Re: Relative Cardinality



Proginoskes wrote:

> > I call 10^10^10^10^10 a number. In fact it is one of the smaller
> > numbers.
>
> You're trying to have your cake and eat it too, here. You want to say
> that you can make arbitrarily large numbers, but you also want to say
> that any number that requires more than 10^100 decimal digits isn't
> really a number.
>
No! No! No! There are numbers, perhaps even irrational ones (I have not
yet figured it out completely) which have never ending n-adic
expansions. One example is 0.333... . We can determine each digit and
know all digits simultaneously.

> Let's define a WM-number to be one that can be expressed by writing
> digits on each piece of matter in the universe, and that there are M
> particles of matter in the universe. (We've been saying M is 10^100,
> but its exact value isn't important.)

The digits need not be written down, if there is a simple rule to
describe them.
>
> THEOREM. There are only a finite number of WM-numbers.

That is correct. There are at most 10^100 different elements of any set
in the universe.
0.333... may have infinitely many digits 3. But the number of different
numbers like 1/3 is less than 10^100.

But that doe not imply a largest number and it does not imply that the
numbers in the set of 10^100 numbers remain always the same. I know it
is a heretic approach, but numbers can be created and can be abolished.
This form of existence has not yet been defined. There are "numbers"
however, which can never be created like floor(pi*10^10^100).

> I mean the set {n+1, n+2, ...} is infinite.

It is potentially infinite. But the set cannot be actually infinite
unless n + omega is reached, i.e., unless there are actually infinite
elements.
>
> > Or easier: Use the unal system, where 7 is represented by IIIIIII.
> > There are infinitely many natural numbers, but thee are only finitely
> > many strokes?
>
> To express ONE natural number, you're allowed a finite number of
> strokes. However, that number of strokes can be a big number -- I'm not
> stopping you from writing down 10^10^100 of them.

As long as the number of strokes is finite, the number n is finite.
There is no actually infinite set.

> > You did not understand my definition. Please refrain from discussing
> > it. The numbers have no upper limit.
>
> Okay, refresh my memory: what IS your definition of a WM-number?

A real number does exist if it is possible to determine its order (<)
with respect to any other existing real number. An n-adic
representation of the two numbers to be compared is in any case
sufficient, if all digits of both are known.

> 8-) means a joke, something not to be taken seriously.

Thank you.

> How much information is too much information?

More than 10^100 bits.
>
> > If I know that all their digits are 1, then this is sufficient.
>
> No, you need to know how many there are. Otherwise there's no
> distinction between 11 and 111. They are both geneated by the rule:
> "All the digits are 1." If you want to differentiate these two numbers,
> you need more information.

I meant the number 0.111... . But you are right, if only a limited
number of 1 is followed by zeros then it is necessary to know how many
1 there are.
>
> BTW, if you use the rule "All the digits are 1", this needs to be kept
> in your head, which is made up of matter, which means it's part of the
> digits written on the particles.

Those digits 111... need not be represented by particles, only the rule
"All the digits are 1" which does not consume much matter.

> There is no definition: You show up "at work", and all those 1's are on
> all particles of matter. What number do they represent?

In order to represent a number the particles representing it must be
put in some order like the digits on a *** of paper or on this
screen. The information for setting up this order, however, does not
consume much matter.
>
> > > This means that your system of recording numbers on particles is not
> > > "well-defined", because you can get more than one number out of a
> > > single representation.
> >
> > 111 can be interpreted as 3 or as onehundred and eleven. It must be
> > definied what is meant.
>
> How do you make the definition? Or can 111 mean 3 today and 111
> tomorrow? Where's the information which tells you which is to be used?
> (Note that it must be written down in the universe somewhere, even if
> it's in your head.)
>
> In other words, suppose I offer you a large monetary reward if you can
> tell me what 111111111 means, based on your definition. You have one
> guess; what do you say?

We have a definition in decimal representation. 111 usually means one
hundred and eleven. We can also easily use definitions like 10^20^30.
However: We have to use a very small amount of the matter of the
universe to set up rules. That is not a new discovery and need not be
discussed in depth when we ponder about the existence of numbers.

> You've shown a part of mathematics "must work inside of the universe",
> but there are other parts which _don't_ have to be a part of the
> universe. What is the phyisical-world interpretation of (first order)
> logic, for instance?

Logic belongs to the stuff which is present in your consciousness which
is present in your head which is made up of matter. No matter ==> No
consciousness ==> No logic.
>
> (Geometry can't measure land, by the way, since it assumes that
> rectangles, triangles, etc., are perfect, which is not the case of
> objects in the physical world, which consist of a finite number of
> particles.)

All application of mathematics to physics (and to mathematics itself,
which is but a small part of physics) is approximate, unless counting
is concerned, which is an exception.

> What exactly is the mechanism that keeps sqrt(2) from existing? Because
> you can't have an object with length sqrt(2)? In that case, 1 doesn't
> either, because there is always error in any measure.
>
A real number does exist if it is possible to determine its order (<)
with respect to any other existing real number. An n-adic
representation of the two numbers to be compared is in any case
sufficient, if all digits of both are known. sqrt(2) and the same
number with the digit number 10^100 exchanged by 5 cannot be put in <
order.

> So maybe you should define "numbers" to be intervals,

No.

> Maybe there isn't a contradiction, but I thought there was one due to
> you using the word "numbers" to mean more than one thing. (This also
> happened when you used the word "cardinality" in your original post
> instead of using another term.) Terminology which already exists should
> refer to the standard definition.

Cardinality is a measure for sets. Cantor defined it by "our thinking".
He then used bijections to measure it but did not explain why.

>
> So you're saying that the numbers 1, 2, 3, etc., off to infinity
> _aren't_ all distinct? Which two are the same?

Some do not exist (see above). Therefore they cannot be neither
different nor the same.
>
> When real numbers were defined (based on sequences of rational
> numbers), great care was taken to show when two of these resulting real
> numbers are the same and when they are different.

In order to define pi by a sequence there is the necessity to show that

I a_n - pi I < epsilon
for any positive epsilon. This is impossible however, because the
approximating terms a_n reach only to at most 10^100 digits. Better is
mpossible. Epsilon cannot get arbitrarily small. This was not known and
culd therefore not be considered by those who took great care.

> Or maybe you are saying that two different _representations_ of an
> object could be the same, but I might count them as being different?

No.

> > But this recognition will spead out nevertheless.
>
> Time will tell if this statement will stand the test of time.

Regards, WM

.