Re: Relative Cardinality


mueckenh@xxxxxxxxxxxxxxxxx wrote:
> 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.

In order to even _talk_ about the notation 0.333..., you need to have
the concept of limits and sums of an arbitrarily large number of
numbers. Since there are only a finite number of Muecken numbers, this
means 0.333... is undefined in your system, so you are not allowed to
use it.

> > 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.

What do you mean by "a simple rule"? Give me a concrete definition.

> > 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.

Well, there goes set theory (which says, for one thing, that if S is a
set with n elements, the set of all subsets of S has 2^n elements).

> 0.333... may have infinitely many digits 3.

_How many_ digits? Infinity is not a number. (Not even in the standard
set of real numbers.)

> But the number of different numbers like 1/3 is less than 10^100.

What do you mean by "numbers like 1/3"? (Your comment is unnecessary,
since the number of Muecken numbers itself has at most 10^100 elements.
Adding more conditions to the set only reduces the number of elements.)

> But that doe not imply a largest number

I never said it did; and in the set of real numbers, there _is_ no
largest number.

> and it does not imply that the
> numbers in the set of 10^100 numbers remain always the same.

If A is the set of 10^100 numbers today, and B is the set of 10^100
numbers tomorrow, then A union B is a set with more than 10^100 numbers
UNLESS A = B. So the (universal/Muecken) set of 10^100 numbers NEVER
CHANGES.

> 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.

Then define it. If there's no definition, it's worthless from a
mathematical point of view (and you may as well move your posts to
alt.fan.numerology).

> There are "numbers"
> however, which can never be created like floor(pi*10^10^100).

What, exactly, is the set of 10^100 numbers that _can_ be created? Give
me a precise definition.

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

If we're talking about Muecken numbers, then, yes, it's not really
infinite (since a lot of numbers n+k are not actual Muecken numbers),
but in terms of natural numbers (as defined in a standard way), it is
infinite because it's not finite. So it depends on what definitions and
frames of reference you're using.

> But the set cannot be actually infinite
> unless n + omega is reached, i.e., unless there are actually infinite
> elements.

This is not true. You are confusing a property of the set with a
property of its elements. The _set_ {Berlin, 23, Mars} is finite, but
you'd (presumably) never say that the _element_ Berlin is finite.

The _set_ {1, 2, 3, ...} in standard number theory is _infinite_, and
its _elements_ are all _finite_.

Infinite means, literally, not finite.

> > > 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.

Yes.

> There is no actually infinite set.

No. If you write down _all_ the positive integers (using the unal
system, or any other system), you get an _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.

Knowing all the digits of the real numbers can be done with arithmetic
(the standard long division algorithm, where you find the integer part
of the first number, the integer part of the second number, and quit if
they're not equal; otherwise, find the tenth digit of the first number
and the tenth digit of the second number, and if they're not equal, you
know which one is bigger; etc. If you have two different real numbers,
this procedure will terminate). This means that real numbers, as
defined in the standard real number system, exist.

> > 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.

which means my point is valid; you do _not_ know all the digits
0.111... unless
you allow arbitrarily large numbers to exist. (I've also warned you
about the notation 0.111..., which in your number system is
meaningless, haven't I? I think so.)

> > 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.

.... as well as the NUMBER of 1's, which can actually be arbitrarily
large.

> > 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.

Okay, let's refine the question: You have a list of particles in the
universe in some order. You show up "at work", with all the 1's on the
particles, in the same order you've established. What number does the
universe represent?

> > > > 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.

The question is not what 111 _usually_ represents, but what it _does_
represent, which you cannot tell me, unless you know the rule.

> 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.

The rules have to give you a unique answer, which you do not get (as
I've shown above, for 111...111, 0.111...111, etc).

There is actually a bit of regression here, once we talk about rules,
because the rules have to be interpretted. (The particles that have
111...111 on them also include the particles that tell what the rules
are.) So you will need to say how to interpret 111..111 (on the "rule"
particles only) first.

All of this boils down to the following: There has to be some rule
which is not written down. This rule should give you a unique
interpretation for every possible representation, which may be simple
or complicated, but you still end up with only 10^100 Muecken numbers.

If the unwritten (literally 8-)) rule changes, the numbers you get can
change, but how do you know which unwritten rule to use, unless it's
written down on some particle? And once again we find that the number
that the universe represents depends on some rule about rules (which
rule do we use?) and what's written on particle X, which may as well
have been stated as an unwritten rule in the first place.

> > 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

How do you know consciousness in physically inside my head? Maybe it
"leaks in" from some hyperspace. (This is an unsettled problem
discussed by philosophers for centures, so I don't really expect an
answer. 8-))

> 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)

Geometry works with what are called "ideal objects": lines that have no
width, points that have no dimension, etc. These objects do not exist
in the physical world, so mathematics cannot be a 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.

Yes, they can; I can perform an algorithm to find any digit of sqrt(2)
in finite time.

> > So maybe you should define "numbers" to be intervals,
>
> No.

Just a thought ...

> > 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.

It is one possible measure for sets. Your "cardinality" is similar, in
that it is one possible measure for sets as well. However, you should
not have said that your "cardinality" is the same as traditional
cardinality, unless you can show that, for any set S, the cardinality
of S is the same as the "cardinality" of S. Only then would you have
had the right to say your "cardinality" is the well-established
cardinality.

In an example which is probably clearer, I could decide to start
calling an apple a banana. There's nothing wrong with this, until I
start talking with other people about fruit. There, my statements that
"some bananas are red" and "bananas are nearly round" make no sense,
even though they are perfectly okay from my point of view. This
confusion results from using standard terminology in a non-standard
way.

> Cantor defined it by "our thinking".
> He then used bijections to measure it but did not explain why.

That's the _definition_ of standard cardinality. It's just like saying
that 3 is an abbreviation for 1 + 1 + 1. Why is "3" defined this way?
You tell me, and you'll understand why Cantor used bijections.

> > 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.

Which in particular don't exist? In particular, what is the smallest
value of N such that 1, 2, 3, ..., N are numbers, but N+1 isn't?

> > 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
>
> | a_n - pi | < epsilon
> for any positive epsilon.

(You've mixed up quantifiers here. Pi can be defined by a sequence of
rational numbers a_n, but the condition on this sequence is that for
every positive epsilon, THERE EXISTS AN N such that |a_n - pi| <
epsilon when n >= N. The value of N changes when epsilon changes; you
do not need to have |a_n - pi| < epsilon for all n and for all epsilon.

An example of another sequence is the sequence a_n = 1/n; then is is
not true that |a_n - 0| < epsilon for every positive epsilon.)

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

Definitely: limits do not work if you only have a finite number of
numbers available, so you shouldn't even be considering them in your
number system. (This is one of the drawbacks to the notation 0.333...,
which is also forbidden, since it's defined as a limit, as opposed to
1/3, which isn't.)

> > 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.

Fair enough. (I was just trying to understand your thought process.)

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

(That's from a Sammy Hagar song, btw.)

--- Christopher Heckman

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