# Re: Multiple infinities - one more look

*From*: "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx>*Date*: Fri, 7 Dec 2007 22:58:18 -0800 (PST)

On Dec 7, 9:59 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Dec 8, 8:33 am, "Mike Terry"

<news.dead.person.sto...@xxxxxxxxxxxxxxxxxxx> wrote:

"Venkat Reddy" <vred...@xxxxxxxxx> wrote in message

news:052a8b26-92db-46a9-8caf-1ba9c7300927@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

I see. I think order is related to how you choose to represent the

number. If the real number is represented by a digit sequence of

infinite length, we immediately have an order for all reals, since

digit position have an order. For example, for the length of 6 digits,

one can generate these sequences by writing all possible permutations

and combinations of digit sequences in an orderly fashion. This can be

continued for lager length of digit sequences without limit. All

possible digit sequences are guaranteed to appear somewhere in the

list.

No. This process only generates finite digit sequences. Real numbers have

infinite digit sequences...

Instead, if you choose to represent a real number by an algebraic

equation, here also we have an order in representing the equation

itself, so the resulting reals have an order. Likewise for

transcendental as well - once you have a way of identifying the

number, we have a way of ordering them.

It seems you are thinking of something like "computable numbers", and these

are indeed countable. Real numbers need not be computable in this sense

(i.e. having a finite program to output their digits in sequence).

You may say you want to represent the real number as a point on the

line such as the interval [0,1]. Here also we have an order. Cut the

line into half and mark the 0.5 as your first real number. Cut the two

pieces in to two equal parts, mark 0.25 and 0.75 as new real numbers.

Continue the process for ever and you have an ordering of all points.

No. Real numbers need not be one of these points you construct, although

your process will construct real numbers arbitrarily close to any given real

number.

Let's identify a real number as a unique label for a cut in a line

segment (spatial continuum) at a random location. The fact is, such

cuts or splits or points exist only after you imagine them. The

continuum doesn't already have them ready for us to count. This is

similar to fact that we also do not have "ones" ready to count as

natural numbers, but only after you imagine some discrete items.

The natural ordering of such cuts, is exactly similar and opposite to

ordering of natural numbers. Natural number sequence keeps introducing

a new "one" in an attempt to fill the "empty continuum". A reciprocal

process for this would be to introduce a new "zero" or split in an

attempt to empty the "full continuum". However, introducing the splits

wouldn't be by adding one at a time, but 2^n of those at a time. This

follows from the observation that in natural number sequence we are

adding a "one" to every "empty continuum", and hence we need to add a

split to every "full continuum", that is, the pieces resulting from

the previous cuts.

This ensures that we can always imagine 2^n number of splits for every

imaginable natural number n. When the natural number hits infinity,

then we have 2^oo as the number of points, all are perfectly ordered

or sequenced.

- venkat

One might be quick to point that this process can only hit the real

numbers of the form of multiples of 1/(2^n), and misses the numbers

like 1/3, 1/pi, 1/sqrt(2). My answer for this is, have a

representation of these numbers as points on the real line and show me

why my process can't hit these points, if you really continue the

process for ever.

Better yet, have a list of all possible representations - digit

sequences, algebraic equations, geometric models etc. For each of

these representations, have a process to generate all possible numbers

in that representation as I have shown above. That should cover all

real numbers in an orderly fashion, though in a 2-dimensional order.

Well, you've given 3 suggestions above, and all 3 suggestions are wrong and

putting them together doesn't make things any better! :-)

Ask for any number, one of these processes is guaranteed to have it in

its list (I would even say every process have that number in its list,

but at different positions, but I don't need to depend on this

anyway).

Also, since each of these processes have a countable list, and the

number of processes is also countable, the reals must be countable.

Did I miss anything?

Each of your processes covers only a proper subset of the real numbers.

Regards,

Mike.

