Re: Orlow cardinality question

Virgil said:
> In article <MPG.1d1b92efea9c82ef989e07@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > Virgil said:
>
> > > The difficulty with subset ordering is that it is only a partial
> > > order leaving lots of sets unable to be compared. Even {1,2} and
> > > {3} cannot be compared by subset ordering. Cardinality certainly
> > > works perfectly for finite sets,
>
>
> > I never claimed that all comparisons can be handled with subset
> > relations. being a proper subset SHOULD make one set smaller than the
> > other, however. Of course, the mere fact that one set is a proper
> > subset of the other doesn't tell us much more than that, but that
> > fact at least should be preserved, and not discarded.
>
> Then it is incomplete, which cardinality is not.
> > >
> > > > In finite sets of naturals, you freely admit that the set size
> > > > can't be larger than the maximum element value, but somehow this
> > > > flies out the window at infinity.
> > >
> > > That only holds when we define the "number" of an initial set of
> > > 1-origin-naturals to be its largest, so is circular.
>
>
> > We don't define that relation to hold. It follows from the definition
> > of the set, and is proven through induction, which means it holds for
> > the infinite set of natural numbers and the sets they each define.
>
> What does the natural number '3' mean? One completely accurate
> definition is that it is the natural nuber that comes immediately after
> '2'. That definition in no way requires any "number" properties. All
> those "number" properties are acquired through deduction.
>
> > > A path is representable by a set of branches, not all such sets
> > > representing paths. So it is not surprising that there are more
> > > such sets of branches than branches when the set of branches is
> > > infinite in a a maximal binary tree. That TO finds it disturbing is
> > > his problem, not ours.
> > What's disturbing is that you can't see that it's incorrect. I am not
> > disturbed by false facts. I am disturbed by trying to imagine the
> > mechanics of a mind that accepts them, and fights to defend them.
>
> I have exactly that problem with WM and TO. They do not even make the
> same mistakes, but each of them fights to defend his own mistakes.
> >
> > Now, you just said something interesting. IF a path were simply ANY
> > set of branches in a tree, THEN the set of paths WOULD be the
> > powerset of the set of branches. HOWEVER, most sets of branches in
> > the tree are NOT paths, but only a very small fraction of them, that
> > represent SEQUENCES of branches from the root and through each level
> > of the tree. THEREFORE, it is IMPOSSIBLE that the set of paths is
> > equivalent to the power set of the branches.
>
> TO has conceded that the set of paths is countable, i.e., can be
> bijected with the naturals.
Conceded? You were the one claimingthe set of paths was UNcountable, while the
set of branches was countable. I always disagreed with that.
>
> I have shown that the set of paths is bijectable with P(N):
> For each path construct the set which contains n if and only if the nth
> branch of the path is a right branch.
That is a different type of tree than the one where the branches are countable.
In the tree where each branch is a digit in a digital number, each path is a
digital number. In the tree you are describing, each path defines a set of
numbers, and each level in the tree corresponds to a number than can be in the
set. So, while you have digits and sets of digits that correspond to numbers,
in the second you have numbers, and sets of numbers. The first tree is the set
of all numbers, and the second is the set of all sets of numbers. By changing
the meaning of the nodes, you derive either countability or uncountability, but
rather than concentrating on the properties of the meanings you assign to the
parts of thetree, you should be concentrating on the properties of trees
themselves, as WM has done. Then you will get solid answers, instead of
convoluted conclusions.
>
> Each branch in this way creates a unique subset of N and every subset of
> N determines a unique path in this way.
>
> So the allegedly "impossible" bijection obviously exists.
I never said the bijection doesn't exist. All sorts of nonsense can be
accomplished with bijections.
>
> And what TO has declared impossible had now been done.
That is not what I declared impossible, but it is what I see as just another
Cantorian feint. If each branch denotes the membership or lack thereof for a
given number in any given set, then for each path you will have two branches,
one that includes the next number and one that won't. it's ultimately the same
thing. Two branches for every infinite path. The only way to get another result
is to change trees midstream.
>
>
>
> > It is a MUCH smaller
> > set. In fact, the set is SMALLER than the set of individual branches,
> > by HALF, as I have shown numerous times.
