Re: abundance of irrationals!)

Virgil said:
> In article <MPG.1ceedc886b8bb200989c50@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > Virgil said:
> > > In article <MPG.1ceea54de94f8258989c39@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> > >
> > > > Virgil said:
> > > > >
> > > > > That is all that is needed for a finite set.
> > > > That's one way of getting one, like proper subset is one kind of smaller
> > > > set.
> > >
> > > And TO claims (below) that what he calls an "indifinite" set can be a a
> > > proper subset of every other set.
> > The null set in particular. So?
>
> So the empty set is NOT finite?
It's infinitesimal, like zero. Set size = 0. Is that a positive number?
> > >
> > > > > > > > Well, I was referring to "definite" sets, as you agreed to call
> > > > > > > > them.
> > > > > > > You want
> > > > > > > > to reserve "finite" for sets with largest members?
> > > > > > >
> > > > > > > I want to reserve "finite" for sets whose members can be
> > > > > > > numbered from 1 to a natural number n.
> > > > > > Same thing. Okay.
> > > > >
> > > > > Not okay. {} is a finite set without a largest member. By TO's
> > > > > definition, it must be an infinite set. Similarly for {"a","b","c"}.
> > >
> > > > No, it would be an indefinite set. That's the term I used for your
> > > > infinite
> > > > set of finite naturals.
> > >
> > > TO now says that {} and {"a","b","c"} are NOT FINITE SETS? How stupid.
> > What is stupid is claiming people have said things that are entirely
> > unrelated
> > to what they have said. I never defined infinite sets the way you are saying.
>
> Unless you are saying that your "indefinite" sets can be finite, you are
> saying that they are NOT finite. Thus you are saying that {} is NOT
> finite.
infinitesimal. This is a minor point and essentially irrelevant. Nice try.
>
>
> > > > > > > >Let's say it's "indefinite", just like its largest
> > > > > > > member.
>
> > > > >
> > > > > "Infinite" means no more than "not finite" so that TO now wants to
> > > > > create a category of sets that are not finite and not not finite at
> > > > > the
> > > > > same time? Does the category of not not not finite follow, and so on ad
> > > > > infinitum (or "ad indefinitum" in Orlow-ese?)?
> > >
> > > > You say it's infinite, I say that's impossible given your restriction
> > > > on the element members, which you say are finite, but without bound.
> > > > Sounds indefinite to me. It's poorly defined.
> > >
> > > The prefix "in" means "not", so that by trying to create a
> > > classification of sets which are neither, TO is saying they are not
> > > finite and not not finite. In my book, that says they do not exist at
> > > all. So that, according to TO's latest ukases, there is no such thing as
> > > {} or {"a","b","c"}.
> >
> > Now you are a half an inch away. Your definition of naturals is self
> > contradictory. It's neither finite nor infinite, because it is both in
> > incompatible ways. That is the only reason I coined "indefinite set", so you
> > can keep your indefinite set with its indefinite largest member, and perform
> > your indefinite operations on it and get your indefinite results.
>
> In my book when something is not finite, it is not finite, which is what
> "in"finite means to everyone except TO.
>
> For TO there is appareantly a claissification that is neither finite nor
> not finite. He must be working with a multivalued logic system where
> things which are not true have alternatives other than false.
I am simply trying to give you a term to describe your problematic definition
of the naturals, so you can talk about your useless set without contradicting
yourself.
>
>
> > > One discards from any discussion issues irrelevant to the subject under
> > > discussion.
> > What if they are relevant? Then you discard them too. And in turn, so are you
> > discarded by me as irrelevant.
> > >
> > > The definition of cardinality depends only on the existence of injective
> > > or bijective functions, and not on the order of the members of the sets
> > > on which those functions are defined, so what is the point on bringing
> > > in the irrelevancy of orderings to an area in which they are not
> > > relevant?
>
>
> > > > Order is inconvenient to numbers when you want to call them numbers but
> > > > treat
> > > > them like they are not numbers.
> > > >
> > > Order is irrelevant in constructing injections and bijections, so is
> > > irrelevant in determining cardinality.
> > > > >
> > > > > The definition of a set, in mathematics dependes only on what are or
> > > > > are
> > > > > not members of that set, and in no way implies any inevitable or
> > > > > required ordering to that set.
> > > > >
> > > > > As sets, {1,2,3} = {1,3,2} = {2,1,3} = {2,3,1} = {3,1,2} = {3,2,1}
>
>
> > > > In most cases, yes.
>
> In all cases a set is determined solely by what are or are not members
> of it, and by nothing else.
And set membership is defined by some function, even if that function is a
simple enumeration.
>
> In the case of infinite sets of numbers, the entire
> > > > set
> > > > can
> > > > only be defined as a function, in which case the values of the elements
> > > > are
> > > > crucial.
> > >
> > > Actually, the definitons of the set of integers, the set of rationals,
> > > and the set of reals can all be made without using a single function.
> > > And in the von Neumann model for the naturals the existence of a
> > > function occurs only as a consequence of the construction itself.
>
> > Function ON those sets, as I said before. Perhaps I needed to say subsets of
> > those sets? There is no point repeating myself. You are playing "who's on
> > first", and I am not a Stooge.
>
> Then stop acting like one.
I am not the one running around in circles. My derivations are rather linear,
with distinct endpoints.
>
> The only time functions are requireed is to compare sets, not to build
> them in the first place, unless one is building a set of functions.