- venkat- Hide quoted text -

- Show quoted text -

In reference to something along the lines of 1, 3, 5, ..., then 2, 4,

6, .... A notion about mapping that to a sequence is to map 1 to 1, 3

to 2, 5 to 3, etcetera, in the order of the natural integer sequence,

then there is a question about all of those values as constants being

used already. Then, with some notion of a transfinite sequence, then

there is a consideration of "omega+1, omega+2", etcetera. So,

consider omega a constant of sorts, yet all the previous constants are

used already. So, omega is some infinitary constant (label), where

each of the first half of the sequence have leading zeros, each of the

second half of the sequence has a leading nonzero constant, an

infinitarily unique constant. Then, the alphabet of values is

infinitary, to well-order the integers in a manner thus that there are

infinitely many elements less than a particular element. Then, in an

infinite base, that is to say where there are infinitely many

constants, similarly to base one, base two, and base three, the

antidiagonal argument doesn't apply, instead simply confirming the

successor.

For any finite list of constants in successive order, the antidiagonal

is the succesor, and order type, and in ubiquitous ordinals,

powerset. (That's again with the notion of successor ordinals formed

that are simply the powerset, the succesor ordinal's mechanistic form

is the powerset.)

That three space dimensions are particularly concise is not obviously

related, yet some people can find a physical parallel to any

mathematical statement. (V = L).

Infinite sets are equivalent. That is so in a theory where all

infinite sets are axiomatized as irregular, in moreso that the are

their own rootsets and powersets.

The powerset has lots use in terms of counting things in the finite

and small. The value 2^n appears in many forms. That's where 2^n

appears in many composite forms. That's how many elements are in the

powerset of the n-set, with size n: 2^n. It might seem interesting

to make tables and then sum the rows of various counts and then assign

them to geometrical figures. Anyways a wide variety of other numbers

are very useful in terms of the cardinalities of combinatorics of

finite sets. (I pronounce finite fi-nit, with no emphasis, not fi-

night, as well, infinite in-fi-nit. ) Consider for example n!, n

factorial, which is the product 1 through n. That's the number of

ways n things can be put in an order, a permutation, that's how many

transitive orders there are of those things. Then using two

variables, the count n of objects and subset's count k of objects,

there are such useful notions as the choice: function, n choose k,

subset function: x subset number k, and cycle function: n cycle number

k, also know as the binomial coefficients and Stirling cycle and

subset numbers, Stirling numbers of the variously first and second

kind.

n!: count of permutations of an n-set

s(n,m): count of permutations of an n-set containing m-many

permutation "cycles",

In Knuth's notation for n cycle k the binomial coefficients is the

stacked numbers in the parentheses, and n cycle k is the stacked

numbers in the square brackets.

http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html (cycle

number)

A nice thing about them is that they can be computed in many ways.

I wonder how many there are of various partial permutation cycles, in

that there aren't complete transitive cycles. Maybe I should

investigate that for more than a minute, not the associated Stirling

numbers.

The Stirling subset number, which is the count of partitions of an n-

set into m-many disjoint subsets, is written in the Knuth notation the

stacked elements in the curly parentheses. Consider for example

pigeonhole problems, where there are n objects and m cubbyholes, how

many ways there are to put a given number of objects into bins so none

of the bins is empty. That's exactly n subset m. Then, there are

more ways than that to put a given number of objects into bins, with

possible some of the bins being empty. Knowing these exact values is

very useful in formulating counting arguments so that then

probabilities are easy to determine, about what happens when a set of

objects is randomly distributed to separate bins.

http://links.jstor.org/sici?sici=0002-9890(199410)101%3A8%3C771%3AATNON%3E2.0.CO%3B2-U

I could easily go on about the tremendous utility of counting

arguments of finite combinatorics. It would really be a reasonable

thing to do to define the counts of the objects of n-sets and

generally multi-sets, or vice versa, as expressions of primary

interest and concise notation in definitions of convenient terms. The

point is that in the consideration of algorithms that would expect to

be useful on finite sets, they often are in the application to systems

modeled by iteratively larger finite sets, infinite sets.

To be putting the points on a line, put them in a line.

Ross

--

Finlayson Consulting

.

**References**:**Multiple infinities - one more look***From:*Venkat Reddy

**Re: Multiple infinities - one more look***From:*FredJeffries@xxxxxxxxx

**Re: Multiple infinities - one more look***From:*Venkat Reddy

**Re: Multiple infinities - one more look***From:*Mike Terry

**Re: Multiple infinities - one more look***From:*Venkat Reddy

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