> >
> > >
> > > >
> > > > I offered a system for sets defined by mapping functions that
> > > > works precisely, both for finite and infinite sets. It wasn't
> > > > welcome.
> > >
> > > It wasn't consistent and it wasn't complete.
> > For sets defined by mapping functions it was both, plus intuitively
> > satisfying in every respect, but i doubt you paid any attention
> > anyway.
> > >
> > >
> > >
> > >
> > >
> > > > You'd rather have your kludge of inconsistencies that violate
> > > > information theory and infinite series
> > >
> > > Those alleged violations of information theory and infinite series
> > > only betray TO's ignorance of both. They are both irrelevant to the
> > > issue of consistency of cardinality, at least as TO has cited them.
> > That is false, but I am not surprised by such a typical spewage on
> > Virgil's part.
> > >
> > >
> > > ...
> > >
> > >
> > > > The size of the set is the number of elements in the set. It
> > > > doesn't get much simpler than that. If that doesn't make sense to
> > > > you, then I don't know what you think we're talking about. A set
> > > > is a unit that consists of a number of units. The size of that
> > > > set is the number of units of which it consists. If nothing is
> > > > know about the units themselves, then this is all we know about
> > > > the set, and is really the only pure measure of a set.
> > > >
> > > And how does one determine the "number of elements" in an infinite
> > > set? One way is by considering injections, surjections and
> > > bijections with other sets, i.e., cardinalities. All the alternate
> > > methods suggested by TO have been found even less satisfactory that
> > > cardinalities.
> > Oh, because they correctly conclude that there are hald as many evens
> > as naturals, infinitely more rationals than naturals, infinitely
> > fewer squares than rationals, etc? Yeah. Whatever. Cling to your
> > flotsam.
> > >
> > > Note that TO's example of the even naturals being "half" of all
> > > naturals relies on structural properties of those sets and their
> > > members other than mere set and member properties, so it is
> > > measuring structural relationships other than mere membership.
> > And your conclusion that they are equivalent in size depends on using
> > mathematical mapping formulas that make use of their properties as
> > numbers as well, to establish tyour sorrespondences and bijections,
> > except that you conveniently forget about that in the last stage of
> > your bijection, and incorrectly declare them equal.
> > >
> > > Since as mere sets, the naturals and the even naturals are in
> > > simple on-to-one correspondence, it must be the additional
> > > not-purely-set structure that leads TO to say that the even
> > > naturals are half of the naturals.
> > It is the nature of numbers in the context of the real line that
> > leads me to conclusions about sets of numbers. If you can create your
> > correspondence between the elements of the two sets without reference
> > to their proeprties as numbers, I'd love to see it.
>
> A: Start with {},
> and for each x create (x union {x})
>
> B: Start with {{}},
> and for each x create (x union {x}) union {x union {x}}
>
> Done!
What does this have to do with the naturals vs. evens?
> > >
> > > There is nothing wrong with such a measure as long as one is aware
> > > of what it is measuring, which TO seems not to be
> .
> > By TO, I take it you mean Virgil.
>
> TO takes it erroneously.
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: infinity
    ... You are constructing a bijection with the elements of the tree and the ... elements of the naturals and of the powerset of the naturals. ... We can also construct a bijection between the paths of the binary tree ...
    (sci.math)
  • Re: infinity
    ... >>> For any given subset defined by a path in this tree, ... I used one maximal binary tree, which has both branches and paths. ... >> showed that the branches biject with the naturals and the paths bijecte ... So that unless TO can construct a bijection between N (the infinite set ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... I consider an infinite tree. ... If you mean that there is a bijection between paths and reals, ... Assume a binary tree in which each node is a parent and has a left-child ...
    (sci.math)
  • Re: infinity
    ... I used one maximal binary tree, which has both branches and paths. ... >>> showed that the branches biject with the naturals and the paths bijecte ... > So that unless TO can construct a bijection between N (the infinite set ...
    (sci.math)
  • Re: infinity ...
    ... it was specifically a bijection between two sets of infinite binary ... >>> from 0, both finite and infinite, in normal binary format, and on the other ... Sets don't have an inherent measure beyond raw size, and the power set is not ... there are not enough naturals to map to every subset ...
    (sci.math)
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