Re: Relative Cardinality

In article <1121616174.789902.256110@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

>
> The power set of the set of all sets does not exist.

The "set" of all sets does not even exist in many axiom systems for set
theory.

> We are in the comfortable situation that, contrary to oil, food, and
> water, the bits have not yet become scarce.
>
> >
> > 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).
>
> I have already defined: A natural number exists if there is a
> fundamental set or if there is an n-adic representation realized.
> >
> > 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_.
>
> That example does not fit the problem. N consists of elements which,
> contrary to Berlin,
> 1) are ordered
> 2) count their initial segment.
>
> >
> > Infinite means, literally, not finite.
>
>
> Yes. But not actually infinite. Actual infinity is a number (a whole
> number according to Cantor).

Which infinity is a number according to Cantor? There are all sorts of
infinities, and they need not all be numbers.

> 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.
> As long as infinity is not actually reached by n it is not reached by
> f(n) and vice versa.
>
> >
>
> > 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.
>
> But it need not terminate before your means of writing down the
> integers is exhausted (including your personal memory).
> >
> > 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.)
>
> It is not meaningless. There are arbitrary many 1's. The limit is 1/9
> because potential infinity is always sufficient for analysis.
> >
> > 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?
>
> You show up at work in front of your screen. It is filled with 1's.
> What number does the screen represent?
>
> You need not write down all the 1's of the prime number 700 of the form
> 111...111. But it is sufficient to encode the information: All digits
> are 1, the number of digits is 10^25. 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.
> > 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.
>
> That is not a particular problem of mine. It is the same with
> mainstream mathematics.
> >
> > 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.
>
> That is all the same with any representation of numbers.
>
> > 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-))
>
> If you are a convinced solipsist you should never attend meetings.
> >
> >
> > 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.
>
> Reality supplies the stuff. A bit of abstraction is fine. But I am
> sure, there is no ideal circle or rectangle in your head.
> >
> >
> > Yes, they can; I can perform an algorithm to find any digit of sqrt(2)
> > in finite time.
>
> Then try to find out and store th firt 10^100 digits.
> >
>
> > 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.
>
> Cardinality was introduced as a measure of the number of elements. And
> when introducing it, Cantor mentioned something like clar thought. That
> is also the foundation of my definition.
> >
> > 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.
>
> I have proved tha there are not more irrational numbers than rational
> numbers.
>
> >
> > 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?
>
> How far does our atmosphere stretch out into the space? Please answer
> in millimeters.
> >
> > > > 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.
>
> You misunderstood me. Perhaps my fault. It must be possible to satisfy
> this inequality for arbitrarily small epsilon. But already for the
> comparatively large eps = 1/10^10^100 this inequality cannot be
> satisfied by any a_n.
>
> Regards, WM
.