Re: Infinite sets.

zuhair wrote:
Jonathan Hoyle wrote:
Hi,

I have a question.
If we can define "infinite set" in the following manner.
X is infinite <-> Ax: x in X -> ES(x):S(x) in X.
were S(x) stands for the successor of x.

If I read this correctly, your definition of "infinite" will not
include the set of even numbers X = { 0, 2, 4, ... } since for no
element x in X is S(x) in X.

Likewise, by your definition, the empty set is infinite, since you
cannot find an x in X where S(x) is missing.

In the end, the question is simply this: why are you trying to create
new "definitions" for mathematical terms that are already appropriately
defined?

IF we can define the term "infinite set" in a simpler manner than that
of Dedekind, and at the same in a non-counter-intuitive manner, then
what is the harm in that.

I have a simpler definition of "infinite set" than that of Dedekind.

Simply first I will derive my definition of infinite set from the Axiom
of infinity.

since Axiom of Infinity states that N exists were N={}eN
/\(Ax:xeN->S(x)eN).

No, first we prove that there is a unique set that is a subset of all
sets that are as described by the axiom of infinity. THEN we define
omega to be that unique set. We've been over this already.

So by axiom of infinity Omega is an infinite set.

Now we define x is infinite as below:

x is infinite<->Ey: y is a subset of x /\ y<<>>N.

Def: x is infinite <-> x is not equinumerous with a natural number.

That definition does not need the axiom of infinity. And it's simpler
than your definition.

And, with the denumerable choice, they're equivalent anyway.

I use the symbole >> for injection so x>>y means x is injective to y.
and I use the symbole
for surjection so x>>y means x is surjective to y. and I use <<>> to mean bijection, so
x<<>>y means that x is bijective to y.

so the essence of this definition of infinite set is: A set is said to
be infinite iff a subset of it is bijectable to N, were N is defined by
axiom of infinity.

It's already a theorem of set theory with denumerable choice. We don't
need it as a definition.

This is better than the dedekindian defintion of "infinite set" since
it is not counter-intuitive.

With the axiom of denumerable choice, it's equivalent to both the
definition I just gave (call it the 'Tarski definition') and to the
Dedekind definition.

and yet applicable to all "infinite sets",
unlike dedekindian definition which I think it is not applicable to all
infinite sets.

With denumerable choice, every infinite set is Dedekind infinite.

Even without denumerable choice, the Tarski definition is especially
simple and does not require the axiom of infinity.

MoeBlee

.

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