Proof that w in w

[Julio Di Egidio]

Proof that w in w
-----------------

With w the first limit ordinal, per definition we have:

w = lim_{n<w} Fn

where Fn is the n-th Fison (finite initial segment of natural numbers).

We prove that w in w.

Since a discreet metric applies, the limit in question, i.e. w, exists only
in so far as eventually all members of Fn contain it as a member. (!)

But the limit does exist, so long as a set of all natural numbers exists
(which is defined to be w itself), hence we have:

exists n in w : w in Fn

OTOH, it is immediate to see that for n < w, Fn is itself finite, i.e. it
cannot contain w as an element, whence:

w in Fn => n in w, not n < w

But w is our *first* infinite ordinal, and, by an application of the pigeon
principle, it must be the case that:

w in Fn <=> n = w

whence w in Fw, where indeed Fw = lim_{n<w} Fn = w.

QED.

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

We could then better write: w =def= lim_{n<=w} Fn. Circular does not hurt, off
by one does! :)

Feedback welcome.

Thank you,

Julio

[William Elliot]

On Sat, 20 Feb 2016, Julio Di Egidio wrote:

> Proof that w in w

What set theory are you using to prove this? Zermelo-Fraenkel (ZF),
Quine's New Foundations (NF) or Von Neumann-Bernays-Godel (NBG)

> With w the first limit ordinal, per definition we have:
> w = lim_{n<w} Fn
>
> where Fn is the n-th Fison (finite initial segment of natural numbers).

How are you defining limit?

> We prove that w in w.

How so? Regularity or foundations prevents that.

> Since a discreet metric applies, the limit in question, i.e. w, exists only
> in so far as eventually all members of Fn contain it as a member. (!)

No, the neighborhoods of w aren't singletons.

> But the limit does exist, so long as a set of all natural numbers exists
> (which is defined to be w itself), hence we have:
>
> exists n in w : w in Fn

I dare you to prove that claim, that w is in F_n for any n in w.

> OTOH, it is immediate to see that for n < w, Fn is itself finite, i.e. it
> cannot contain w as an element, whence:
>
> w in Fn => n in w, not n < w

Since w not in F_n, assuming w in F_n gives you any conclusion you want.

> But w is our *first* infinite ordinal, and, by an application of the pigeon
> principle, it must be the case that:
> w in Fn <=> n = w

The pigeon principle isn't needed to show
that two false statements are equivalent.

> whence w in Fw, where indeed Fw = lim_{n<w} Fn = w.

Why all this on the other handed stuff when you
previously proclaimed that w in Fw is obvious.

> QED.

Thou doest prove thee makest baffoonery.

[Virgil]

In article <685f7319-ba16-4055...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> Proof that w in w
> -----------------
>
> With w the first limit ordinal, per definition we have:
>
> w = lim_{n<w} Fn
>
> where Fn is the n-th Fison (finite initial segment of natural numbers).

A Fison is a finite initial SET of natural numbers, and your
"w = lim_{n<w} Fn", being a circular definition, does not define anything

> Since a discreet metric
No metric applies

> exists n in w : w in Fn

Nonsense!

> But w is our *first* infinite ordinal, and, by an application of the pigeon
> principle, it must be the case that:
>
> w in Fn <=> n = w

Thus for every n in |N, w not in Fn and n =/= w
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)

[Julio Di Egidio]

On Sunday, February 21, 2016 at 8:31:06 AM UTC, William Elliot wrote:
> On Sat, 20 Feb 2016, Julio Di Egidio wrote:
>
> > Proof that w in w
>
> What set theory are you using to prove this? Zermelo-Fraenkel (ZF),
> Quine's New Foundations (NF) or Von Neumann-Bernays-Godel (NBG)

I'd venture nothing more than Peano Axioms and standard discrete topology is needed for this proof. Note that we are only dealing with the set of ordinals up to and including w.

> > With w the first limit ordinal, per definition we have:
> > w = lim_{n<w} Fn
> >
> > where Fn is the n-th Fison (finite initial segment of natural numbers).
>
> How are you defining limit?

I am using the following set-limit definitions, with a discrete metric:
<https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior#Generalized_definitions>

> > We prove that w in w.
>
> How so? Regularity or foundations prevents that.

The present proof may also be an argument against foundation and pro anti-foundation: but these would be the consequences of the proof.

> > Since a discreet metric applies, the limit in question, i.e. w, exists only
> > in so far as eventually all members of Fn contain it as a member. (!)
>
> No, the neighborhoods of w aren't singletons.

Of course they are, a set of ordinals is a discrete set.

Julio

[Martin Shobe]

On 2/21/2016 12:59 AM, Julio Di Egidio wrote:
> Proof that w in w
> -----------------
>
> With w the first limit ordinal, per definition we have:
>
> w = lim_{n<w} Fn
>
> where Fn is the n-th Fison (finite initial segment of natural numbers).
>
> We prove that w in w.
>
> Since a discreet metric applies, the limit in question, i.e. w, exists only
> in so far as eventually all members of Fn contain it as a member. (!)

Doesn't seem likely. Can you prove this?

Martin Shobe

[Julio Di Egidio]

Prove what, that a discreet metric applies or that I am applying it correctly? The former again is because any set of ordinals is a discrete set, the latter is your home work if you don't know.

Julio

[Ross A. Finlayson]

Some other notions why (or where)
omega is extra-ordinary or compact
or "infinity contains itelf":

1) In a theory with quantification
over finite ordinals, and where a
collection of ordinals < alpha = alpha,
then omega is all of them, and also
Russell's set of all sets which contain
themselves, and does contain itself.

There are other examples in various
set- and number-theoretic contexts
(the compactness of N, point-at-
infinity, etcetera). There are
examples where half or most of the
integers are infinite, eg "bounded
infinity" or the hyperintegers.

[Julio Di Egidio]

On Sunday, February 21, 2016 at 7:08:34 PM UTC, Ross A. Finlayson wrote:
> On Saturday, February 20, 2016 at 10:59:54 PM UTC-8, Julio Di Egidio wrote:

<snip>

> > We could then better write: w =def= lim_{n<=w} Fn.
> > Circular does not hurt, off by one does! :)
>

> Some other notions why (or where)
> omega is extra-ordinary or compact
> or "infinity contains itelf":
>
> 1) In a theory with quantification
> over finite ordinals, and where a
> collection of ordinals < alpha = alpha,
> then omega is all of them, and also
> Russell's set of all sets which contain
> themselves, and does contain itself.
>
> There are other examples in various
> set- and number-theoretic contexts
> (the compactness of N, point-at-
> infinity, etcetera). There are
> examples where half or most of the
> integers are infinite, eg "bounded
> infinity" or the hyperintegers.

Thank you, though note that we have more than just yet another non-standard theory here: if the proof is valid, it *rules out* any set theory where w not in w.

Julio

[Virgil]

In article <nad04u$iql$1...@dont-email.me>,

Martin Shobe <martin...@yahoo.com> wrote:

> On 2/21/2016 12:59 AM, Julio Di Egidio wrote:
> > Proof that w in w
> > -----------------
> >
> > With w the first limit ordinal, per definition we have:
> >
> > w = lim_{n<w} Fn
> >
> > where Fn is the n-th Fison (finite initial segment of natural numbers).
> >
> > We prove that w in w.

If w is a set, then w is always a subset of w but never a member of w.

[Ross A. Finlayson]

"Regular" ("ordinary", well-founded, often
well-ordered) theories with infinity can
have bounded fragments of arithmetic then
also quantification over elements with a
"rather regular transfer principle", then
for what those are, as they are within a
greater theory that has an extra-ordinary
omega.

This can be seen as the difference between
numbering, as we have here, and counting,
which is as of a regular arrangement (that
there are more fundamentally numerical
arrangements).

Then for something like the reals as both
complete ordered field and partially ordered
ring of iota values with rather restricted
transfer principle, it is in these terms.

I agree that the real omega is like the
Ding-an-Sich (Kant).

Then that N = N+ and that N e N and w e w,
these are not unknown concerns, rather
they're extra-ordinary the usual regular.

Regular logical theories are within an
axiomless (logical) system of natural
deduction.

Use an appropriate tool for the domain,
the theory of everything is logical and
scientific.

[Moufang Loop]

What is a well-ordered theory?

[Ross A. Finlayson]

Theories with the well-ordering principle,
or an axiom, or Zorn's lemma or Ono, Axiom
of Choice, that there is a well-ordering for
the elements of any object, for induction,
these are well-ordered theories.

Some might have that a theory is more originally
well-"ordered" than well-"founded". In set
theories like ZF, well-ordering follows well-
founding as ZFC, where ZF first seeks to
establish a well-founded infinity. Zermelo
for example was a strong advocate of the
well-ordering principle as "structurally
self-evident", ZF "would be" ZFC.

So, well-order the reals. There are
various eventually compatible ways
they can be constructed or derived.

[Moufang Loop]

Ross A. Finlayson wrote:

> So, well-order the reals.

After you.

[Ross A. Finlayson]

There aren't uncountably many standard reals
in their normal ordering in a well-ordering.
(This is where then induction would find a
rational between each pair of standard reals.)
Then, the "natural/unit equivalency function"
or "sweep" or "drawing the line" is well-ordering
the unit line segment. You'll excuse me as this
is off-topic to JdE's so you can further look
into well-ordering the reals or my opinion
separately, about a geometry of points and spaces.

[Jim Burns]

On 2/21/2016 1:59 AM, Julio Di Egidio wrote:

> Proof that w in w
> -----------------
>
> With w the first limit ordinal, per definition we have:
>
> w = lim_{n<w} Fn
>
> where Fn is the n-th Fison
> (finite initial segment of natural numbers).
>
>
> We prove that w in w.
>
> Since a discreet metric applies, the limit in question,
> i.e. w, exists only in so far as eventually all members of Fn
> contain it as a member. (!)

I am reading your "eventually" in your definition of the limit
as saying (more generally):

if the set Flim = lim_{n<w} Fm exists, then
x e Flim iff for some m e w, all k > m, x e Fk

That sounds familiar to me. I have no problem with that as
a definition of a set limit.

I don't see where you get your next statement, that (more generally)
Flim exists (or maybe Flim is the limit?) only if
for some m e Flim, all k > m, Flim e Fk

That is the requirement for Flim (or w ) being a _member_ of Flim,
not for it being the limit or for it existing.

