Re: Relative Cardinality



Proginoskes wrote:

> But you will need an arbitrarily large number of arrows (or squares) to
> do this, and you don't have that; the set of all Muecken numbers is
> finite, so you can only do arrow notation or squaring a finite number
> of times, resulting in a finite number (no matter how large it may be).

Every bunch of arrows can be abbreviated by another symbol. And so on.
There is no upper limit of values as long as an ingenious being is
interested to express large numbers. You were right, if the rules were
fixed. But they are not.


>
> > I have a limit,
> > namely there is always a (decadic) terminating rational fraction a
> > such that | a - 1/3 | < eps for any eps > 0 which can be expressed.
>
> In the true
> definition of a limit, epsilon is chosen before a.

In normal every-day-use of language there is no difference, in
particular if one adds "always". I did not think it was necessary to
teach you the Cauchy criterion.


> How is the information encoded?

Uninteresting. The one who does it may decide.

> How do you express "potential infinity" in the universe?

By never ending but always finite entities: numbers, time, distance.

> But, as I've mentioned, you still only have a FINITE amount of storage.
> This means there are certain things which you are unable to do (which
> is why not every real number is a Muecken number).

Of course, irrational numbers like pi do not exist.

> What is a "fundamental set"? When is a number "realized"? Is e
> "realizable"?

A fundamental set is a set which realizes that number in one or more
ways. e is not realizable as a number.

> No, it fits your problem precisely. You are confusing the properties of
> a set with the properties of the elements. I showed you this in a
> fundamental way, as an analogy.

I have not a problem. The "set" of natural numbers consists of natural
numbers and nothing else. The numbers have a natural order, contrary to
the elements of your set. The properties which this set N has "as a
set" are uninteresting for my arguing and any mathematics.
>
> > N consists of elements which, contrary to Berlin,
> > 1) are ordered
> > 2) count their initial segment.
>
> Your two properties are only further properties that _can_ be assigned
> to the set; they are not _necessary_ for the set itself.

Your "set" consists of natural numbers which do not have different
magnitudes, implying an order?
When I speak of natural numbers, then I mean in the first line the
order by magnitude. That is the main property of these numbers. The set
is uninteresting, a cheap coat at most.

> > Yes. But not actually infinite.
>
> Infinite means not finite.

Infinite means without end, but every number is finite.

> As in the number of prime numbers,

Each of the numbers is finite. Each set of numbers which really can be
proven to be primes is finite.

> or the
> number of rational numbers, or the number of points in the plane. None
> of these sets is finite, so each of these sets is infinite. Period.
> There is no "actually" about it.

Here you are mistaken: Cantor to Illgens, 16.5.1886: Will man in der
Materie des Unendlichen überzeugend wirken, so muß man vor allem
zwischen dem potenzialen und actualen Unendlichen so streng
unterscheiden, dass keinerlei Verwechslung dabei vorkommen kann.
And here he is right with no doubt.

>
> (When you say a set is "potentially infinite", I am understanding this
> to mean that there is no upper bound on how big the numbers can
> actually be. The term for this situation in standard mathematics is
> "unbounded".)

No, the term is infinite.
>
> > Actual infinity is a number (a whole number according to Cantor).
>
> In some systems, yes, it is allowed to be a number. But when dealing
> with integers, rational numbers, real numbers, complex numbers,
> infinity is NOT allowed to be a number. So for the purposes here, no
> real number, and no natural number, can equal infinity, ever.
>
> (In the surreal numbers, or the ordinals, infinity IS allowed to be a
> number, and it has certain necessary properties. But that's not the
> case here.)

In the surreal numbers even the ordinals can be no-integers. Cantor's
ordinals are always whole numbers, like his cardinals.
>
> > Now draw the graph of a simple function f(n) = n where n represents the
> > natural numbers and f(n) the numbers {1,2,3,...,n}. This function is
> > the diagonal of the first quadrant of the Cartesian co-ordinate system.
>
> (To get that, you would have to let n be any _real_ number. That's not
> important right now, though.)
>
> > As long as infinity is not actually reached by n it is not reached by
> > f(n) and vice versa.
>
> So you would say that this set of points is "potentially infinite", but
> not "actually infinite", right?

