From: Virgil on
In article <460e5198(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> Virgil wrote:
> > In article <1175275431.897052.225580(a)y80g2000hsf.googlegroups.com>,
> > "MoeBlee" <jazzmobe(a)hotmail.com> wrote:
> >
> >> On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote:
> >>
> >>> They
> >>> introduce the von Neumann ordinals defined solely by set inclusion,
> >> By membership, not inclusion.
> >
> > By both. Every vN natural is simultaneously a member of and subset of
> > all succeeding naturals.
> >
>
> Yes, you're both right. Each of the vN ordinals includes as a subset
> each previous ordinal, and is a member of the set of all ordinals.

In ZF and in NBG, there is no such thing as a set of all ordinals.
In NBG there may be a class of all ordinals, but in ZF, not even that.





>
> Anyway, my point is that the recursive nature of the definition of the
> "set" introduces a notion of order which is not present in the mere idea
> of membership. Order is defined by x<y ^ y<z -> x<z.

That is only a partial ordering on sets, and on ordinals is no more than
the weak order relation induced by membership, namely:
x < y if and only if either x e y or x = y.



This is generally
> interpreted as pertaining to real numbers or some subset thereof, but if
> you interpret '<' as "subset of", then the same rule holds. I suppose
> this is one reason why I think a proper subset should ALWAYS be
> considered a lesser set than its proper superset. It's less than the
> superset by the very mechanics of what "less than" means.

But your version of "less than" is is only a partial order

> >
> > On the other hand, Tony Orlow is considerably less of an ignoramus than
> > Lester Zick.
>
> Why, thank you, Virgil. That's the nicest thing you've ever said to me.

You would not think so if you knew my opinion of Zick.
From: Lester Zick on
On Fri, 30 Mar 2007 12:17:31 -0600, Virgil <virgil(a)comcast.net> wrote:

>In article <1175275431.897052.225580(a)y80g2000hsf.googlegroups.com>,
> "MoeBlee" <jazzmobe(a)hotmail.com> wrote:
>
>> On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote:
>>
>> > They
>> > introduce the von Neumann ordinals defined solely by set inclusion,
>>
>> By membership, not inclusion.
>
>By both. Every vN natural is simultaneously a member of and subset of
>all succeeding naturals.
>
>> > and
>> > yet, surreptitiously introduce the notion of order by means of this set.
>>
>> "Surreptitiously". You don't know an effing thing you're talking
>> about. Look at a set theory textbook (such as Suppes's 'Axiomatic Set
>> Theory') to see the explicit definitions.
>
>On the other hand, Tony Orlow is considerably less of an ignoramus than
>Lester Zick.

Who is considerably less of an ignoramus than you.

~v~~
From: Lester Zick on
On 30 Mar 2007 11:29:29 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote:

>On Mar 30, 11:17 am, Virgil <vir...(a)comcast.net> wrote:
>> In article <1175275431.897052.225...(a)y80g2000hsf.googlegroups.com>,
>>
>> "MoeBlee" <jazzm...(a)hotmail.com> wrote:
>> > On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote:
>>
>> > > They
>> > > introduce the von Neumann ordinals defined solely by set inclusion,
>>
>> > By membership, not inclusion.
>>
>> By both. Every vN natural is simultaneously a member of and subset of
>> all succeeding naturals.
>
>Of course. I just meant as to which is the primitive, in the sense
>that the definitions revert ultimately solely to the membership
>relation.

How about the "domain of discourse" relation, Moe(x)?

~v~~
From: Lester Zick on
On Sat, 31 Mar 2007 07:18:15 -0500, Tony Orlow <tony(a)lightlink.com>
wrote:

>Virgil wrote:
>> In article <1175275431.897052.225580(a)y80g2000hsf.googlegroups.com>,
>> "MoeBlee" <jazzmobe(a)hotmail.com> wrote:
>>
>>> On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote:
>>>
>>>> They
>>>> introduce the von Neumann ordinals defined solely by set inclusion,
>>> By membership, not inclusion.
>>
>> By both. Every vN natural is simultaneously a member of and subset of
>> all succeeding naturals.
>>
>
>Yes, you're both right. Each of the vN ordinals includes as a subset
>each previous ordinal, and is a member of the set of all ordinals. In
>this sense, they are defined solely by the "element of" operator, or as
>MoeBlee puts it, "membership". Members are included in the set. Or,
>shall we call it a "club"? :)
>
>Anyway, my point is that the recursive nature of the definition of the
>"set" introduces a notion of order which is not present in the mere idea
>of membership. Order is defined by x<y ^ y<z -> x<z. This is generally
>interpreted as pertaining to real numbers or some subset thereof, but if
>you interpret '<' as "subset of", then the same rule holds. I suppose
>this is one reason why I think a proper subset should ALWAYS be
>considered a lesser set than its proper superset. It's less than the
>superset by the very mechanics of what "less than" means.
>
>>>> and
>>>> yet, surreptitiously introduce the notion of order by means of this set.
>>> "Surreptitiously". You don't know an effing thing you're talking
>>> about. Look at a set theory textbook (such as Suppes's 'Axiomatic Set
>>> Theory') to see the explicit definitions.
>>
>> On the other hand, Tony Orlow is considerably less of an ignoramus than
>> Lester Zick.
>
>Why, thank you, Virgil. That's the nicest thing you've ever said to me.

Virgil speaketh in tongues and truisms.

~v~~
From: Virgil on
In article <460e5476(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> Virgil wrote:
> > In article <460d4813(a)news2.lightlink.com>,
> > Tony Orlow <tony(a)lightlink.com> wrote:
> >
> >
> >> An actually infinite sequence is one where there exist two elements, one
> >> of which is an infinite number of elements beyond the other.
> >
> > Not in any standard mathematics.
>
> Well, "actually infinite" isn't a defined term in standard mathematics.

If it is outside the pale, why bother with it at all?
>
> >
> > In standard mathematics, an infinite sequence is no more than a function
> > whose domain is the set of naturals, no two of which are more that
> > finitely different. The codmain of such a function need not have any
> > particular structure at all.
>
> That's a countably infinite sequence. Standard mathematics doesn't allow
> for uncountable sequences like the adics or T-riffics, because it's been
> politically agreed upon that we skirt that issue and leave it to the
> clerics.

What sect would have been so foolish as to have ordained TO into its
priesthood?



However, where every element of a set has a well defined
> successor and predecessor, it's a sequence of some sort.

Every such "sequence" set must be representable as a function from the
from the integers to that set.