If instead of this sequence of sets of naturals Fn,
we had a sequence of sets of reals An, where An = ( 1/n, 1)
then we could, using the same definition of limit, say
lim_{n<w} ( 1/n, 1 ) = ( 0, 1 )

But the open interval (0,1) is not an _element_ of (0,1)
nor of any set in the sequence. It is not a real number,
it is a set of real numbers.

I think the reason it is less clear in your case is that
the elements of the sets in the sequence can also be sets
in the sequence, being ordinals. But w being the limit
of the sets is not a reason for it to be _in_ any set,
as far as I can see.


And this is a good way to show that Flim (or w ) is _not_ in Flim.
For all k, k < w (that is the limit stated, after all)
and each Fk is defined as { j | j =< k } .
Therefore, for all k, all j, j e Fk, j =< k < w, and w ~e Fk
For all k, w ~e Fk, therefore w ~e Flim = w

> But the limit does exist, so long as a set of all natural numbers
> exists (which is defined to be w itself), hence we have:
>
> exists n in w : w in Fn

Sorry, I don't see how you got here.

[Jim Burns]

On 2/21/2016 3:50 PM, Ross A. Finlayson wrote:
> On Sunday, February 21, 2016 at 12:28:24 PM UTC-8,
> Moufang Loop wrote:
>> Ross A. Finlayson wrote:

>>> So, well-order the reals.
>>
>> After you.
>
> There aren't uncountably many standard reals
> in their normal ordering in a well-ordering.

This is a contradiction. the normal ordering of the reals
is not a well-ordering.

The well-ordering guaranteed by the axiom of choice
is not the normal ordering of the reals.

[Ross A. Finlayson]

It seems the contrapositive,
he establishes a symmetry about
the e-minimal element for some
e-maximal element, then fills
in the balance that both are members.

It's a matter of deduction from
some usual notion of the integers
or ordinals as a "continuum" (or
"contiguum") a la Bruch Spinoza.

You can clearly denote what features
have been assigned to the elements
that their regular (another sense,
uniform, having unitarity) progression
sees symmetry between "none" and "all".

[Moufang Loop]

I think that what Ross might have meant is that any subset of the reals
in their normal order that also happens to be well-ordered (by that
normal order) can't be uncountable.

[Ross A. Finlayson]

It was not so stated. I think you've
jumped from an internalization of sweep
where I claim the usual or normal or
natural ordering is a well-ordering of
ran(sweep). It would be a contradiction
for an uncountable well-ordering (of the
ordinal that is mapped to the elements to
so provide) to have uncountably many in
their normal ordering, but, subsets of
pairs of elements that are a well-ordering,
are a well-ordering, there would be
uncountably many following, then, in the
normal ordering (contradiction). So,
the existence of the well-ordering points
back to sweep, the line-drawing function.

That said then, and that's the summary of
my opinion, please follow on this thread
the development of: omega in omega.

[Virgil]

In article <56CA24...@att.net>, Jim Burns <james....@att.net>
wrote:

And while such a well-ordering of |R may have been proved to exist,
it has not yet been proved to be findable!

[Virgil]

In article <nad8ql$iqa$1...@news.albasani.net>,

But if RAF meant that why did he not say that?

[Virgil]

In article <56CA242D...@att.net>,

Jim Burns <james....@att.net> wrote:

> On 2/21/2016 1:59 AM, Julio Di Egidio wrote:
>
> > Proof that w in w
> > -----------------
> >
> > With w the first limit ordinal, per definition we have:
> >
> > w = lim_{n<w} Fn
> >
> > where Fn is the n-th Fison
> > (finite initial segment of natural numbers).

By what definition of "limit"?
Certainly not any delta-epsilon definition!

Note that there is no universally accepted definition for "the limit" of
a sequence of sets

[Waldek Hebisch]

Julio Di Egidio <ju...@diegidio.name> wrote:

> On Sunday, February 21, 2016 at 8:31:06 AM UTC, William Elliot wrote:
> > On Sat, 20 Feb 2016, Julio Di Egidio wrote:
> >
> > > Proof that w in w
> >

<snip>

> > > Since a discreet metric applies, the limit in question, i.e. w, exists only
> > > in so far as eventually all members of Fn contain it as a member. (!)
> >
> > No, the neighborhoods of w aren't singletons.
>
> Of course they are, a set of ordinals is a discrete set.

Why do you think that set of ordinals is a discrete set? You
observe that lim_{n<w} n = w plus assumption that this
limit is in discrete topplogy leads to:

exists n < w such that n = w

However n < w and n = w is clear contradiction. So you
should doubt your assumptions. And in fact definition
of set limit (any definition with nontrivial limit!)
lead to topology which is not discrete.

--
Waldek Hebisch

[Julio Di Egidio]

On Monday, February 22, 2016 at 12:33:58 AM UTC, Waldek Hebisch wrote:

> Why do you think that set of ordinals is a discrete set?

Isn't it?

> You
> observe that lim_{n<w} n = w plus assumption that this
> limit is in discrete topplogy leads to:
>
> exists n < w such that n = w

I do not observe any such thing. What I wrote is:

exists n in w : w in Fn

Julio

[Virgil]

In article <a59050ba-41a3-450f...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

But every member of any Fn is an n in |N.
So w in Fn implies w in |N, either as a member or a subset

[Julio Di Egidio]

On Monday, February 22, 2016 at 1:00:09 AM UTC, Virgil wrote:
> In article <a59050ba-41a3-450f...@googlegroups.com>,
> Julio Di Egidio <ju...@diegidio.name> wrote:
> > On Monday, February 22, 2016 at 12:33:58 AM UTC, Waldek Hebisch wrote:
> >
> > > Why do you think that set of ordinals is a discrete set?
> >
> > Isn't it?
> >
> > > You
> > > observe that lim_{n<w} n = w plus assumption that this
> > > limit is in discrete topplogy leads to:
> > >
> > > exists n < w such that n = w
> >
> > I do not observe any such thing. What I wrote is:
> >
> > exists n in w : w in Fn
>

> But every member of any Fn is an n in |N.

Which is what we immediately disprove.

Thanks for the spam.

Julio

[Jim Burns]

On 2/21/2016 4:05 PM, Moufang Loop wrote:
> Jim Burns wrote:
>> On 2/21/2016 3:50 PM, Ross A. Finlayson wrote:
>>> On Sunday, February 21, 2016 at 12:28:24 PM UTC-8,
>>> Moufang Loop wrote:
>>>> Ross A. Finlayson wrote:

>>>>> So, well-order the reals.

Ross: I've never figured out if you are asking us to
do something you think we should be able to do.

>>>>
>>>> After you.
>>>
>>> There aren't uncountably many standard reals
>>> in their normal ordering in a well-ordering.
>>
>> This is a contradiction. the normal ordering of the reals
>> is not a well-ordering.
>>
>> The well-ordering guaranteed by the axiom of choice
>> is not the normal ordering of the reals.
>
> I think that what Ross might have meant is that any subset
> of the reals in their normal order that also happens to be
> well-ordered (by that normal order) can't be uncountable.

Looking again, I see that might be a reading of what Ross wrote.

However, I think the reading I gave it is also possible, and
my reading is consistent with Ross's history of trying to
prove the reals are countable, by means of various transformations,
such as sweep: N -> [0,1] which is order-preserving and with
an image sweep[0,1] dense in [0,1] -- and which also does not
exist (Sorry, Ross. No, it doesn't.)

I think the claim you are reading is not so clearly wrong,
at least. But is it true?

Can we construct a well-ordered subset of reals
(in the usual order) which is uncountable? I don't know.

I think I see how to construct a set of reals with
w*w*w*w*... elements, fractally defined. That may not
be enough, though.

[William Elliot]

On Sun, 21 Feb 2016, Julio Di Egidio wrote:
> On Sunday, February 21, 2016 at 8:31:06 AM UTC, William Elliot wrote:
> >
> > > Proof that w in w
> >
> > What set theory are you using to prove this? Zermelo-Fraenkel (ZF),
> > Quine's New Foundations (NF) or Von Neumann-Bernays-Godel (NBG)
>
> I'd venture nothing more than Peano Axioms and standard discrete topology is
> needed for this proof. Note that we are only dealing with the set of
> ordinals up to and including w.
>

> > No, the neighborhoods of w aren't singletons.
>
> Of course they are, a set of ordinals is a discrete set.

Within a discrete space a sequence will converge iff
it's eventually constant which lim(n<w) n isn't.
Hence that limit doesn't exist except within your imagination.

Do you want to learn some math or just fart idioticy?

[Virgil]

In article <56CA66E5...@att.net>,

Jim Burns <james....@att.net> wrote:

> Can we construct a well-ordered subset of reals
> (in the usual order) which is uncountable? I don't know.

Not with the usual ordering in |R, since that would require uncountably
many successive non-successor elements with strictly positive distances
between successive non-successors!

[Virgil]

In article <01dedba7-ae2d-416a...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> On Monday, February 22, 2016 at 1:00:09 AM UTC, Virgil wrote:
> > In article <a59050ba-41a3-450f...@googlegroups.com>,
> > Julio Di Egidio <ju...@diegidio.name> wrote:
> > > On Monday, February 22, 2016 at 12:33:58 AM UTC, Waldek Hebisch wrote:
> > >
> > > > Why do you think that set of ordinals is a discrete set?
> > >
> > > Isn't it?
> > >
> > > > You
> > > > observe that lim_{n<w} n = w plus assumption that this
> > > > limit is in discrete topplogy leads to:
> > > >
> > > > exists n < w such that n = w
> > >
> > > I do not observe any such thing. What I wrote is:
> > >
> > > exists n in w : w in Fn
> >
> > But every member of any Fn is an n in |N.
>
> Which is what we immediately disprove.

F1 = {1},
F2 = {1,2},
F3 = {1,2,3}
...
Fn = {1..2.3.,,,.n}
...


And thus every Fn is a subset of |N,
So which member(s) of which Fn(s) does JDE claim are not in |N?

[Ross A. Finlayson]

Perhaps this organization of infinitesimals
is more familiar from its central placement
in the development of the integral calculus
(the infinitesimal analysis) as of the method
of fluxions or the differential.

That is to say, sweep as a function directly
founds the Intermediate Value Theorem and
thusly the Fundamental Thms. of Calculus,
without, necessarily, for example, modern
abstract algebra's usual formation as of
equivalence classes of sequences that are
Cauchy.

Then it's also central in probability as the
CDF and a pdf of the natural integers at
uniform random. (Surprisingly, it's not
unique as a pdf.)

It's shown as a unique counterexample to
uncountability.

