Re: Orlow cardinality question

In article <MPG.1d2a15247d3c84f6989ef1@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > 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.
> Except that that definition also applies to {000...000,000...001, ...
> , 999...998,999...999}, which has a smallest and a largest element,
> is totally ordered, and is infinite. Hmmm......

In order for any abc...xyz to exist as a number in an _ordered_ set,
one must be able to index the digits by a discretely and totally
ordered index set so that one can tell by comparing the indices of the
non-zero digits which of, say, 000...050...000 or 000...060...000 is
actually larger. Note that for finite strings, the number of zero's
represented by different ellipses, "...", makes a difference in which
number would be larger.

As TO has not done that, there is no saying whether his "set" is
totally ordered by the supposedly indicated ordering or only partially
orderable. If TO's "set" cannot be totally ordered, and shown to be
totally ordered, then TO's objection is out of order.



> > 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.

> No, of course you can imagine anything you want. Just don't call it
> "correct".

Nor incorrect, without better reason than TO has come up with.

The issue is not correctness anyway , but self-consistency. math that
is self-consistent is good math. Math that is not self-consistent is
bad math and gets tossed out.

No one, least of all TO, has \ been able to show that math as it is now
is not self-consistent, while many have shown that TO's constructions
are not self-consistent.

> >
> > 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.
> No, you are the one who conflates largest members to signify finite
> sets, which is unwarranted. You must have just pulled that out of
> your ass, or if not, you should be able to prove it.

I have proved it indirectly by proving the the equivalent
contrapositive statement, namely that an ordered set having any
non-empty subset, including itself, without both a maximim and a
minimum member allows an injection ointo a proper subset, and is
therefore, of infinite cardinality.



> > > > 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.
> So, that's why you don't believe in infinite numbers. I see. Except,
> how do you even know what they are enough to not believe in them?

I don't believe in Santa Claus, but that doesn't mean I am totally
unaware of what a Santa Claus would have to be like if there were one.
> >
> > > Does {{}} really contain any natural numbers
> >
> > It does by definition!
> A set containing only the null set contains natural numbers?

I did not say that, what I said was that it contained at least one (any
means at least one, but does not require more than one), and in the NBG
formulation, {} is a natural number, so {{}} contains one natural number.
.

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