Re: Orlow cardinality question

In article <MPG.1d24d5a92ab5c385989eb6@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> stephen@xxxxxxxxxx said:

> > First, we need to define "finite" and "infinite".
> >
> > A set X is infinite if there exists a bijection from X to a proper
> > subset of X.
> >
> > A set is finite if it is not infinite.


Alternately, a set is finite if for any ordering of its members every
non-empty subset, including the set itself, has a greatest or last
member (this also implies existence of a first member since the reverse
ordering has a last member).

Then a set is infinite if it is not finite, i.e., it can be ordered so
that some non-empty subset does not have a last member (and therefore
the set itself does not have a last member).

The above definition of finiteness can be used to show that a set is
finite if and only if it has no injection into any proper subset, so
that it is a mere matter of convenience which definitions one chooses to
use.

> >
> > We now need to describe the natural numbers in terms of sets.
> > First, let S(x) = { x + {x}}, where "+" is the union operator.

That should be S(x) = x + {x}, so that S(x) will contain all the
members of x and one new object, x itself.

> > We define N, the set of natural numbers, recursively:
> > 1. {} is in N
> > 2. if n is in N, then S(n) is in N
> > 3. nothing else is in N

Instead of condition 3, one might equally well say that N is the
intersection of all sets satisfying conditions 1 and 2.
> >
> > We can now prove that each n in N is finite using induction.
> >
> > Base Case. {} is finite. {} has no proper subsets, so there
> > cannot exist a bijection from {} to one of its proper subsets.
> >
> > Inductive Step. If n is finite, then S(n) is finite. Assume that
> > S(n) is infinite. This means there exists a bijection f from S(n)
> > to some subset V of S(n). We can assume that V does not contain
> > {n}. Given f we can construct a bijection from n to a proper
> > subset of n. Simply remove f({n}) from f, and we know have a
> > bijective function from n to V-(f{n}). This means n is infinite,
> > which contradicts the inductive hypothesis.
> >
> > Therefore, each n in N is finite.
> >
> > N itself is infinite because S(x) is a bijection from N to a proper
> > subset of itself, namely N-{{}}.
> >
> > So which part of this proof do you not accept?
> >
> > Stephen
> >
> > Base Case. {} is finite. {} has no proper subsets, so there
> > cannot exist a bijection from {} to one of its proper subsets.
> >
> > Inductive Step. If n is finite, then S(n) is finite.
> >
> >
> >
> >
> > From: stephen@xxxxxxxxxx Subject: Re: Orlow cardinality question
> > Newsgroups: sci.math References:
> > <MPG.1d013904c3d6d9bf989d74@xxxxxxxxxxxxxxxxxxxxxxxxx>
> > <MPG.1d1b8bc5ac093dc0989e03@xxxxxxxxxxxxxxxxxxxxxxxxx>
> > <ITSnetNOTcom#virgil-6226B7.16004816062005@xxxxxxxxxxxxxxxxxxxxxxxx>
> > <MPG.1d1cc41db70e5519989e18@xxxxxxxxxxxxxxxxxxxxxxxxx>
> > <d8uu03$fbq$4@xxxxxxxxxxxx>
> > <MPG.1d1cf7572a24b6ed989e2a@xxxxxxxxxxxxxxxxxxxxxxxxx>
> > <d8vcrk$t13$3@xxxxxxxxxxxx>
> > <MPG.1d20c0b0a420a4e8989e44@xxxxxxxxxxxxxxxxxxxxxxxxx>
> > <d975m3$7no$3@xxxxxxxxxxxx> <MPG.1d220acc6032fcf6989e62
> @newsstand.cit.cornell.edu>
> >

> Within the world of mathematics, one can compare results from
> different areas for consistency. That's the immediate environemtn of
> cardinality. Mathematics exists within a larger world of science and
> logic, and should agree with those larger areas, which in turn should
> agree with observed phenomena in the world. It's a matter of levels
> of abstraction from concrete reality.


The world of imagination, in which numbers exist if they are to exist
at all, is not constrained by the need to correspond to any physical
reality at any level of abstraction.
>

>
> Maybe I am too philosophical for this group, but math and science
> have grown out of philosophy in the past, so maybe I'm just what you
> need.

Not hardly!
> >
> > <snip>

> > But you want to limit everyone else by what you can't fathom.


> What is it I can't fathom?

The way that mathematics works, and has to work in order to be workable.


> > I have never seen anyone use set theory to draw a conclusion about
> > " numbers and strings and trees that violate their properties",
> > other than you and some other cranks. You are the one who insists
> > that for some finite n, n+1 is infinite.

> No, I never claimed that

Consider the definitions of finite and infinite above. TO claims that
the ordered set of "finite" naturals is finite but has no largest
element. These two claims are mutually exclusive. So that by claimimg
that the set of finite naturals has no largest member, he is admitting
that it is not a finite set even while he is claiming that it is a
finite set.


> >
> > I would agree that a set IS the elements in the set. But what does
> > that have to do with anything? We can still look at properties of
> > the set that do not depend on all the properties of the elements of
> > the set. It is known as abstraction, and it is a useful skill in
> > mathematics and in the real world.
>
> But, there is nothing you can say about the size of an infinite set
> without looking at the properties of the elements. A set is simply a
> collection of things, so with that simple abstract definition, there
> is really no other measure of the set than that number. What other
> properties of a finite set can you discuss without reference to
> properties of the elements? How can you even discuss relative
> infinite set sizes without reference to those properties?

>
> Incorrect. Convergent series have a definite finite sum.

Only in the sense that there is a limit to the sequence of finite
partial sums, but this limit is not an element of that sequence.


> >
> > There are no infinite natural numbers. Every natural number is
> > finite, and every natural number has a finite successor.

> mantra...mantra....

if so it is a mantra that leads to truth, unlike TO's mantras.
> >
> > Part of your problem is that you do not even have a working
> > definition of "finite" or "infinite". Elsewhere you said that a
> > number is finite if it is smaller than every infinite value. What
> > then is your definition of "infinite"?

> Without end or bound. What's your general definiton of infinite?

Only defined for sets, and not otherwise, and for a suitable definition,
see above.

> Does {{}} really contain any natural numbers

It does by definition!
.

Relevant Pages

  • Re: Orlow cardinality question
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