Z.F.C. Axiom Of INFINITY!

[Graham Cooper]

EXIST(N):
0eN & Ax xeN->s(x)eN


--------------------------


I have to side with W.M. on this one!

This is just Peano's Postulates with '&"
put between them!

Is Induction supposed to hold here?

Ax xeN!



-----------------------------

All A.O.I. states is that

0 e N
s(0) e N
s(s(0)) e N
s(s(s(0))) e N
and so on...

So N has un-bounded size.

Please state where INFINITY EXISTS here!


Herc
--
www.BLoCKPROLOG.com

[Virgil]

In article
<ddf02189-88d4-4dfa...@vx13g2000pbb.googlegroups.com>,

Graham Cooper <graham...@gmail.com> wrote:

> I have to side with W.M. on this one!

The more fool you!
--

[Graham Cooper]

On May 17, 10:23 am, Virgil <vir...@ligriv.com> wrote:
> In article
> <ddf02189-88d4-4dfa-82d6-d78630a82...@vx13g2000pbb.googlegroups.com>,

>  Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > I have to side with W.M. on this one!
>
> The more fool you!
>

WM's main point is very clear on this particular issue.

[fom]

First, you should actually take the time
to look at a book on set theory and cite
the axiom precisely.

What you are calling the axiom of infinity
is a fantasy of a self-described "smarter than
the rest of us".

Second, you should actually understand what
the axioms of first-order logic presume when
a theory states that an object exists in the
theory.

Third, you should ask yourself why your self-esteem
apparently depends on whether or not an *abstract*
"object" has a metaphysical existence.

Finally, you should start posting all of your
comments in machine language until such time as
you come to understand that "computerese" is
a false paradigm that has an entire hierarchy of
presuppositions that runs backward through compilers,
assemblers, architectures, solid state physics, and
field theories.

You are like the do-gooder on a street corner criticizing
women wearing mink stoles without concern for the habitat
destruction caused by the cotton fields that provide the
cloth for your animal-friendly clothing.

Grow up.

[Graham Cooper]

It's just a simple Post on UNBOUNDED vs INFINITE size!

It made me think of a defn of infinity since ZG asked for one.

----INFINITY-----

If a TERM exists - a
and a FUNCTION exists - f
with the term as its domain

and the FUNCTION is also in the domain of it's own function
and a constant EPSILON exists > 0

such that

| f(Z) | > | Z | + Epsilon

where Z is of type

Z ::= term
or
Z ::= f( Z )

then the closure of f on Z has | Z | = Infinity


---------------

Herc
--
www.BLoCKPROLOG.com

[Virgil]

In article
<2ad3740b-d2fe-477a...@ua8g2000pbb.googlegroups.com>,

IN ZF by axiom!
--

[Graham Cooper]

On May 17, 10:58 am, Virgil <vir...@ligriv.com> wrote:
> In article

> <2ad3740b-d2fe-477a-810b-a58a97325...@ua8g2000pbb.googlegroups.com>,

>  Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > On May 17, 10:23 am, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <ddf02189-88d4-4dfa-82d6-d78630a82...@vx13g2000pbb.googlegroups.com>,
> > >  Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > > I have to side with W.M. on this one!
>
> > > The more fool you!
>
> > WM's main point is very clear on this particular issue.
>
> > So N has un-bounded size.
> > Please state where INFINITY EXISTS here!
>
> IN ZF by axiom!
>

Where is INFINITE SIZE defined

EXIST(N):
0eN & Ax xeN->s(x)eN

I see PEANO POSTULATE 1 & PEANO POSTULATE 2

-------------------

1 nat 0
2 nat [ s X ]
nat X


These are on the Front page of www.BLoCKPROLOG.com too!

Herc

[Virgil]

In article
<710803a5-5aa9-4af7...@oy6g2000pbb.googlegroups.com>,

Graham Cooper <graham...@gmail.com> wrote:

> On May 17, 10:58�am, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <2ad3740b-d2fe-477a-810b-a58a97325...@ua8g2000pbb.googlegroups.com>,
> > �Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> > > On May 17, 10:23�am, Virgil <vir...@ligriv.com> wrote:
> > > > In article
> > > > <ddf02189-88d4-4dfa-82d6-d78630a82...@vx13g2000pbb.googlegroups.com>,
> > > > �Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> > > > > I have to side with W.M. on this one!
> >
> > > > The more fool you!
> >
> > > WM's main point is very clear on this particular issue.
> >
> > > So N has un-bounded size.
> > > Please state where INFINITY EXISTS here!
> >
> > IN ZF by axiom!
> >
>
>
> Where is INFINITE SIZE defined

Lots of places. One definition says that a non-empty set is finite if
every ordering of it has a last element and is infinite if some ordering
of it does not have last element.

The general ordering above can be replaced by well-ordering.

An alternate, but equivalent definition says a non-empty set is finite
if every injection from it to itself is a surjection and is infinite if
some injection to itself is not a surjection.

I am not offhand aware of any definition of being countably infinite
that does not involve sets in the definition.
>
--

[fom]

On 5/16/2013 7:39 PM, Graham Cooper wrote:
>>
>
> It's just a simple Post on UNBOUNDED vs INFINITE size!
>
> It made me think of a defn of infinity since ZG asked for one.
>

Herc,

Please accept my apologies.

I simply find these arguments tedious.

There are mathematical approaches to
these matters. The Russian school of
constructive mathematics shows how it
is done.

But, this is different from classical
mathematics. The fact that things are
different does not make either of them
"wrong".

[Graham Cooper]

You didn't comment on my posts

DEFINITIONAL SET THEORY

or

6 QUANTIFIER ELIMINATION METHODS

If you pick and choose posts I'm sure everybody can find threads on
usenet they are NOT interested in.

But that would be like constructing NON_MEMBERS of a universal set.

If JACK always says "I DONT LIKE POSTS"

then how can you be universally likable?

Herc

[Virgil]

In article
<2c4aa273-09fc-4c87...@ve4g2000pbb.googlegroups.com>,

Graham Cooper <graham...@gmail.com> wrote:

> >
> > Please accept my apologies.
> >
> > I simply find these arguments tedious.
> >
> > There are mathematical approaches to
> > these matters. �The Russian school of
> > constructive mathematics shows how it
> > is done.
> >
> > But, this is different from classical
> > mathematics. �The fact that things are
> > different does not make either of them
> > "wrong".
>
> You didn't comment on my posts

Such lack of comment is a comment.
--