If the set is defined by a function, like the set of all squares, then that's
the definition of the set, which can be used as a mapping function. What else
defines the set you are talking about?

> > >
> > >
> > >
> > > > >
> > > > > We shall denote the initial segment deterimed by n as N_n.
> > > > >
> > > > > For example N_3 = {1,2,3} = {1,3,2} = {2,1,3}
> > > > > = {2,3,1} = {3,1,2} = {3,2,1}.
> > > > >
> > > > Finite initial segment, you mean.
> > >
> > > No! By my definition, all initial SEGMENTS are necessarily finite.
> > > The term "segment" in mathematics usually refers to something with two
> > > specific endpoints and contains all the points in between.
> > > An initial segment starts at the first natural and ends at some other
> > > natural reached in a finite number of steps of adding one.
> > No! The number of steps can be infinite and still terminate. That your system
> > doesn't allow for that is a consequence of your intellectual myopia.
>
> > >
> > > Which bijections are not valid, and why?
> > Already explained.
>
> Not to any mathematician's satisfaction.
> > > In mathematics, a bijection is a bijection is a bijection.
>
> > > > >
> > > > > Note that, by this definition, each N_n is finite but that N is
> > > > > infinite.
> > > >
> > > > Yes, it's poorly defined. Shall I repeat that again?
>
> Repeating it will not make it any less false.

And apparently showing you exact math that contradicts your vague derivations
doesn't make any difference either.

> > >
> > > Not unless you can proof your silly statement.
> > been there done that. Buy another ticket or get off the ride.
> > > >
> > > > >
> > > > > Note also: an equivalent definition for finite and infinite is:
> > > > > S is finite if there does NOT exist any injection from S to any proper
> > > > > subset of S, and is infinite if there does exist some such injection
> > > > > from S to some proper subset of S.
> > > > > NOTE: a function f:A -> B is injective if
> > > > > whenever f(x) = f(y) then also x = y.
> > > > > (distinct elements have distinct images)
> > > > >
> > > > >
> > > > >
> > > > > > If that's the way you define a finite set, then your
> > > > > > naturals
> > > > > > are not a finite set.
> > > > >
> > > > > Right!
> > > > >
> > > > > > But if the elements are all finite, then the set cannot be infinite
> > > > >
> > > > >
> > > > > The way we define infinite, it can be and is.
> > > > It's the way you define finite that is at issue, and the generalizations
> > > > you
> > > > make about bijections that are not universal.
> > >
> > > Since "in" is a standard prefix indicating "not", that is how I define
> > > it.
> > "you" don't define anything.
> Most of what I state as definitions here are generally accepted
> mathematical definitions, so not "mine" in that sense.
>
> If TO can't hack math, nobody is forcing him into it.
I am doing real math. You are blindly accepting definitions which upon close
strutiny are inherently inconsistent, and placing beliefs in a system which you
admit has nothing to do with any other math, or with reality.
> > >
> > > Definition (1) A set is "finite" if there does not exist any injection
> > > from the set to any of its proper subsets.
> > > Definition (2) A set is finite it is either empty or if there is a
> > > bijection between it and some intitial segment (as defined above) of N.
> > > In this latter definition, the cardinality of the set is the largest
> > > value in the initial segment of n as defined above or zero if the set is
> > > empty.
> > >
> > > These two definitions may be proved equivalent.
> > > You keep saying that bijections are not all bijections, but you give no
> > > examples nor any reasons. Such handwaving is not convincing.
> > > > >
> > > > >
> > > > > > > No you didn't, but you have now. Fine. By the way,
> > > > > > > "mapping function" is completely general. It doesn't have
> > > > > > > to be easily expressible in one line. Any set we
> > > > > > > call countable has such a mapping function.
> > > > > > Do those multiline functions tend to have inverse functions?
> > > > >
> > > > > Bijections have inverse functions, non-bijections do not.
> > > > >
> > > >
> > > > Then, where there is a bijection, use the function to calculate relative
> > > > set
> > > > sizes. Is that so hard?
> > >
> > > Except that there are bijections between some sets and some of their
> > > proper subsets, Like set N and the subset, E, of even naturals.
> > > According to the proper subset definition, Card(E) < Card(n), but
> > > according to the bijection definition, Card(E) = Card(N).
>
> > Interesting inconsistency. What does that say to you? Does cardinality agree
> > with ANYTHING else? no.
>
> If cardinality is consistent with itself, that is all that is needed,
> and neither TO nor anyone else has shown it not to be.

