Re: Cantor and the binary tree

In article <d7s9d8$vne$03$1@xxxxxxxxxxxxxxxxx>, helms@xxxxxxxxxxxxx
says...
> Well some thoughts, even if it is a bit late...
Actually, Gottfried, your comments are good, even if they point in a
different direction than I am proposing. They show that you are thinking
about the subject, and seeing the possibility that our understanding and
approach may be improved. That's a good thing, in my mind. I don't see
cardinality as at all sufficient. Thanks for your input.
>
> Am 29.05.05 17:40 schrieb mueckenh@xxxxxxxxxxxxxxxxx:
>
> >>
> >>No number can have the property of "being uncountable".
> >
> >
> > The number of elements of a set can be finite or infinte. It can be
> > countable (finite or aleph_0) or uncountable. In set theory such
> > numbers are defined, although you are correct.
> >
> It seems to me, that this is a good point to pause and think about
> thinking itself (I hope my english is sufficient to express things
> right/understandable in the following), and not to think primarily
> about the binary tree.
>
> If we start thinking about
>
> - the property of infinitude of that mental object, that we
> create by counting, which means here the idea of an infinite
> set of numbers, which is thought to satisfy the requirement,
> that -unboundedly- for any element n there is another element n+1,
> (different from n, call n and n+1 neighbours),
>
> - and on the other hand start studying the properties of another
> infinity, where neighbours cannot be identified, such as the
> idea of a continuum (*1, and see further remarks below),
>
> then we might assign to those types of infinities a "name",
> a nominal category.
I agree. To me, there is a discrete infinity and a continuous infinity
which are distinct. The discussion of the binary tree is purely about
the discrete, or countable, infinity. It really doesn't even touch on
the questions of the continuum, which also needs to be considered, along
with the relationship between the two.
>
> If we further think, we are able even to put that types of
> infinities (and maybe some more) in an ordered system,
> we even may assign them a numerical index instead of a "name".
> Now this numerical index appears as a number, but is basically
> an index only establishing an order.
>
> Next step one can study, whether the assigned index-value
> correlates with the properties of the different types
> of infinity in such a way, that even arithmetical operations in
> them make sense (how it is done historically). But before
> adding "alephs" or creating thoughts like "2^aleph_1"
> I guess it is important to consider your remark (or such like):
>
> > This property of a set is expessed by or as a transfinite number. (In
> > German: transfinite Zahl)
>
> the introduction of the word "number" in this context, (even when
> specified as "transfinite number") may be a source of confusion.
>
> I always understood "transfinite number" not being a "count-of-elements"
> (which would be impossible with uncountable sets) but being an index of
> the *type* of infinity (already in mind, that that types have
> somehow an order).
To me, a set size IS a number of elements. For it to have some other
meaning when the sets have an infinite number of elements makes no sense
to me. They are infinite numbers.
> That this index is assumed to agree with numerical operations on
> it, prominently in its extended interpretation as a size of finite
> sets, but also with its interpretation for types of infinite sets
> (aleph_1, Aleph_2,...) may be the reason to call it "transfinite
> *number*" instead of, for instance "infinity type index", but that
> may be unfortunately misleading, if not always deliberately considered.
Again, I agree, I think. I believe that arithmetic can easily be
developed using both the discrete and continuous infinities as units,
but one DOES need to be very careful about the generalizations they
accept and those they question or reject. As far as type of infinities,
I really see only those two types, but within those two classes of
infinities there can be a full spectrum of particular values, and more
complex infinities can be built from them, the way I see it.
>
> It is (different to the "elements" of the set of numbers) a "property"
> and not just another element, just another "number" and hence not
> comparable to that of the elements of the set.
>
> ---------------
>
> Put it in slightly other words:
>
> If we establish a "dimension", which covers
>
> - from the first, naive infinity (the infinity of our
> mental object of the natural counting)
>
> - up to that type of infinity, which we assume to be a
> "continuum"(*1), (where we cannot single out neighbours),
>
> then the use of the concept of "number of elements" for all
> of these sets is in itself misleading. It focuses the mind
> to a property, which is not given for the continuum (since
> we cannot single out neighbours). So -maybe- prominently
> the use of that conception in questions like:
>
> "what is the number of elements of this infinite set
> compared with the number of that other infinite set"
>
> is surely an important source of miscommunication about the
> more basic (and yet more interesting) properties of comparisions
> of types of infinities.
Oh, here I disagree. There are only so many classes of infinities, and
that's been done, for the most part. To me, the much more interesting
question is whether we can do better, which we can, and whether we can
relate these infinities in exact ways that lead to interesting
applications in the real world that work. I see this as entirely
possible.
>
> One aspect is, to give it another name, a terminus technicus,
> like "cardinality" ("Mächtigkeit" german), to overcome the
> connotation of "numbering","counting" (which fixes oneself
> to only one type of infinities, as I suggested above), and to make
> it clear, that the symbols, which denote different cardinalities
> (aleph_xy), are essentially not numbers, but firstly *indices*.
Well, I use the term Bigulosity to distinguish my methods from
cardinality, which seemed necessary to avoid confusion. It also sounds
good. :) I still see these indices as numbers, but I come from a
computer science background, where indices are numbers that denote the
position of an array element, which transaltes into a number that
denotes a memory location.
>
>
> For instance, that following statement induces confusion:
>
> > We can compare the number of nodes between level 0 and level n with the
> > number of all nodes.
>
> in the -possible- case, that your tree represents a rule to describe
> a continuum(*1): since then it is referring to a "number of all nodes",
> which cannot be given, if it possibly is "un-countable".
But the binary tree is a discrete example, and really not related at all
to the continuum.
>
> The concept "number of" may be assigned to describe an (important)
> property of a finite set, but if we deal with mathematical objects,
> which are constructed to be infinite, it is better to switch to the
> terminus "cardinality" (or maybe a better one), to not apriori throw
> away the ability to include uncountable(*1) infinities in our
> considerations.
Yes, we need to preserve that distinction, unless we can find some way
to express one in terms of the other, but I don't think that's possible
in any finite formula.
>
> That may be an amateurish view of things; it again may mix the
> non-neighbour-property of rational numbers with that of real
> numbers, and thus is not sufficient to characterize continuum,
> but I think it is at least a necessary criterion (and it may
> be more helpful after that to line out another, more powerful
> distinction)
The rationals are an interesting problem. They seem to be a dense
enumeration of reals of some sort, and yet any way to enumerate them
leads to some discrete infinity, much like the digital enumeration of
reals. They seem to be enumerations, since they get arbitrarily close to
any real. Personally, I consider them enumerations, even though they
require potentially infinitely many digits to represent any given real
number, because I know that to enumerate an infinite set of natural
numbers, one also needs infinite digits. That doesn't bother me. Still,
ANY enumeration is a subset of some sort of the continuum, and I think
the continuum itself cannot be formulaically expressed in any way that I
see, except by assigning it a unit value.
>
> Gottfried Helms
>
>
> *1: "Uncountable", for instance "continuum", where we cannot specify
> a number and its neighbour, which, for instance occurs by the
> introduction of a rule, that we allow a third number always between
> two numbers, for instance by the non-bounded application of the
> computation of a mean.
> If we have such a continuum, we cannot "count" its elements.
>
But that is also true of the rationals. This area needs careful
consideration, in my opinion.
.

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