Well, there was a fence, and a sign on the
fence said "there's no other side to the
fence", and people were discouraged to
look, and for real analysis they were told
to wear blindfolds coming and going from
this walled Eden of set theory, for where
calculus is done in the fields of the real numbers.
Now there's a bridge.

Excuse me, that's a usual summary that I
regularly emit - let is contain our concern
on this thread to "omega in omega", or
of similar results as of the compactification
of the natural integers, the sputnik of
quantification of the integers or Russell's,
and otherwise these variously concerns of
what it means for mathematics that "real"
infinity is "extra-ordinary".


There are some few thousands of posts here
on this subject matter, or, more than a few.

[Ross A. Finlayson]

It's always been so-
it just has to be "said".
It's not that that is false -
it's that this is true.

[Julio Di Egidio]

On Sunday, February 21, 2016 at 8:08:48 PM UTC, Ross A. Finlayson wrote:

> Then that N = N+ and that N e N and w e w,
> these are not unknown concerns, rather
> they're extra-ordinary the usual regular.

If w in w, the standard approach is proven *inconsistent*: not even the standard ordinal construction is valid. On a side, I consider strict finitism a legitimate finitary theory, and there is no infinite set in that theory, period. On the other side, as for infinitary theories, we have now *proved* (still in predicato) that these *cannot* be extensions from the finitary, there cannot be any "transfer". In practice, to have a properly infinitary theory, we have to *first* *establish*, pretty much a la Peano, that there is an infinite set such and such, and that set already contains its limit point. Then, and only then, we rather define the limit of the points to be the limit point: i.e. not the other way round! Then I suppose we can even define a set w\{w}, the set of all and only the finite ordinals, but only then. -- We do start from basic principles: infinite is just not finite.

Julio

[Julio Di Egidio]

On Sunday, February 21, 2016 at 8:08:48 PM UTC, Ross A. Finlayson wrote:

> This can be seen as the difference between
> numbering, as we have here, and counting,
> which is as of a regular arrangement (that
> there are more fundamentally numerical
> arrangements).

I would agree, most fundamental is labelling and arranging (your "counting", I take), then comes arithmetic (your "numbering"). I'd just find your terminology a bit unfortunate, to me it is all already "numbers", there is no pre-mathematical in mathematics: and, I concur, that is a crucial distinction.

In that sense, the 'w' I am contending must be established since the outset is indeed the prototypical "infinite index", or "infinite label", as well as "infinite arrangement", not yet any infinity in an arithmetic or set theory, rather the very establishment of the possibility of operating with infinite numbers and sets: as such, an already given, and, provably (per the OP), not otherwise obtainable.

Along that line, if you'll allow some speculation, Peano's set is an arrangement of labels, it is an infinite set, it does contain the limit label, and it is the most numerous set we can ever construct: omega is, already, the continuum.

Julio

[Martin Shobe]

On 2/21/2016 12:50 PM, Julio Di Egidio wrote:
> On Sunday, February 21, 2016 at 6:40:23 PM UTC, Martin Shobe wrote:

>> On 2/21/2016 12:59 AM, Julio Di Egidio wrote:
>>> Proof that w in w

>>> -----------------
>>>
>>> With w the first limit ordinal, per definition we have:
>>>
>>> w = lim_{n<w} Fn
>>>
>>> where Fn is the n-th Fison (finite initial segment of natural numbers).
>>>

>>> We prove that w in w.
>>>

>>> Since a discreet metric applies, the limit in question, i.e. w, exists only
>>> in so far as eventually all members of Fn contain it as a member. (!)
>>

>> Doesn't seem likely. Can you prove this?
>
> Prove what, that a discreet metric applies or that I am applying it correctly?

Neither. I want you to prove the statement above.

> The former again is because any set of ordinals is a discrete set,

That too needs to be proven.

> the latter is your home work if you don't know.

Which leads me to believe that you can't.

Martin Shobe

[Moufang Loop]

That I don't know!

[Moufang Loop]

What you could do is say whether Jim's or my (or some other)
interpretation of what you said is correct.

[Julio Di Egidio]

On Monday, February 22, 2016 at 12:32:22 PM UTC, Martin Shobe wrote:
> On 2/21/2016 12:50 PM, Julio Di Egidio wrote:
> > On Sunday, February 21, 2016 at 6:40:23 PM UTC, Martin Shobe wrote:
> >> On 2/21/2016 12:59 AM, Julio Di Egidio wrote:
> >>> Proof that w in w
> >>> -----------------
> >>>
> >>> With w the first limit ordinal, per definition we have:
> >>>
> >>> w = lim_{n<w} Fn
> >>>
> >>> where Fn is the n-th Fison (finite initial segment of natural numbers).
> >>>
> >>> We prove that w in w.
> >>>
> >>> Since a discreet metric applies, the limit in question, i.e. w, exists only
> >>> in so far as eventually all members of Fn contain it as a member. (!)
> >>
> >> Doesn't seem likely. Can you prove this?
> >
> > Prove what, that a discreet metric applies or that I am applying it correctly?
>
> Neither. I want you to prove the statement above.

That statement is a direct application of the definition of limit of a sequence in the case of the discreet metric: the limit point of a sequence exists if the terms of the sequence are eventually equal to that limit. In formulae, x_n->x iff exists m : x_n=x for all n>=m. There exist slightly different formulations, say in terms of "all but finitely many" and similar, but the effect is the same.

> > The former again is because any set of ordinals is a discrete set,
>
> That too needs to be proven.

Not really, and if I didn't know you are just trolling, I'd say you are just clueless. Anyway, that a set of ordinal numbers is discrete is again a direct consequence of the definition, of discrete set, and if you don't know what that definition is, YOU ARE NOT QUALIFIED.

> > the latter is your home work if you don't know.
>
> Which leads me to believe that you can't.

*Plonk*

Julio

[Moufang Loop]

I shall ask in a new thread.

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Sunday, February 21, 2016 at 8:31:06 AM UTC, William Elliot wrote:

>> On Sat, 20 Feb 2016, Julio Di Egidio wrote:
>>
>> > Proof that w in w
>>

>> What set theory are you using to prove this? Zermelo-Fraenkel (ZF),
>> Quine's New Foundations (NF) or Von Neumann-Bernays-Godel (NBG)
>
> I'd venture nothing more than Peano Axioms and standard discrete
> topology is needed for this proof. Note that we are only dealing with
> the set of ordinals up to and including w.

This is not the topology that the standard story associates with
the limit in question.

See "ordinals as topological spaces" on wonkypedia:

https://en.wikipedia.org/wiki/Order_topology


>
> Julio

--
Alan Smaill

[Julio Di Egidio]

"It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets." I.e. it is totally irrelevant and totally beside the point.

That said, you and co. are truly sick...

*Plonk*

Julio

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Monday, February 22, 2016 at 2:10:07 PM UTC, Alan Smaill wrote:
>> Julio Di Egidio <ju...@diegidio.name> writes:
>> > On Sunday, February 21, 2016 at 8:31:06 AM UTC, William Elliot wrote:
>> >> On Sat, 20 Feb 2016, Julio Di Egidio wrote:
>> >>
>> >> > Proof that w in w
>> >>
>> >> What set theory are you using to prove this? Zermelo-Fraenkel (ZF),
>> >> Quine's New Foundations (NF) or Von Neumann-Bernays-Godel (NBG)
>> >
>> > I'd venture nothing more than Peano Axioms and standard discrete
>> > topology is needed for this proof. Note that we are only dealing with
>> > the set of ordinals up to and including w.
>>
>> This is not the topology that the standard story associates with
>> the limit in question.
>>
>> See "ordinals as topological spaces" on wonkypedia:
>>
>> https://en.wikipedia.org/wiki/Order_topology
>
> "It is a natural generalization of the topology of the real numbers to
> arbitrary totally ordered sets." I.e. it is totally irrelevant and
> totally beside the point.

Well, no.

You have *chosen* to use the discrete topology.
That is a choice, not something to be proved or disproved.
It is not the only possible choice, however.

The comment on reals on wonkypedia is just a passing
comment, there is no requirement that any reals are around
or that any theory of reals is accepted or rejected.

> That said, you and co. are truly sick...
>
> *Plonk*

Have a good day.

>
> Julio

--
Alan Smaill

[Julio Di Egidio]

On Monday, February 22, 2016 at 3:30:09 PM UTC, Alan Smaill wrote:
> Julio Di Egidio <ju...@diegidio.name> writes:
> > On Monday, February 22, 2016 at 2:10:07 PM UTC, Alan Smaill wrote:
> >> Julio Di Egidio <ju...@diegidio.name> writes:
> >> > On Sunday, February 21, 2016 at 8:31:06 AM UTC, William Elliot wrote:
> >> >> On Sat, 20 Feb 2016, Julio Di Egidio wrote:
> >> >>
> >> >> > Proof that w in w
> >> >>
> >> >> What set theory are you using to prove this? Zermelo-Fraenkel (ZF),
> >> >> Quine's New Foundations (NF) or Von Neumann-Bernays-Godel (NBG)
> >> >
> >> > I'd venture nothing more than Peano Axioms and standard discrete
> >> > topology is needed for this proof. Note that we are only dealing with
> >> > the set of ordinals up to and including w.
> >>
> >> This is not the topology that the standard story associates with
> >> the limit in question.
> >>
> >> See "ordinals as topological spaces" on wonkypedia:
> >>
> >> https://en.wikipedia.org/wiki/Order_topology
> >
> > "It is a natural generalization of the topology of the real numbers to
> > arbitrary totally ordered sets." I.e. it is totally irrelevant and
> > totally beside the point.
>
> Well, no.

Oh, yes.

> You have *chosen* to use the discrete topology.
> That is a choice, not something to be proved or disproved.
> It is not the only possible choice, however.

Nonsense: that set is discrete, so no other metric applies.

> The comment on reals on wonkypedia is just a passing
> comment, there is no requirement that any reals are around
> or that any theory of reals is accepted or rejected.
>
> > That said, you and co. are truly sick...
> >
> > *Plonk*
>
> Have a good day.

It is not about the mathematics, it is about being a decent human being.

