Re: abundance of irrationals!)

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.

> > > > > 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.
> > > >
> > > > > Fine. We play the name game.
> > > > > The set of all finite naturals is not infinite, and not finite by
> > > > your
> > > > > definition. Let's say it's "indefinite", just like its largest
> > > > member. It's
> > > > > certainly not infinite.
> >
> > "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"}.
> >
> > > > If it stops, call it "finite".
> > > > If it doesn't stop, call it "infinite".
> > > > Is there a third possibility?
> >
> > > Well, yes.
> >
> > Note: TO revokes and replaces the law of the excluded middle.
> > It will, no doubt, be labeled the law of the included muddle.

> > TO's system is irrelevant if it prevents reordering sets to suit one's
> > aims.
> Yes, and the rules of poker are irrelevant if they prevent you from sneaking
> in
> aces from your pocket when you wish.

Axiom systems are the "games" of matnematics, and there are no rules
that do not belong to some axiom system. TO cannot show me where TO's
rules appear in any axiom system, so, mathematically speaking, there are
no such rules.
> >
> > > It's a non-quantitative reordering
> >
> > So what? If it establishes the desired bijection, it is acceptable.
> > No set of more than one element has any required ordering AS A SET.
> > Orderings, if any, are extra add-ons which may be discarded whenever
> > inconvenient.
> It may be acceptable to cardinality, but so are the erroneous conclusions it
> produces. That's because it concentrates more on convenience than
> consistency,

The only inconsistencies are TO's rules which are not consistent with
the axioms of any known system, and are, therefore, irrelevant to
mathemaics.


> because this permission to discard whatever is inconvenient is the ONLY thing
> that allows anyone to draw ANY conclusions from this system. It's like your
> concvenient distinction between the size of the set of naturals and the range
> of its values, or the convenient discarding of any other logic that might be
> brought to bear on the questions at hand, like information theory or infinite
> series.

One discards from any discussion issues irrelevant to the subject under
discussion.

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 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.



> >
> > 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.
>
> > Definitions of "finite" versus "infinite":
> > An arbitrary set, S, is "finite" if and only if there is some n in N for
> > which there is a bijection between S and N_n. Otherwise, S is infinite.
>
> No bijections, please, without specifics on the function. Not all bijections
> are valid.

Which bijections are not valid, and why?
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?

Not unless you can proof your silly statement.
>
> >
> > 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.

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).

So which is it?

The injection/bijection definition has the virtue of creating
cardinalities which, assuming choice, satisfying the order properties of
a total order:

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?

We wait with unbated breath.
.

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