ball.
Here are the Tony gobbledgook answers:
(1) noon - 1/oo
(2) Ball number omega
In TO-matics, one can confidently give an answer like
number 2 despite the fact that one can also agree
that no ball numbered omega is ever put into the
vase.
In TO-matics, it is also possible to end up with
an empty vase by simply adding balls. According to TO-matics

..1111111111 = 1 + 1 + 1 + 1 + ...

and ..1111111111 + 1 = 0

So if you just keep on adding balls one at a time,
at some point, the number of balls becomes zero.
You have to add just the right number of balls. It is not
clear what that number is, but it is clear that it
exists in TO-matics.

But in mathematics and logic, we don't get to
keep a set of self-contradictory assumptions around,
only using the ones we want as needed.
- Randy
Where's the fun in that? :)

Stephen
You drew that from my suggestion of the number circle, and that ...11111 could be considered equal to -1. Since then, I looked it up. I'm not the first to think that. It's one of two perspectives on the number line. It's either really straight, or circular with infinite radius, making it infinitesimally straight. The latter describes the finite universe, and the former, the limit. But, you knew that, and are just trying to have fun.
Tony
I am just pointing out that according to your mathematics
that if you keep adding balls to the vase, you can end up
with an empty vase. The fact that other people may have considered a number circle does not change the fact that the number circle implies that if you keep on adding balls, eventually you will have zero balls.
That's a bastardization of the concept. There are two ways to look at the number circle, and you are combining them in mutually contradictory ways.
How is it a bastardization of the concept? You claim that
1+1+1+1+ ... = ..11111111

I wouldn't put it that way. ...1111 can be interpreted as the largest binary natural, if you claim all bit positions are finite.

I do claim that all bit positions are finite, but there is no
largest binary natural. That is something you have invented.


That is what that binary string would represent, but of course it's not any exact number, being a boundless string.

So this "largest binary natural" of yours is not equal to 1+1+1+...
for some number of 1's? In other words, it is not in the
chain of successors? In what sense then is it a "natural number"?

You can think of it that way if you want, but since this number doesn't exist as any specific quantity, I wouldn't try to do any math with it. That's why the concept of omega leads to so much hocus pocus.


and that
..11111111 + 1 = 0

...11111 can be interpreted indeed as -1, as is done every millions of times per microsecond all over the world in computers.

I know of now computer that can hold ...111111 let alone
intreprets it as -1. Remember, ...111111 is an unending
string of 1's.

If you fill the 2's complement register with 1 bits, it's equal to -1. This is true for any size register, and can be assumed true for an infinite register in that context.


Those are two different interpretations of -1. They aren't really compatible, as far as I can see, although Ross used to like to point out that temperatures below 0 Kelvin were somehow hotter than any positive temperature. There could be a connection.

The moment you mention physics or Ross it is clear you are
just rambling.


If you read what I wrote, it's on-topic.

Why does that not apply to balls in the vase? Each ball
is a 1. If a add balls, I add 1's. Do I not eventually get 0?
If it does not apply to balls, what does it apply to,
and how do you determine when it applies?


They are two different interpretations of the string. Do you honestly think that I think sum(x=1->oo: 1)=0? For god's sake, how long have I been saying it's infinite, and that's why any infinite set of naturals would have to include infinite naturals? Oy. Like I said, you're combining concepts that are mutually contradictory, from two different number systems. And, people complaint hat I try to use normal operations normally on infinite numbers....

You are the one who is combining things. In "my" system ...111111 does
not exist. Yet it does in your system, and it apparently is
a natural number, and it apparently equals 1+1+1+.... (does it
not need to be a successor of a successor of a successor and
so on of 1), and it apparently sometimes equals -1. Can you clearly and concisely state the rules that govern ...11111111? If not,
please never mention it again.


....1111 in binary means a 1 in every bit position, the value being sum(x=0->n: 2^x). The question is, for which n does the value become infinite? It is finite for all finite bit positions. Unless n becomes infinite, the sum is finite. So, if you insist that all bit positions are finite and the string countable, then the value is countable too, and finite. It's not a specific value, really, because there is no specific n when the bit positions are defined as all the "finite" positions.

So why is it okay to end up with zero balls, when you never remove
any at all, but it is not okay to end up with zero balls when
each ball is clearly removed at a definite time?
Because the model of the number circle where all strings are positive is incompatible with the number circle where any string with a left-unending string of 1's (in binary) is negative. Duh.
That seems to be a non-answer. The most I can glean from
that is that the number circle is not relevant to the balls
in the vase problem. Is the number circle relevant to anything? And
how does one determine when it is relevant? If it is not
relevant to anything, why did you bring it up in
the first place?

There are an number of topics going on here besides the crazy vase. It's relevant to number systems, and came up initially with my suggestion that ...1111 represents the largest finite if all bit positions are finite. I only mentioned the alternative interpretation of ...1111 being -1 as a sidebar. It's an *alternative* interpretation of the *string*.

Okay. So the number circle is not relevant to this problem,
and ...11111 is not equal to -1 in this context. I have no
problem with that. However elsewhere you seemed to be arguing
that there was only one correct system, that there was a reality
to numbers that must be obeyed. Of course the modern position
is that the axioms are the only thing that must be obeyed,
but one is always free to choose different axioms.


Languages are tools used for math and other communication. Number systems are very structured languages, of which there are a wide variety. There is no contradiction in stating that the same string of symbols may mean different things in different number systems. There is a contradiction in saying that the real line is twice its own length.

Like most of the other cranks, your "system" is only
usable by yourself, as the only way to know when
one of your "rules" applies is by asking you.

Not if you read more carefully.

So apparently sometimes 1+1+1+1+ ... = ...1111111
and sometimes ...1111111+1 = 0 but at other times
they equal something different and there appears
to be know way to know which is which.

Stephen


Sum(x=1->n: 1) = n.
"...1111" can mean a number of things.

Tony

Yes, the string can be used to represent a number of things,
but within a single system it should only represent a single
thing.

Yes, it cannot be the largest positive and -1 without causing a paradox or contradiction. That doesn't necessarily mean that our understanding isn't at fault, and that the two interpretations can coexist, but for the time being, consider them mutual alternatives.


In standard mathematics, "..111111" does not represent anything.
In the p-adics it can represent something. I believe it is
equivalent to -1 in the 2-adics. It is something different
in the 10-adics. I still do not know what it represents in your system, or whether or not your system has a number circle or not.

Yes, in the n-adics, the string of digits n-1 can be considered equal to -1 and arithmetic can be preserved. So, if ...9999 is -1 in 10-adic and that infinite string is allowed, and can be counted down (has negative successors), then perhaps strings such as ....11111 can have meaning in 10-adics as well. What would that meaning be?


Of course the real gist of all this was that I still
find it odd that someone who thinks the number circle
is intuitive, that overflow can occur in the natural numbers,
would somehow object to an increasing sum ended up as zero.

Stephen

In order for that to happen, one would have to make the transition from oo to -oo through succession, but one will never get to that "point". Still, it exists conceptually in the number system, which indicates there's a concept there. The number circle leads to concepts regarding the shape of space and time. It's intuitive in the context of the Tao, and it's implemented in the computer, using 1's and 0's, like yins and yangs. Intuition is like that. :)

Tony
.

Quantcast