Goodbye,

Julio

[Martin Shobe]

On 2/22/2016 7:17 AM, Julio Di Egidio wrote:
> On Monday, February 22, 2016 at 12:32:22 PM UTC, Martin Shobe wrote:
>> On 2/21/2016 12:50 PM, Julio Di Egidio wrote:
>>> On Sunday, February 21, 2016 at 6:40:23 PM UTC, Martin Shobe wrote:
>>>> On 2/21/2016 12:59 AM, Julio Di Egidio wrote:
>>>>> Proof that w in w
>>>>> -----------------
>>>>>
>>>>> With w the first limit ordinal, per definition we have:
>>>>>
>>>>> w = lim_{n<w} Fn
>>>>>
>>>>> where Fn is the n-th Fison (finite initial segment of natural numbers).
>>>>>
>>>>> We prove that w in w.
>>>>>
>>>>> Since a discreet metric applies, the limit in question, i.e. w, exists only
>>>>> in so far as eventually all members of Fn contain it as a member. (!)
>>>>
>>>> Doesn't seem likely. Can you prove this?
>>>
>>> Prove what, that a discreet metric applies or that I am applying it correctly?
>>
>> Neither. I want you to prove the statement above.
>
> That statement is a direct application of the definition of limit of a sequence in the case of the discreet metric: the limit point of a sequence exists if the terms of the sequence are eventually equal to that limit. In formulae, x_n->x iff exists m : x_n=x for all n>=m. There exist slightly different formulations, say in terms of "all but finitely many" and similar, but the effect is the same.

The topology for which w = lim_{n<w} Fn isn't the discrete topology.
Neither is the topology you linked to. You appear to be equivocating
over what topology you are using.

>>> The former again is because any set of ordinals is a discrete set,
>>
>> That too needs to be proven.
>
> Not really, and if I didn't know you are just trolling, I'd say you are just clueless.

It appears that you can't prove that either.

> Anyway, that a set of ordinal numbers is discrete is again a direct consequence of the definition, of discrete set, and if you don't know what that definition is, YOU ARE NOT QUALIFIED.

A subset of a topological space is a discrete set if and only if the
subspace topology on that set is the discrete topology. Using the
topology that you used to get w = lim_{n<w} Fn is not the discrete
topology and therefore not every subset of it is going to be discrete.

Martin Shobe

[Waldek Hebisch]

Julio Di Egidio <ju...@diegidio.name> wrote:

> On Monday, February 22, 2016 at 12:33:58 AM UTC, Waldek Hebisch wrote:
>
> > Why do you think that set of ordinals is a discrete set?
>
> Isn't it?

Well, you can put whatever topology you wish on ordinals,
including discrete one. But the result

lim_{n<w} n = w

is proved using topology which is not discerte. So you should
either accept use of non-discerte topology on ordinals or
reject the limit statemet above as lacking proof...

BTW: You seem to prefer write

lim_{n<w} Fn = w

This does not change fact that you need non-discrete topology.

> > You
> > observe that lim_{n<w} n = w plus assumption that this
> > limit is in discrete topplogy leads to:
> >
> > exists n < w such that n = w
>
> I do not observe any such thing. What I wrote is:
>
> exists n in w : w in Fn

OK, you do not want to obseve this. That is clever, since
having contadiction in your hand you can prove anything.
But do not be surprised that people here do not want to
play such games.

BTW2: Do you realize that w in w is contradiction by the
definition of ordinal?

--
Waldek Hebisch

[Virgil]

In article <nagbgl$qpf$1...@z-news.wcss.wroc.pl>,

Waldek Hebisch <heb...@antispam.uni.wroc.pl> wrote:

> Julio Di Egidio <ju...@diegidio.name> wrote:
> > On Monday, February 22, 2016 at 12:33:58 AM UTC, Waldek Hebisch wrote:
> >
> > > Why do you think that set of ordinals is a discrete set?
> >
> > Isn't it?
>
> Well, you can put whatever topology you wish on ordinals,
> including discrete one. But the result
>
> lim_{n<w} n = w
>
> is proved using topology which is not discerte. So you should
> either accept use of non-discerte topology on ordinals or
> reject the limit statemet above as lacking proof...
>
> BTW: You seem to prefer write
>
> lim_{n<w} Fn = w
>
> This does not change fact that you need non-discrete topology.
>
> > > You
> > > observe that lim_{n<w} n = w plus assumption that this
> > > limit is in discrete topplogy leads to:
> > >
> > > exists n < w such that n = w

For all n < w, not n = w.

And there is no w in |N for which lim_{n<w} n = w.

[Julio Di Egidio]

On Tuesday, February 23, 2016 at 1:09:48 AM UTC, Waldek Hebisch wrote:
> Julio Di Egidio <ju...@diegidio.name> wrote:
> > On Monday, February 22, 2016 at 12:33:58 AM UTC, Waldek Hebisch wrote:
> >
> > > Why do you think that set of ordinals is a discrete set?
> >
> > Isn't it?
>
> Well, you can put whatever topology you wish on ordinals,
> including discrete one.

No, we cannot, and on two accounts:

First of all, I am considering the ordinals up to w, then reread e.g. that article on the Order Topology: "When lambda = w (the first infinite ordinal), the space [0,w) is just N with the usual (still discrete) topology, while [0,w] is the one-point compactification of N." <https://en.wikipedia.org/wiki/Order_topology>

So, I might of course be missing something, but it seems to me that you and co. are simply wrong on that point. Which is significant: no other objections have been really raised, indeed my proof is otherwise pretty elementary.

But there is an even more important point that maybe I have not stressed enough, although I did elaborate on it in a previous reply to Ross:

1) We are building a system of mathematics *from scratch* here!

2) The "ordinals" in the proof are *not* the standard ordinals!

3) The "ordinals" in the proof are just a collection of *labels*!!

4) Of these labels we require and only require that:

4.1) They are distinguishable from one another;
4.2) They in fact come with an intrinsic order that let's us tell, for every pair of such ordinals, whether one is greater than, equal, or less than the other.
4.3) They include a constant for the infinite ordinal (again, distinguishable and comparable to all other ordinals).

5) With that, *and with that only*, we get to prove what we prove.

> BTW: You seem to prefer write
>
> lim_{n<w} Fn = w

On that too I have already elaborated: it is not a matter preferences and it is not just an equality, I have claimed that it must be lim =def= w, which is about the order in which you get to construct things, what comes first and what comes later. Specifically, I am contending that in any correct infinitary theory of sets, the limit is defined after omega and not the other way round as per the standard constructions (structural induction is *wrong* in this sense!).

> > > You
> > > observe that lim_{n<w} n = w plus assumption that this
> > > limit is in discrete topplogy leads to:
> > >
> > > exists n < w such that n = w
> >
> > I do not observe any such thing. What I wrote is:
> >
> > exists n in w : w in Fn
>
> OK, you do not want to obseve this.
> That is clever, since having contadiction in your hand you can prove anything.

It's you disingenuous, to say the least...

> BTW2: Do you realize that w in w is contradiction by the
> definition of ordinal?

Carts before horses: unless you disprove me, it is your dogma that is proven inconsistent, not the other way round...

Julio

[Waldek Hebisch]

Julio Di Egidio <ju...@diegidio.name> wrote:
> On Tuesday, February 23, 2016 at 1:09:48 AM UTC, Waldek Hebisch wrote:
> > Julio Di Egidio <ju...@diegidio.name> wrote:
> > > On Monday, February 22, 2016 at 12:33:58 AM UTC, Waldek Hebisch wrote:
> > >
> > > > Why do you think that set of ordinals is a discrete set?
> > >
> > > Isn't it?
> >
> > Well, you can put whatever topology you wish on ordinals,
> > including discrete one.
>
> No, we cannot, and on two accounts:
>
> First of all, I am considering the ordinals up to w, then reread e.g. that article on the Order Topology: "When lambda = w (the first infinite ordinal), the space [0,w) is just N with the usual (still discrete) topology, while [0,w] is the one-point compactification of N." <https://en.wikipedia.org/wiki/Order_topology>
>
> So, I might of course be missing something, but it seems to me that you and co. are simply wrong on that point.

You seem to be missing simple thing: by definition of limit in
topological space S statement

lim a_n = b

implies b in S. So if you take as S interval without upper bound
[0, w), them lim a_n = w implies w in [0, w) (which in turn means
w < w). To have w as topological limit you need to already have
w in your space. In particular the correct space is [0,w] which
in order topology is non-discrete.

> Which is significant: no other objections have been really raised, indeed my proof is otherwise pretty elementary.
>
> But there is an even more important point that maybe I have not stressed enough, although I did elaborate on it in a previous reply to Ross:
>
> 1) We are building a system of mathematics *from scratch* here!

It is strange that you claim to work *from scratch* and accept
Wikipedia as authority. Ignoring other defects, Wikipedia is
majority view, using all assumptions of classical math.
Starting from scratch imply that you first need to specify
rules of the game: what kind of statements are legal, what
is valid proof, etc... There is nothing wrong with saying
that you are using classical logic, but some folks have
doubts about for example law of exluded middle so it is
better to be explicit here. Now one would state axioms,
write definitions and derive conseqences.

Experience has shown that changing little detail can break
proof and make results invalid, so do not be surprised that
people demand from you proofs of something "obvious": such
proofs have been extensively worked out for classical math,
but if you touch foundations then you need to redo this
work.

> 2) The "ordinals" in the proof are *not* the standard ordinals!
>
> 3) The "ordinals" in the proof are just a collection of *labels*!!
>
> 4) Of these labels we require and only require that:
>
> 4.1) They are distinguishable from one another;
> 4.2) They in fact come with an intrinsic order that let's us tell, for every pair of such ordinals, whether one is greater than, equal, or less than the other.
> 4.3) They include a constant for the infinite ordinal (again, distinguishable and comparable to all other ordinals).
>
> 5) With that, *and with that only*, we get to prove what we prove.

AFAICS you are in trouble here: in classical approach ordinals are
well-ordered. Without assumption of well-order
than anyting you prove about ordinals will apply to real numbers
(and rational numbers). Even if you add extra assumption of
well-order you still need to say what "infinite ordinal" means,
otherwise your statements about ordinals will apply to integers
with 1 beeing "infinite ordinal". Worse, where do you assume
that 0 is not the "infinite ordinal"?

In other words, your assumptions above allow to conclude that
thare is at least one ordinal (since we have a constant naming
it), but nothing more.

> > BTW: You seem to prefer write
> >
> > lim_{n<w} Fn = w
>
> On that too I have already elaborated: it is not a matter preferences and it is not just an equality, I have claimed that it must be lim =def= w, which is about the order in which you get to construct things, what comes first and what comes later. Specifically, I am contending that in any correct infinitary theory of sets, the limit is defined after omega and not the other way round as per the standard constructions (structural induction is *wrong* in this sense!).