Sure, and if Cantor agrees with himself, that is all that he needs as well,
despite the fact that everyone else in the world disagrees. If his thoughts are
consistent with themselves, well, what better measure of sanity could there
possibly be? Except, maybe, if his thoughts actually were consistent with the
world around him. That might have kept him out of the looney bin.

>
> > >
> > > So which is it?
> > the first
> > >
> > > The injection/bijection definition has the virtue of creating
> > > cardinalities which, assuming choice, satisfying the order properties of
> > > a total order:
> > assuming......
> > >
> > > A relation is a total order if the following properties hold.
> > > 1. Reflexivity: a <= a for all a.
> > > 2. Antisymmetry: a <= b and b <= a implies a = b.
> > > 3. Transitivity: a <= b and b <= c implies a <= c.
> > > 4. Comparability (trichotomy law): For any a and b,
> > > either a <= b or b <= a.
> > >
> > > One can prove that cardinalities, as defined by Cantor and assuming the
> > > axiom of choice, satisfy these properties.
> > >
> > > For any alternative definition of the "size" of sets to be acceptable,
> > > it must also satisfy these properties.
> > >
> > > No such definition which requires a proper subset to be of smaller
> > > "size" than its superset has ever been found satisfactory, since there
> > > are always either sets which such rules fail to compare, so that rule 4
> > > above cannot be satisfied, or one of the other 3 rules gets violated.
> > >
> > > Can Orlow find what better minds have often sought?
> > yup
>
> Orlow has yet to present any definition of "size" of sets that can be
> shown to satisfy all four the properties of a total order above. The
> subset ordering satisfies properties 1,2 and 3 but not 4. Every
> extension of ordering by subset extended to other sets that is
> sufficient to establish 4 has been shown to violate one of the others.
>
> Given the axiom of choice, Cardinality has been proved to have them all.
>
That's very nice for you.

--
Smiles,

Tony
.

Relevant Pages

  • Re: abundance of irrationals!)
    ... >> That's one way of getting one, like proper subset is one kind of smaller set. ... I never defined infinite sets the way you are saying. ... >> It may be acceptable to cardinality, but so are the erroneous conclusions it ... >> No bijections, please, without specifics on the function. ...
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  • Re: abundance of irrationals!)
    ... >>> That's one way of getting one, like proper subset is one kind of smaller ... Unless you are saying that your "indefinite" sets can be finite, ... >> Order is irrelevant in constructing injections and bijections, ... >> irrelevant in determining cardinality. ...
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  • Re: Distinct linear orderings on Z
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  • Re: infinity ...
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  • Re: Distinct linear orderings on Z
    ... What was the central idea of cardinality again? ... > Of course cardinality is defined by bijections. ... My point was that a bijection does not "prove that infinite ... For instance, with the naturals ...
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