Yes. But mainly that the abscissa (x) value cannot get infinite unless
the ordinate (y) value does it and vice versa. Therefore it is absolute
nonsense to talk of an infinite set of finte natural numbers.

> Your set of numbers does not allow the concept of "arbitrarily many"
> objects; you can only count up to H, where H is the largest Muecken
> number. "Arbitrarily many" means that for any number N, there are more
> than N objects. So you can only say that there are more than H 1's, so
> you can't tell the difference between a string of H+1 of them or an
> "infinite" number of them.

Wrong. There is no largest number. If your H is defined by 50 % of all
memory available then I abbreviate it by a symbol using only small
memory and go on.

By the way, if there was a largest encodable number then it would also
apply to mainstream mathematics, because you have no other means to
express large numbers than I. Do you believe that there is a largest
number?


> But there's no way for you to distinguish 0.111... from 0.111...111
> (where there are H + 1 1's, H being the largest Muecken number).

you need to do some off-line thinking for a while.

> But you can't say this, because you have to have some "rule" which
> decides this. And this rule will take one configuration of the universe
> and give you a unique number; there is no possibility for more than one
> interpretation.

Why? We can fix many different languages. There need not be a single
rule. A fundamental set can contain many different objects. 2 = {II,
1+1, mother & father, .., !!,--, sun & moon, me & you}. We can agree
that & connects two objects. We can also agree that & is an object such
that && represents 2.

>
> > You need not write down all the 1's of the prime number 700
>
> 700 is not prime, not in any base.

Really? Are you pretending to not understand again, or don't you?
See the text continued below. I considered the sevenhundredst prime
number of the form 111...111 (if here are so many of that form. Perhaps
we will never know. Perhaps one can never know because it is not
fixed.).

>
> > of the form
> > 111...111. But it is sufficient to encode the information: All digits
> > are 1, the number of digits is 10^25.
>
> But what if 10^25 isn't representable? This encoding won't work in that
> case.

10^10^10^10^10 is representing a number with one 1 and as much zeros as
are written down.
>
> > As i have written down here. Only
> > for numbers which do not follow any simple rule, like pi, we must write
> > down all the digits.
>
> What about e-1 ? How do you express that number?

Not at all. It is not a number.

>
> In mainstream mathematics, there is an established difference between
> the number (the object) and the mathematician (the observer).

That was the same in physics. It is a relative primitive world view.

> In your
> physical description, these two are the same thing, since you're using
> every particle in the universe.

90 % would be sufficient.

> This is the difference between the two,
> and why mainstream mathematics doesn't really have this problem.

Say: It does not yet know to have it.

> > Reality supplies the stuff. A bit of abstraction is fine. But I am
> > sure, there is no ideal circle or rectangle in your head.
>
> Not in my head, but it might be in my mind (which need not be
> restricted to the physical universe).

That may be your opinion. If I consider how the personality of people
changes in case their brain is hurt, I cannot believe in your idea.
>

> > Cardinality was introduced as a measure of the number of elements. And
> > when introducing it, Cantor mentioned something like clear thought. That
> > is also the foundation of my definition.
>
> But your "clear thought" definition has different properties than
> Cantor's definition, and should not be treated the same way, unless you
> know it results in the same concept.

It compares the number of rationals and the number of irrationals. That
is all which is necessary. No other aspects are involved.
>
> > How far does our atmosphere stretch out into the space? Please answer
> > in millimeters.
>
> what determines whether a number is "realizable"?
> If you can't answer that, then your definition is meaningless. Maybe I
> should have asked:

Start with the easy problem: If someone does know the place of n in the
order of the natural numbers, then n does exist. If this is impossible
to kow, then does not exist.
>
> If a_n is defined by:
>
> a_1 = 3.1
> a_2 = 3.14
> a_3 = 3.141
> a_4 = 3.1415
> ...
>
> then a_n is pi rounded to the nth decimal place, which makes
> | a_n - pi | <= 1/10^n.
>
> So I can guarantee | a_n - pi | < epsilon = 1/10^10^100 if I insist
> that n >= 10^100 (maybe plus one).

You cannot realize a fraction a_n for n>10^100 for pi. Nobody can do
that!
>
>
> One thing that I've wondered about (which no one seems to have touched
> upon) is how you do any calculation in your system of numbers. If the
> universe can hold only one number at a time

The universe obviously can hold many numbers.

Regards, WM

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