I am not sure what you call "standard constructions". In von Neumann
approach omega is defined without any mention of limit. Similarely
in other approaches. Perhaps you mean "standard handwaving" used
to explain what rigorous definitions mean?

And definitions (there are various definitions) of limit either may
be independent of omega (like definiton of limit of a function on
topological space or limit of pointed sequence), use notion of finite
set (half of definition of omega) or plainly use omega (limit of usual
sequence). Only in last case you need to define omega before defining
limit.

> > > > You
> > > > observe that lim_{n<w} n = w plus assumption that this
> > > > limit is in discrete topplogy leads to:
> > > >
> > > > exists n < w such that n = w
> > >
> > > I do not observe any such thing. What I wrote is:
> > >
> > > exists n in w : w in Fn
> >
> > OK, you do not want to obseve this.
> > That is clever, since having contadiction in your hand you can prove anything.
>
> It's you disingenuous, to say the least...

Assumption that limit is with respect to discrete topology directly
gives you (using your notation):

exists n in w : Fn = w

Classical definition have n not in Fn and n in w, so we get clear
contradiction.

You wrote:

exists n in w : w in Fn

Again, we have n in w and n not in Fn. Also, classically x in Fn
implies x subset Fn. So w in Fn implies w subset Fn, so together
with n in w this implies n in Fn. So we got both n in Fn and
n not in Fn -- a contradiction.

This uses standard von Neumann definion, in particular
"x in a impiles x subset a" for ordinal a is part of
von Neumann definiotin. One consequence of von Neumann
definion is a = {b < a} for any ordinal a.

You claim to use different defintion. However, as I explained
it is hard to prove anything interestig (even things like
"there are two ordinals x and y such that not (x = y)" require
unstated assumptions) using your definitions.
Now you refuse to provide _your_ proof of "exists n in w : w in Fn",
yet all you can _directly_ get from having discrete topology is
"exists n in w : Fn = w". To pass between them is easy using
von Neumann definiotin but how _you_ are doing this
(as you apparently reject von Neumann)?

> > BTW2: Do you realize that w in w is contradiction by the
> > definition of ordinal?
>
> Carts before horses: unless you disprove me, it is your dogma that is proven inconsistent, not the other way round...

OK, let it be my dogma (the points above are more important).

--
Waldek Hebisch

[Julio Di Egidio]

On Tuesday, February 23, 2016 at 4:04:42 PM UTC, Waldek Hebisch wrote:
> Julio Di Egidio <ju...@diegidio.name> wrote:

<snipped>

> To have w as topological limit you need to already have
> w in your space.

That is *my* point, that w must be in the domain of the sequence to begin with! And please note that I am saying domain, not range: I am talking of w the index, or, label.

> In particular the correct space is [0,w] which
> in order topology is non-discrete.

Sorry, I do not believe that, it just does not sound right... but thanks for helping narrowing this contention down, of course I'll try to dig further on that.

> It is strange that you claim to work *from scratch* and accept
> Wikipedia as authority.

Still trying to put stupidities in my mouth? That is a fallacious as well as an awful rhetorical habit.

> Experience has shown that changing little detail can break
> proof and make results invalid

And I am perfectly aware of that, too: I may be a rookie in mathematics, but e.g. I fvcking happen to know logic pretty well. Indeed, I could do without, not only the fallacies and, in some cases, not you specifically, the plain offences, but I can also do without this patronising, which is anyway idiotic and, in its own right, offensive.

> > 5) With that, *and with that only*, we get to prove what we prove.
>
> AFAICS you are in trouble here: in classical approach ordinals are
> well-ordered.

Yes, and we have just presented a proof that that is inconsistent, in case you hadn't realised.

> Without assumption of well-order
> than anyting you prove about ordinals will apply to real numbers
> (and rational numbers).

Nonsense, rather keep in mind which "ordinals" we are talking about.

> I am not sure what you call "standard constructions". In von Neumann
> approach omega is defined without any mention of limit. Similarely
> in other approaches. Perhaps you mean "standard handwaving" used
> to explain what rigorous definitions mean?

You are not sure, but you end up calling it hand-waving: that is where I cannot but conclude that you are yet another clown, regardless of how much math you may know.... And, honestly, wouldn't you do the same if you were in my shoes?

> And definitions (there are various definitions) of limit either may
> be independent of omega (like definiton of limit of a function on
> topological space or limit of pointed sequence), use notion of finite
> set (half of definition of omega) or plainly use omega (limit of usual
> sequence). Only in last case you need to define omega before defining
> limit.

Well, glad to hear that what I have presented is at least not outright unmathematical.

> > > > exists n in w : w in Fn
>

> Assumption that limit is with respect to discrete topology directly
> gives you (using your notation):
>
> exists n in w : Fn = w

I wouldn't think so, that needs proof: I'd rather venture you are too used to accept dogmas on faith to see what actually needs or does not need to be proved.

> Classical definition have n not in Fn and n in w, so we get clear
> contradiction.

Yep, that is the bottom line.

> OK, let it be my dogma (the points above are more important).

All the above said, I do appreciate your feedback: I had to reply to some of your comments above even just to put an end to it, but really I'd propose that we just cut the personal compliments and insinuations and all that crap once and for all: I respect you, you respect me, and then we talk mathematics, our present interest...

Thank you,

Julio

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Monday, February 22, 2016 at 3:30:09 PM UTC, Alan Smaill wrote:
>> Julio Di Egidio <ju...@diegidio.name> writes:
>> > On Monday, February 22, 2016 at 2:10:07 PM UTC, Alan Smaill wrote:
>> >> Julio Di Egidio <ju...@diegidio.name> writes:
>> >> > On Sunday, February 21, 2016 at 8:31:06 AM UTC, William Elliot wrote:
>> >> >> On Sat, 20 Feb 2016, Julio Di Egidio wrote:
>> >> >>
>> >> >> > Proof that w in w
>> >> >>
>> >> >> What set theory are you using to prove this? Zermelo-Fraenkel (ZF),
>> >> >> Quine's New Foundations (NF) or Von Neumann-Bernays-Godel (NBG)
>> >> >
>> >> > I'd venture nothing more than Peano Axioms and standard discrete
>> >> > topology is needed for this proof. Note that we are only dealing with
>> >> > the set of ordinals up to and including w.
>> >>
>> >> This is not the topology that the standard story associates with
>> >> the limit in question.
>> >>
>> >> See "ordinals as topological spaces" on wonkypedia:
>> >>
>> >> https://en.wikipedia.org/wiki/Order_topology
>> >
>> > "It is a natural generalization of the topology of the real numbers to
>> > arbitrary totally ordered sets." I.e. it is totally irrelevant and
>> > totally beside the point.
>>
>> Well, no.
>
> Oh, yes.

Historically, ordinals were characterised in terms of linear
well-ordered sets, with the idea of least upper bound being used
rather than limit. The article talks about a general way
of characterising least upper bounds in terms of a topology
corresponding to the order. Why is that irrelevant to the use
of the limit notion when talking about ordinals?

>> You have *chosen* to use the discrete topology.
>> That is a choice, not something to be proved or disproved.
>> It is not the only possible choice, however.
>
> Nonsense: that set is discrete, so no other metric applies.

The ordered set of natural numbers with the usual ordering is
just that, an ordered set and nothing more. Looking at the set

{ 0, 1 - 1/2, 1 - 1/3, 1 - 1/4, ... }

and using the order on the real numbers, we have an ordered set that is
for the purpose of ordinals just the same as the naturals. It is also a
discrete set if we use the usual metric on the reals (or the usual
topology).

It is a different metric from the one you use, but why outlaw this
alternative?

> Goodbye,
>
> Julio

--
Alan Smaill

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Tuesday, February 23, 2016 at 1:09:48 AM UTC, Waldek Hebisch wrote:
>> Julio Di Egidio <ju...@diegidio.name> wrote:
>> > On Monday, February 22, 2016 at 12:33:58 AM UTC, Waldek Hebisch wrote:
>> >
>> > > Why do you think that set of ordinals is a discrete set?
>> >
>> > Isn't it?
>>
>> Well, you can put whatever topology you wish on ordinals,
>> including discrete one.
>
> No, we cannot, and on two accounts:
>
> First of all, I am considering the ordinals up to w, then reread
> e.g. that article on the Order Topology: "When lambda = w (the first
> infinite ordinal), the space [0,w) is just N with the usual (still
> discrete) topology, while [0,w] is the one-point compactification of
> N." <https://en.wikipedia.org/wiki/Order_topology>
>
> So, I might of course be missing something, but it seems to me that
> you and co. are simply wrong on that point.

What message do you think I myself should take from the quote above from
wikipedia?

According to the article, if I follow, the topology for [0,w)
is discrete (which does not uniquely define a metric); and
the topology for [0,w] is not, since otherwise we would have
an infinite set with a discrete metric, and so not a compact set.

>
> Julio

--
Alan Smaill

[Virgil]

In article <fweegb6...@foxtrot.inf.ed.ac.uk>,

A common topology for the set of all naturals has all co-finite sets
being open and it is also a common topology for the set of naturals plus
oo.
Thus the infinite sets containing oo are the only neighborhoods of oo.

[Ross A. Finlayson]

"1.The empty set and X itself belong to τ."

https://en.wikipedia.org/wiki/Topological_space

The empty set is not co-finite in N (that its
complement is finite).

Perhaps you have in mind a particular specialization
of a topological space, thus that omitting the very
finest of detail still gives that the remaining
bulwark of derived results yet hold.

"Subsets" is a usual topology (sets are open).

"Subsets" also has a variety of quite trivial
functions defining neighborhoods, for example
"identity" and all the elements of the subset
being neighbors. Then, in that sense, N is
itself quite neighborly.

[Julio Di Egidio]

On Saturday, March 19, 2016 at 9:15:07 PM UTC, Alan Smaill wrote:
> Julio Di Egidio <ju...@diegidio.name> writes:
> > On Tuesday, February 23, 2016 at 1:09:48 AM UTC, Waldek Hebisch wrote:
> >> Julio Di Egidio <ju...@diegidio.name> wrote:
> >> > On Monday, February 22, 2016 at 12:33:58 AM UTC, Waldek Hebisch wrote:
> >> >
> >> > > Why do you think that set of ordinals is a discrete set?
> >> >
> >> > Isn't it?
> >>
> >> Well, you can put whatever topology you wish on ordinals,
> >> including discrete one.
> >
> > No, we cannot, and on two accounts:
> >
> > First of all, I am considering the ordinals up to w, then reread
> > e.g. that article on the Order Topology: "When lambda = w (the first
> > infinite ordinal), the space [0,w) is just N with the usual (still
> > discrete) topology, while [0,w] is the one-point compactification of
> > N." <https://en.wikipedia.org/wiki/Order_topology>
> >
> > So, I might of course be missing something, but it seems to me that
> > you and co. are simply wrong on that point.
>
> What message do you think I myself should take from the quote above from
> wikipedia?

