Re: infinity

Virgil said:
> In article <MPG.1d6d6d461dbe3aa498a0fc@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > > What, nothing but numbers can be elements of an infinite set?
>
> > When working with infinite sets, there is really no way to compare
> > the sizes of sets without resorting to properties of their elements.
>
> There is, but TO doesn't like to admit it. Bijections and injections
> work without reference to any properties except those already needed to
> determine the sets themselves.
Yes, the properties of the elements used in defining the set.
>
> > Numeric sets are one of the most common, but discussions of infinite
> > sets also involve structures such as trees, processors such as Turing
> > machines, and infinite systems such as symbolic languages, which
> > include symbolic number systems.
>
> > When drawing a bijection using an
> > arithmetic formula, we are dealing with sets of quantities, and
> > inverse functions are the way to compare those kinds of sets.
>
> If the functions are bijections, no inverse functions are needed.
The inverse functions are what cause them to be bijections. The inverse of the
function which generates a set from another describes it's size relative to the
other.
>
> > When
> > working with symbolic systems of any sort, the formula N=S^L is used
>
> Only by TO and only to confuse the real issue. Unless the symbolic
> language has some finite limit on string length and number of characters
> in its alphabet, that formula is irrelevant.
No, it is relevant to whether a language is infinite if it only consists of
finite length words from a finite alphabet.
>
> > In general, this yields a maximum size
> > of the language.
>
> Only for computer languages for which there is always some limit on both
> S and L. For non-computer languages, there is no inherent limit on
> either S or L, so no inherent limit on N.
> That TO limits himself does not mean that everyone is similarly limited.
If you limit your words to a finite length while using a finite alphabet, then
you have limited the language as a whole to a finite maximum size.
>
> > For binary trees, we need to closely
> > examine the structure of the tree at the node/branch level to gauge
> > the relationship between infinite nodes, branches and paths, as we
> > discussed at length in Meuckenheim's Cantor and the Binary Tree
> > thread (where he was more or less correct, by the way, despite being
> > accused of quantifier dyslexia and general idiocy).
>
> Again TO is limited by his vision of the world through computer
> limitations. WM's analysis of maximal binary trees was as willfully
> blind as TO's. The set of nodes easily bijects with the set of finite
> naturals and the set of maximal paths with the power set of the set of
> finite naturals. Both bijections have been demonstrated several times by
> several different people, and neither TO nor WM had any coherent
> objection to them, though both objected incoherently at great leagth.
I have already pointed out the willfull dishonesty in your proofs. You use two
different trees, interpreted differently as the set of naturals and the set of
subsets of the naturals. You could just have easily proven there are countable
paths and uncountable nodes with that trickery, which tells me you have proven
nothing with it. The fact is, in a maximal binary tree of any depth, including
infinite, there are always HALF as many paths as there are branches, not
infinitely more, since it takes two branches to create a new path from an
existing one. Your analysis of the tree is just bass-ackwards, and WM may have
his finitist limitations, but much of what he was saying there was essentially
correct, and you were in lala land. You think just because there are a bunch of
you in the gang that might makes right in the world of math? I scoff at such a
notion.
>
> > So, in my opinion, the desire to have one simple method for comparing
> > all infinite sets is unreasonable, since there ARE different kinds of
> > infinite sets with different element properties that need to be
> > examined.
>
> There are many ways of comparing both finite and infinite sets which
> distinguish different properies of those sets.
>
> Sticking only to measuring subsets of the set of finite reals, there
> are, among others, diameters, cardinalities, outer-measure based on open
> covers and the corresponding inner-measure, and for measurable sets (for
> which outer and inner measure agree) "measure". And I have no doubt
> omitted a few.
>
> Each of these measures something different.
None of them compares set size for infinite sets properly, in my opinion.
>

--
Smiles,

Tony
.

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