Re: Definition of finite.

On May 3, 6:06 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On May 3, 4:01 pm, zuhair <zaljo...@xxxxxxxxx> wrote:

On May 3, 5:57 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

In Z without the axiom of infinity, the 'R and conv(R)' definition is
equivalent to Tarski's 'subset maximal' definition, which is
equivalent to other definitions, and, in Z, all of these are
equivalent to 'member of w'.

Meanwhile your 'essentially circular' and 'quasi-circular' are your
own self-distractions.

Nup.

Check your keyboard. It seems to have crossed 'N' for 'Y'.

MoeBlee

Ok, Moe let me put matters in this way.

Which constitutes a more plausable appraoch?

To define 'natural number' in terms in finitude
Or
To define 'finitude' in terms of 'natural number'.

Lets say that definition D1 is simpler than definition D2
if D1 requires less number of axioms then D2 requires.

Lets say that a definition D1 is more stable than definition D2
if D1 require more number of axioms to change before it changes
than do D2 require.

Let me clarify.

you see lets take the definition of 'x is a natural number'.

How do you define this in Z-I-R in an independent manner
from finitude.

I think the definition would be

x is a natural number <-> x is an ordinal.

were 'ordinal' is defined in the standard manner.

Now how do you define it in Z-R

you see the definition will CHANGE to the following:

x is a natural number <-> xew

While lets take the opposite approach

x is a natural number <-> x is a finite ordinal.

were

x is finite <-> EREconvR( R well order x & convR well order x).

and ordinal defined in the standard manner.

Now this definition is the same in Z-I-R and in Z-R

So the later definition is more stable than the first.

Not only this, the later definition is Simpler than the first
since it requires Z-I-R only

while the other is more complex it is actually two definitions
depending on the presence of infinity or not.

From that I see that it is easier to define

x is finite , in a manner that doesn't depend on 'natural number'

and THEN we define 'natural number' as a finite ordinal.

And not the opposite manner of Tarski's.

Tarski is defining the Hen after the egg it layed,
While my approach define the Hen after the egg it hatched from.

I don't know, it seems common sense for me, to
define 'finite set' first in a manner that is independent
of 'natural number'.

And after that we proceed to define 'natural number'
as a special case of finite set like natural number is a finite
ordinal.

This constitutes a simpler, more plausable and more stable
approach than the opposite one.

I don't know, it seems a matter of preference.

Zuhair




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