I suppose I just misunderstood it: I am a logician, you are the mathematician: in general, *you* should *not* ask *me* to make sense of the plethora of mathematical definitions, para-definitions, and contra-definitions, and actually sorting out for you what comes first and what comes next, and what really is relevant here.

> According to the article, if I follow, the topology for [0,w)
> is discrete (which does not uniquely define a metric); and
> the topology for [0,w] is not, since otherwise we would have
> an infinite set with a discrete metric, and so not a compact set.

Then let's drop compactness: does that help? To me, w is *first and foremost*, by my construction, an *isolated point* (just adjoined to the domain, and for all n finite, w-n indeed never vanishes!). "We say that X is topologically discrete but not uniformly discrete or metrically discrete", maybe?

OTOH, if you use "metric spaces" (maybe), I think essentially you assume properties of completeness of the real numbers that here are literally carts before horses! But of course, my terminology and my grasp of the technical definitions may be off mark, but the point again is, I gave a construction, its properties, and its consequences: how that is called mathematically and what we keep and what we drop is more for you to say...

Julio

[Alan Smaill]

I simply wondered why you quoted from it.

>> According to the article, if I follow, the topology for [0,w)
>> is discrete (which does not uniquely define a metric); and
>> the topology for [0,w] is not, since otherwise we would have
>> an infinite set with a discrete metric, and so not a compact set.
>
> Then let's drop compactness: does that help?

It's not the main issue.

> To me, w is *first and
> foremost*, by my construction, an *isolated point* (just adjoined to
> the domain, and for all n finite, w-n indeed never vanishes!).

I agree that it is an extra point added to the domain
The "one-point" in the "one-point compactification" is a single
point that is not a member of the given domain.
(I don't see what you are getting at in the second comment,
though.)

> "We
> say that X is topologically discrete but not uniformly discrete or
> metrically discrete", maybe?

The enlarged domain will not be topologically discrete, however, if
limit is to coincide with least upper bound, as in the original
intention.

> OTOH, if you use "metric spaces" (maybe), I think essentially you
> assume properties of completeness of the real numbers that here are
> literally carts before horses! But of course, my terminology and my
> grasp of the technical definitions may be off mark, but the point
> again is, I gave a construction, its properties, and its consequences:
> how that is called mathematically and what we keep and what we drop is
> more for you to say...

Yes, using some particular instantiation of a set of order type w,
and then using convergence in the reals is simply a possible
choice.

On the other hand, when you assume that the enlarged space is still
discrete (where therefore any convergent sequence must be eventually
constant) -- that is another choice.

If we go back to thinking about ordinals as ordered sets of a certain
sort, then we can have omega + 1 even within potential infinity by
taking the potentially infinite set { 1, 2, 3, ... } with a different
ordering, namely

x < 1 iff x =/= 1
1 < y false
x < y if x=/=1, y=/=1, and x < y in normal order.

Now, the set S = { 2, 3, 4, ... } in this order is exactly the same
as the set { 1, 2, 3, ... } with the usual order. And the element 1
now plays the role of w, and is indeed the least upper bound
of the set S for the new order (it is >= each element of S;
and if z >= each element of S, then z <= 1).

Do you find this presentation problematic, understood as just involving
potential infinity?


>
> Julio

--
Alan Smaill

[Julio Di Egidio]

On Sunday, March 20, 2016 at 4:05:06 PM UTC, Alan Smaill wrote:
> Julio Di Egidio <ju...@diegidio.name> writes:

<snip>

> > To me, w is *first and
> > foremost*, by my construction, an *isolated point* (just adjoined to
> > the domain, and for all n finite, w-n indeed never vanishes!).
>
> I agree that it is an extra point added to the domain
> The "one-point" in the "one-point compactification" is a single
> point that is not a member of the given domain.
> (I don't see what you are getting at in the second comment,
> though.)

I am trying to stress that w is an isolated point: I think that is the key issue.

> > "We
> > say that X is topologically discrete but not uniformly discrete or
> > metrically discrete", maybe?
>
> The enlarged domain will not be topologically discrete, however, if
> limit is to coincide with least upper bound, as in the original
> intention.

Can you please flesh that out in some formal detail (even just give a link to the relevant definition(s), if you think that should be enough), otherwise I am not understanding why that is so.

> > OTOH, if you use "metric spaces" (maybe), I think essentially you
> > assume properties of completeness of the real numbers that here are
> > literally carts before horses! But of course, my terminology and my
> > grasp of the technical definitions may be off mark, but the point
> > again is, I gave a construction, its properties, and its consequences:
> > how that is called mathematically and what we keep and what we drop is
> > more for you to say...
>
> Yes, using some particular instantiation of a set of order type w,
> and then using convergence in the reals is simply a possible
> choice.

But it is a choice that presupposes what here I am rather trying to "reconstruct" (the carts before horses): IOW, there is a lot to do before we go from the infinite set of labels I am talking about (and the basic arithmetic thereof) to anything like the standard real numbers and their properties. No?

> On the other hand, when you assume that the enlarged space is still
> discrete (where therefore any convergent sequence must be eventually
> constant) -- that is another choice.

OK, that's great, at least it is possible. Of course, more than that, I have claimed it is the *only* thing possible: as (I claim) is evident when starting from scratch.

> If we go back to thinking about ordinals as ordered sets of a certain
> sort, then we can have omega + 1 even within potential infinity by
> taking the potentially infinite set { 1, 2, 3, ... } with a different
> ordering, namely
>
> x < 1 iff x =/= 1
> 1 < y false
> x < y if x=/=1, y=/=1, and x < y in normal order.
>
> Now, the set S = { 2, 3, 4, ... } in this order is exactly the same
> as the set { 1, 2, 3, ... } with the usual order. And the element 1
> now plays the role of w, and is indeed the least upper bound
> of the set S for the new order (it is >= each element of S;
> and if z >= each element of S, then z <= 1).
>
> Do you find this presentation problematic, understood as just involving
> potential infinity?

Yes, because potential infinity is not infinity proper, and I claim that that is exactly the essential problem with the standard, that it conflates the two. Namely, your set above is *not* potentially infinite, you just you use the label '1' instead of 'w' (and offset the other labels by one), but it is still an omega. Labels are arbitrary, arithmetic properties aren't.

Julio

[Me]

On Monday, March 21, 2016 at 12:26:58 AM UTC+1, Julio Di Egidio wrote:
>
> Yes, because potential infinity is not infinity proper,
>

Agree.

> and I claim that that is exactly the essential problem with the standard,
> that it conflates the two.

I won't think so.

"Cantor's work was well received by some of the prominent
mathematicians of his day, such as Richard Dedekind. But his
willingness to regard infinite sets as objects to be treated in
much the same way as finite sets was bitterly attacked by others,
particularly Kronecker. There was no objection to a 'potential
infinity' in the form of an unending process, but an 'actual
infinity' in the form of a completed infinite set was harder to
accept."

(Herb Enderton, Elements of Set Theory)

Might be interesting for you:
http://www.math.vanderbilt.edu/~schectex/courses/thereals/potential.html

[Julio Di Egidio]

On Monday, March 21, 2016 at 10:37:06 AM UTC, Me wrote:
> On Monday, March 21, 2016 at 12:26:58 AM UTC+1, Julio Di Egidio wrote:
> >
> > Yes, because potential infinity is not infinity proper,
> >
> Agree.
>
> > and I claim that that is exactly the essential problem with the standard,
> > that it conflates the two.
>
> I won't think so.

I wouldn't think so: otherwise it reads like a commitment not to ever listen... ;)

> "Cantor's work was well received by some of the prominent
> mathematicians of his day, such as Richard Dedekind. But his
> willingness to regard infinite sets as objects to be treated in
> much the same way as finite sets was bitterly attacked by others,
> particularly Kronecker. There was no objection to a 'potential
> infinity' in the form of an unending process, but an 'actual
> infinity' in the form of a completed infinite set was harder to
> accept."
>
> (Herb Enderton, Elements of Set Theory)
>
> Might be interesting for you:
> http://www.math.vanderbilt.edu/~schectex/courses/thereals/potential.html

As I am contending standard mathematics just gets that distinction wrong, what is your point? Indeed, Cantor *claims* to be using actually infinite sets, but he *does not*, in fact he rather invalidly mixes the two to reach invalid conclusions. Do you have any actual objection or counter-argument to my contention, i.e. not just a quote from the very source I am criticising?

Thanks,

Julio

[Me]

On Monday, March 21, 2016 at 12:01:13 PM UTC+1, Julio Di Egidio wrote:
>
> As I am contending standard mathematics just gets that distinction wrong,
> what is your point? Indeed, Cantor *claims* to be using actually infinite
> sets, but he *does not*, in fact he rather invalidly mixes the two to
> reach invalid conclusions.

And you REALLY think no one (other than you) ever noticed? Does that sound REASONABLE?

Especially when considering the following facts (?):

"Cantor's [...]

willingness to regard infinite sets as objects to be treated in
much the same way as finite sets was bitterly attacked by others,
particularly Kronecker. There was no objection to a 'potential
infinity' in the form of an unending process, but an 'actual
infinity' in the form of a completed infinite set was harder to
accept."

(Herb Enderton, Elements of Set Theory)

Just my 2 cents, btw.

[Julio Di Egidio]

On Monday, March 21, 2016 at 11:11:47 AM UTC, Me wrote:
> On Monday, March 21, 2016 at 12:01:13 PM UTC+1, Julio Di Egidio wrote:
> >
> > As I am contending standard mathematics just gets that distinction wrong,
> > what is your point? Indeed, Cantor *claims* to be using actually infinite
> > sets, but he *does not*, in fact he rather invalidly mixes the two to
> > reach invalid conclusions.
>
> And you REALLY think no one (other than you) ever noticed? Does that sound REASONABLE?

It is not impossible, which is what matters: you know, sometimes we even discover new things... OTOH, note that your objection is just a basic fallacy.

> Especially when considering the following facts (?):

Which facts?? I just see another, reiterated fallacy.

