Re: infinity

Virgil said:
> In article <MPG.1d73c56abbd0d05498a11a@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > 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.
>
> Then, according to TO, the set {1,2,3} and the set {a,b,c} cannot be
> compared by looking merely at positions in the listings of members, but
> depend in some way on the actual nature of the members themselves?
Of course finite sets are easily compared. Infinite sets require some mapping
between members to be compared. But, you already knew that.
> > >
> > > > 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.
>
> That they are bijections is what causes them to have inverses.
No kidding. Same thing.
>
> > The inverse of the function which generates a set from another
> > describes it's size relative to the other.
>
> If there is an inverse, they are of the same size (cardinality).
If there is a bijection, then they are the same cardinality. If there is a
bijection from the naturals to a set, then the inverse of the function that
forms the bijection determines the size of that set relative to N.
>
> > > > 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.
>
> WRONG AGAIN! TO has this persistent delusion that infinite sets must
> contain infinite objects. He is wrong, and has been wrong since the
> beginning of this thread, and is still obstinate in his error.
Hand waving, whining, and weeping won't change the facts.
> > >
> > > > In general, this yields a maximum size
> > > > of the language.
>
> Only for limited minds like TO's whose delusions require that there be a
> largest natural number and no largest natural number simultaneously.
How do YOU define the size of a language in terms of the alphabet and maximum
word length allowed? Let's see your formula, O Unlimited Virgil.
> > >
> > > 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.
>
> See, TO is doing it again. WRONG AGAIN, TO!
Actually I am exactly right, and you offer no alternative formula or
interpretation. You can way "wrong" till your teeth fall out, but your
declarations don't convince anyone of anything.
> > >
> > > > 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.
>
> TO has claimed so, but never established it to the satisfaction of
> anyone exzcept himself, and possibly WM.
>
> > You use two different trees,
> WRONG! One single maximal binary tree for both bijections.
Ahem, wrong.
>
> > interpreted differently as the set of
> > naturals and the set of subsets of the naturals.
>
> Different parts of the tree: branches corresponding to natural and
> maximal paths to sets of naturals.
No, you used one tree where branches were bits and paths were strings of bits
representing numbers, and another where branches denoted set membership for
numbers, and paths represented subsets of N. Using the first, you "proved" that
the branches are "countable", and using the second you "proved" that the paths
are "uncountable".
>
>
> > You could just have
> > easily proven there are countable paths and uncountable nodes with
> > that trickery,
>
> That TO refuses to admit that there the only "trickery" involved was a
> better undersading of the facts involved, does not refute the clear and
> valid proofs I made.
Hahahaha oh God. No comment.
>
>
> > which tells me you have proven nothing with it.
>
> TO's "understanding" tells hhim only what he wants to hear, which never
> includes the fact that he is so often wrong.
If you say so, Bergil.
>
>
> > The fact is
> If TO has any notion of what the facts really are, then he is a
> masterful troll.
If you still think I am a troll, then you are a trollop. :)
> > >
> > > > 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
.

Relevant Pages

  • Re: infinity
    ... >>> infinite sets also involve structures such as trees, ... >>> way to compare those kinds of sets. ... >> If the functions are bijections, ... >> the set of finite naturals. ...
    (sci.math)
  • Re: abundance of irrationals!)
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  • Re: Skolems Paradox and why is math the way it is?
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  • Re: Orlow cardinality question
    ... >>> Perhaps he's one of those guys that believes bijections between sets ... > regardless of whether that listing corresponds to any other sort of ... I would like to see your formula that maps the naturals to the rationals. ...
    (sci.math)
  • Re: infinity ...
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    (sci.math)
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