Re: Definition of finite.

On May 5, 11:34 am, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 3, 7:38 pm, hru...@xxxxxxxxxxxxxxxxxxxx (Herman Rubin) wrote:
Now it would be interesting to see how one can define
natural number in Z-I-R

We don't need axioms to make these definitions. I keep explaining that
to you and you keep ignoring what I wrote. And I've already said that
we don't even have to mention omega to define 'natural number'.

n is a natual number <-> (n is finite & n is an ordinal)

where 'finite' is defined as any of the equivalents of 'well ordered
by an R and its converse', and 'ordinal' is defined as 'well ordered
by epsilon and is epsilon-transitive'.

I propose the followingdefinitionof x is a natural number in Z-I-R

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

I think this would do the job.

That's IDIOTIC.

And I ALREADY addressed that. You just skip what I wrote.

Are you thinking that because Z-R-I can't prove the existence of an
infinite set (thus, a fortiori, of an infinite ordinal) it follows
that we might as well take the naturals to be the ordinals?

If that's what you're thinking, it's IDIOTIC and you are RELAPSING
into illogical thinking that we ALREADY have been through with you.
What I thought you finally understood by this time is that even though
a theory might not prove the existence of objects having a certain
property, it doesn't follow that the theory RULES OUT that there are
such objects. Even though Z-R-I doesn't prove the existence of any
infinite ordinals, it still doesn't RULE OUT that they exist. So the
definition of 'natural number' still must be stated so as to rule out
that a natural number could be an infinite ordinal, otherwise it is
left UNdetermined whether there are natural numbers that are infinite
when what we WANT to do is make it DETERMINED that there are NOT
natural numbers that infinite.

However in this theory there is no need to define x isfinite, since
every x in Z-I-R isfinite.

NO!!! NO!!! NO!!!

We went through that with you about a year ago!!!

In Z-I-R it is UNDETERMINED whether every set is finite. In Z-I-R it
is NOT determined that every set is finite.

A theory that DOES determine that every set is finite is Z-R-I + ~I.
That is, Z-R-I plus the NEGATION of the axiom of infinity. There, yes,
all sets are finite. But in just Z-R-I, it is UNDETERMINED whether
there are infinite sets, thus UNDETERMINED whether all sets are
finite.

Come on, zuhair, we spent a lot of work a year ago to finally get you
to understand that but now you're repeating your old misconceptions
again.

MoeBlee

.

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