> "Cantor's [...]
> willingness to regard infinite sets as objects to be treated in
> much the same way as finite sets was bitterly attacked by others,
> particularly Kronecker. There was no objection to a 'potential
> infinity' in the form of an unending process, but an 'actual
> infinity' in the form of a completed infinite set was harder to
> accept."
>
> (Herb Enderton, Elements of Set Theory)
>
> Just my 2 cents, btw.

Better keep your cents: "Do you have any actual objection or counter-argument to my contention, i.e. not just a quote from the very source I am criticising?" was the essence of my earlier reply to the same. I must conclude that you don't, that you are simply yet another parrot incapable of any actual reasoning or arguments.

*Plonk*

Julio

[Virgil]

In article <4b5b5622-762d-4df0...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> As I am contending standard mathematics just gets that distinction wrong,
> what is your point? Indeed, Cantor *claims* to be using actually infinite
> sets, but he *does not*, in fact he rather invalidly mixes the two to reach
> invalid conclusions. Do you have any actual objection or counter-argument to
> my contention, i.e. not just a quote from the very source I am criticising?

Two of Cantor's main results involving infinite sets may be expressed as

[1] Card(Q) = Card(N), the set of rationals can be bijected with the set
of naturals, and

[2] Card(N) < Card(R),the set of naturals cannot be surjected onto the
ret of reals.

Do you regard either of these conclusions of his as less than true?

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Sunday, March 20, 2016 at 4:05:06 PM UTC, Alan Smaill wrote:
>> Julio Di Egidio <ju...@diegidio.name> writes:
> <snip>
>> > To me, w is *first and
>> > foremost*, by my construction, an *isolated point* (just adjoined to
>> > the domain, and for all n finite, w-n indeed never vanishes!).
>>
>> I agree that it is an extra point added to the domain
>> The "one-point" in the "one-point compactification" is a single
>> point that is not a member of the given domain.
>> (I don't see what you are getting at in the second comment,
>> though.)
>
> I am trying to stress that w is an isolated point: I think that is the
> key issue.

Yes, it is the key jssue.
I am saying that yours is not the only view possible, the extra point
does not have to be isolated (and your view does not agree with the view
of "limit" ordinals as least upper bounds).

>> > "We
>> > say that X is topologically discrete but not uniformly discrete or
>> > metrically discrete", maybe?
>>
>> The enlarged domain will not be topologically discrete, however, if
>> limit is to coincide with least upper bound, as in the original
>> intention.
>
> Can you please flesh that out in some formal detail (even just give a
> link to the relevant definition(s), if you think that should be
> enough), otherwise I am not understanding why that is so.

In the discrete topology, every set (every subset of the domain) is an
open set. The general topological definition of convergence says that a
sequence (xn)_{n in |N} converges to x if and only if for *every* open
set O containing x, the members of the sequence are eventually in O (ie,
for some N, for every m > n . xm is in O). In the discrete case the set
{x} is open, so for some N, for every m > n . xm = x.

>> > OTOH, if you use "metric spaces" (maybe), I think essentially you
>> > assume properties of completeness of the real numbers that here are
>> > literally carts before horses! But of course, my terminology and my
>> > grasp of the technical definitions may be off mark, but the point
>> > again is, I gave a construction, its properties, and its consequences:
>> > how that is called mathematically and what we keep and what we drop is
>> > more for you to say...
>>
>> Yes, using some particular instantiation of a set of order type w,
>> and then using convergence in the reals is simply a possible
>> choice.
>
> But it is a choice that presupposes what here I am rather trying to
> "reconstruct" (the carts before horses): IOW, there is a lot to do
> before we go from the infinite set of labels I am talking about (and
> the basic arithmetic thereof) to anything like the standard real
> numbers and their properties. No?

Yes, that's right.
But there is no need to go that far; the topology prescribed for the
one-point compactification of |N is all that is needed, in other words a
topology defined directly over the set |N U {w}, so no need to bring
reals into the picture.

>> On the other hand, when you assume that the enlarged space is still
>> discrete (where therefore any convergent sequence must be eventually
>> constant) -- that is another choice.
>
> OK, that's great, at least it is possible. Of course, more than that,
> I have claimed it is the *only* thing possible: as (I claim) is
> evident when starting from scratch.

See above; there are alternatives.

>> If we go back to thinking about ordinals as ordered sets of a certain
>> sort, then we can have omega + 1 even within potential infinity by
>> taking the potentially infinite set { 1, 2, 3, ... } with a different
>> ordering, namely
>>
>> x < 1 iff x =/= 1
>> 1 < y false
>> x < y if x=/=1, y=/=1, and x < y in normal order.
>>
>> Now, the set S = { 2, 3, 4, ... } in this order is exactly the same
>> as the set { 1, 2, 3, ... } with the usual order. And the element 1
>> now plays the role of w, and is indeed the least upper bound
>> of the set S for the new order (it is >= each element of S;
>> and if z >= each element of S, then z <= 1).
>>
>> Do you find this presentation problematic, understood as just involving
>> potential infinity?
>
> Yes, because potential infinity is not infinity proper, and I claim
> that that is exactly the essential problem with the standard, that it
> conflates the two.

I agree that that problem arises -- but not immediately when all
that is under consideration is countable, even effective, ordinals,
which is all that is immediately at issue here.

> Namely, your set above is *not* potentially
> infinite, you just you use the label '1' instead of 'w' (and offset
> the other labels by one), but it is still an omega. Labels are
> arbitrary, arithmetic properties aren't.

First, the arithmetic properties are not immediately at issue
(we are still just talking about ordered sets and limits/lubs).
If you prefer, take the usual set |N and a new element w,
and define the order analogously.

why is there not a potentially infinite set of natural
numbers (if that is what you are suggesting)? If there are any
potentially infinite sets, then surely this would be one of them.


>
> Julio

--
Alan Smaill

[Virgil]

On Monday, March 21, 2016 at 12:01:13 PM UTC+1, Julio Di Egidio wrote:
>
> As I am contending standard mathematics just gets that distinction
> wrong, what is your point? Indeed, Cantor *claims* to be using
> actually infinite sets, but he *does not*, in fact he rather
> invalidly mixes the two to reach invalid conclusions.

One of Cantor's theorems says that it is possible to define a
surjection from the set of natural numbers onto the set of rational
numbers.

Are you claiming that all of the many such bijections that have been
invented, including Cantor's, are flawed?

Another of Cantor's theorems says that it is impossible to define a
surjection from the set of natural numbers onto the set of real numbers.

Are you claiming that it is possible to define such a surjection?

Or if you do not object to either of these theorems of Cantor's, can
you cite some theorem of his that you do object to, together with some
evidence of its falsehood?

[Ross A. Finlayson]

There's a constructive example of the points
in a line segment: in a line (and as a line).

By many or most accounts as you might aver
the rationals are quite larger than the
integers, in the rationals, for example as
of their values, per H. Friedman, their
asymptotic density being zero, infinitely
many subsets being supersets of the integers,
and the various other reasonings why the
rationals as a set (and their structure)
are so much larger in those ways than the
integers.

Or, way to go, Galileo.

Galileo has that the n'th square is quite
larger than the n'th integer, that squares
are sparse in integers, and, there's a bijection
between them as sets, for what it is.

Back now to w in w or a one-point compactification
of the naturals, and some have that the naturals
are compact already, here in the context of Julio
and Alan's discussion, some point to Russell for
that the "actual infinite" set is thusly (in accord
with his antinomy) non-well-founded, and that
_Axiomatization_ (and definition, which it is) as
actual _and_ well-founded, is contradictory. That
doesn't say it doesn't exist, just that it's an
implicit subset of the extra-ordinary integers and
that they are compact.

[Ross A. Finlayson]

(Of course, I'm not the first person to note
points in a line segment as forming it, and
can refer to examples of Newton and Leibniz
and the founders of the calculus. What I do
declare is that in a post-modern sense, that
there is a constructive development of the
points of the line, in a line, in a modern
set theory, for a rigorous and novel deductive
system about definitions on continuity. This
provides constructive means for a continuity
from first principles and more-of-less free
from definition. So, in that sense continuity
is properly fundamental, mathematically.)

[Julio Di Egidio]

On Monday, March 21, 2016 at 2:45:10 PM UTC, Alan Smaill wrote:
> Julio Di Egidio <ju...@diegidio.name> writes:
> > On Sunday, March 20, 2016 at 4:05:06 PM UTC, Alan Smaill wrote:

> > <snip>

> > > The enlarged domain will not be topologically discrete, however, if
> > > limit is to coincide with least upper bound, as in the original
> > > intention.
> >
> > Can you please flesh that out in some formal detail (even just give a
> > link to the relevant definition(s), if you think that should be
> > enough), otherwise I am not understanding why that is so.
>
> In the discrete topology, every set (every subset of the domain) is an
> open set. The general topological definition of convergence says that a
> sequence (xn)_{n in |N} converges to x if and only if for *every* open
> set O containing x, the members of the sequence are eventually in O (ie,
> for some N, for every m > n . xm is in O). In the discrete case the set
> {x} is open, so for some N, for every m > n . xm = x.

I was hoping you would flesh out "the view of "limit" ordinals as least upper bounds" (the "original intention") and, in particular, why the domain in that case would not be discrete.

> > > Yes, using some particular instantiation of a set of order type w,
> > > and then using convergence in the reals is simply a possible
> > > choice.
> >
> > But it is a choice that presupposes what here I am rather trying to
> > "reconstruct" (the carts before horses): IOW, there is a lot to do
> > before we go from the infinite set of labels I am talking about (and
> > the basic arithmetic thereof) to anything like the standard real
> > numbers and their properties. No?
>
> Yes, that's right.
> But there is no need to go that far; the topology prescribed for the
> one-point compactification of |N is all that is needed, in other words a
> topology defined directly over the set |N U {w}, so no need to bring
> reals into the picture.

Along the same lines as above, consider again my labels, our "extended domain": it seems to me that our domain is compact and complete while I still do not see why it "will not be topologically discrete".

> > > If we go back to thinking about ordinals as ordered sets of a certain
> > > sort, then we can have omega + 1 even within potential infinity by
> > > taking the potentially infinite set { 1, 2, 3, ... } with a different
> > > ordering, namely
> > >
> > > x < 1 iff x =/= 1
> > > 1 < y false
> > > x < y if x=/=1, y=/=1, and x < y in normal order.
> > >
> > > Now, the set S = { 2, 3, 4, ... } in this order is exactly the same
> > > as the set { 1, 2, 3, ... } with the usual order. And the element 1
> > > now plays the role of w, and is indeed the least upper bound
> > > of the set S for the new order (it is >= each element of S;
> > > and if z >= each element of S, then z <= 1).
> > >
> > > Do you find this presentation problematic, understood as just involving
> > > potential infinity?
> >
> > Yes, because potential infinity is not infinity proper, and I claim
> > that that is exactly the essential problem with the standard, that it
> > conflates the two.
>
> I agree that that problem arises -- but not immediately when all
> that is under consideration is countable, even effective, ordinals,
> which is all that is immediately at issue here.

My *domain* of labels certainly cannot be associated with anything effective or countable: that is the whole point, of an infinite ordinal and of actual infinity. That the standard is invalid re infinities remains immediately at issue here.

Julio

[Julio Di Egidio]

On Monday, March 21, 2016 at 2:45:10 PM UTC, Alan Smaill wrote:
> Julio Di Egidio <ju...@diegidio.name> writes:
<snip>

> > Namely, your set above is *not* potentially
> > infinite, you just you use the label '1' instead of 'w' (and offset
> > the other labels by one), but it is still an omega. Labels are
> > arbitrary, arithmetic properties aren't.
>
> First, the arithmetic properties are not immediately at issue
> (we are still just talking about ordered sets and limits/lubs).

I call those properties arithmetical already: what I have in mind is that these belong to the semantic side of the syntactic vs. semantic characterisation of a mathematical theory-system.

> why is there not a potentially infinite set of natural
> numbers (if that is what you are suggesting)? If there are any
> potentially infinite sets, then surely this would be one of them.

Because there is no such thing as a potentially infinite set, all infinite sets are actually infinite sets. I do not yet have a convincing argument and even less a proof of this (though I'd say Cantor himself advocates for this), but I am envisioning it has to do with what I am proving here (in particular, re an axiom of infinity), as well as with considerations on the notion (and axiom) of choice and how that is guaranteed/needed for infinite sets.

OTOH, to add even more vague speculation (and further and further OT), back to the syntactic vs. semantic levels and interplay: I would forward that the collection of *labels*, which belongs to the syntax, is (and only is) potentially infinite, which now literally means *as many as one needs*. So, reversing the usual idea that "actual infinite is bigger than potential", labels (thinking, for choice, at least as many as there are elements!) are more numerous than any domain we can define: as with the proverbial balls and vase, we are to assume that there are as many balls available as we can ever have to put into any vase, and potentially more...

Julio

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Monday, March 21, 2016 at 2:45:10 PM UTC, Alan Smaill wrote:
>> Julio Di Egidio <ju...@diegidio.name> writes:
>> > On Sunday, March 20, 2016 at 4:05:06 PM UTC, Alan Smaill wrote:
>> > <snip>
>> > > The enlarged domain will not be topologically discrete, however, if
>> > > limit is to coincide with least upper bound, as in the original
>> > > intention.
>> >
>> > Can you please flesh that out in some formal detail (even just give a
>> > link to the relevant definition(s), if you think that should be
>> > enough), otherwise I am not understanding why that is so.
>>
>> In the discrete topology, every set (every subset of the domain) is an
>> open set. The general topological definition of convergence says that a
>> sequence (xn)_{n in |N} converges to x if and only if for *every* open
>> set O containing x, the members of the sequence are eventually in O (ie,
>> for some N, for every m > n . xm is in O). In the discrete case the set
>> {x} is open, so for some N, for every m > n . xm = x.
>
> I was hoping you would flesh out "the view of "limit" ordinals as
> least upper bounds" (the "original intention") and, in particular, why
> the domain in that case would not be discrete.

Let's see, take the sequence (n)_{n in |N} taking values in the enlarged
domain with extra element w. Using the discrete topology, this sequence
does not have a limit, since it does not become eventually constant (as
above):

* no n in |N can be the limit (sequence takes larger values)
* w cannot be the limit (the sequence never takes the value at all).

But w is the least upper bound of the ordered set |N in the enlarged
domain (it is greater than any n in |N by definition; and it is the
smallest element with this property).

>> > > Yes, using some particular instantiation of a set of order type w,
>> > > and then using convergence in the reals is simply a possible
>> > > choice.
>> >
>> > But it is a choice that presupposes what here I am rather trying to
>> > "reconstruct" (the carts before horses): IOW, there is a lot to do
>> > before we go from the infinite set of labels I am talking about (and
>> > the basic arithmetic thereof) to anything like the standard real
>> > numbers and their properties. No?
>>
>> Yes, that's right.
>> But there is no need to go that far; the topology prescribed for the
>> one-point compactification of |N is all that is needed, in other words a
>> topology defined directly over the set |N U {w}, so no need to bring
>> reals into the picture.
>
> Along the same lines as above, consider again my labels, our "extended
> domain": it seems to me that our domain is compact and complete while
> I still do not see why it "will not be topologically discrete".

If it is topologically discrete, then every set is an open set.
In particular, for every point x, {x} is open. So the whole
set X is the union of the infinite set I = { {x} : x in X }. If X is to
be compact, there must be a finite subset of I whose union is also X.
(Definition of compact). But no finite set will cover X in this way.

>> > > If we go back to thinking about ordinals as ordered sets of a certain
>> > > sort, then we can have omega + 1 even within potential infinity by
>> > > taking the potentially infinite set { 1, 2, 3, ... } with a different
>> > > ordering, namely
>> > >
>> > > x < 1 iff x =/= 1
>> > > 1 < y false
>> > > x < y if x=/=1, y=/=1, and x < y in normal order.
>> > >
>> > > Now, the set S = { 2, 3, 4, ... } in this order is exactly the same
>> > > as the set { 1, 2, 3, ... } with the usual order. And the element 1
>> > > now plays the role of w, and is indeed the least upper bound
>> > > of the set S for the new order (it is >= each element of S;
>> > > and if z >= each element of S, then z <= 1).
>> > >
>> > > Do you find this presentation problematic, understood as just involving
>> > > potential infinity?
>> >
>> > Yes, because potential infinity is not infinity proper, and I claim
>> > that that is exactly the essential problem with the standard, that it
>> > conflates the two.
>>
>> I agree that that problem arises -- but not immediately when all
>> that is under consideration is countable, even effective, ordinals,
>> which is all that is immediately at issue here.
>
> My *domain* of labels certainly cannot be associated with anything
> effective or countable: that is the whole point, of an infinite
> ordinal and of actual infinity.

I'm all in favour of finding other ways of doing maths.
(I do think that some ordinals do fit within the potentially infinite
story, but let's leave that for the moment.)

> That the standard is invalid re
> infinities remains immediately at issue here.

I'm not in the business of defending the established status quo.

>
> Julio

--
Alan Smaill

[WM]

Am Dienstag, 22. März 2016 19:50:13 UTC+1 schrieb Alan Smaill:


> But w is the least upper bound of the ordered set |N in the enlarged
> domain (it is greater than any n in |N by definition; and it is the
> smallest element with this property).

That is possible. But it is absolutely wrong to claim that aleph_0 (or omega) is the number of natural numbers.

Regards, WM

[Dan Christensen]

In analysis, aleph_0 is defined to be the cardinality of the set of natural numbers N. Likewise for any set that is bijectable to N, e.g. the set of even numbers, the integers or the rational numbers.

Informally, it can be said that the "number of natural numbers" is aleph_0. I hope this helps.


Dan

Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[Alan Smaill]

WM <wolfgang.m...@hs-augsburg.de> writes:

> Am Dienstag, 22. März 2016 19:50:13 UTC+1 schrieb Alan Smaill:
>
>
>> But w is the least upper bound of the ordered set |N in the enlarged
>> domain (it is greater than any n in |N by definition; and it is the
>> smallest element with this property).
>
> That is possible.

Thus spake The Prophet.

> But it is absolutely wrong to claim that aleph_0 (or
> omega) is the number of natural numbers.

Just as well that no such claim was made in this thread, then.

>
> Regards, WM

--
Alan Smaill

[Virgil]

In article <48b97fa1-4206-404a...@googlegroups.com>,

Says WMe who also implies there are more rational numbers than reals by
claiming both more rationals than naturals and as many naturals as reals!

When WM claims something, one is far more likely to be right by
rejecting it than by believing it.

[Virgil]

In article <d25f277f-2016-4d2e...@googlegroups.com>,

Thus WM is absolutely wrong to claim it absolutely wrong!

[Julio Di Egidio]

On Tuesday, March 22, 2016 at 6:50:13 PM UTC, Alan Smaill wrote:
> Julio Di Egidio <ju...@diegidio.name> writes:
> > On Monday, March 21, 2016 at 2:45:10 PM UTC, Alan Smaill wrote:
> >> Julio Di Egidio <ju...@diegidio.name> writes:
> >> > On Sunday, March 20, 2016 at 4:05:06 PM UTC, Alan Smaill wrote:
> >> > <snip>
> >> > > The enlarged domain will not be topologically discrete, however, if
> >> > > limit is to coincide with least upper bound, as in the original
> >> > > intention.
> >> >
> >> > Can you please flesh that out in some formal detail (even just give a
> >> > link to the relevant definition(s), if you think that should be
> >> > enough), otherwise I am not understanding why that is so.
> >>
> >> In the discrete topology, every set (every subset of the domain) is an
> >> open set. The general topological definition of convergence says that a
> >> sequence (xn)_{n in |N} converges to x if and only if for *every* open
> >> set O containing x, the members of the sequence are eventually in O (ie,
> >> for some N, for every m > n . xm is in O). In the discrete case the set
> >> {x} is open, so for some N, for every m > n . xm = x.
> >
> > I was hoping you would flesh out "the view of "limit" ordinals as
> > least upper bounds" (the "original intention") and, in particular, why
> > the domain in that case would not be discrete.
>
> Let's see, take the sequence (n)_{n in |N} taking values in the enlarged
> domain with extra element w.

You use that "domain" for the range but not for the domain of the sequence, then we are back to square one and what I have tagged incongruent.

> Using the discrete topology, this sequence
> does not have a limit, since it does not become eventually constant (as
> above):

As said, what you write here is along the lines of what I am objecting to with my proof: in particular having to do with the fact that I rather *start* from an extended domain, and that the limit *does* exist, that is a given as we are defining it, not proving it... and so on: I won't try to be more precise, in the meantime I have anyway realised, thanks of course to the feedback, that I shall seriously reconsider, review, and rewrite the whole thing: i.e. when I manage to get my head around the remaining issue of topology...

For some reason I cannot get my head around the definition of compact (FYI, I am using Wikipedia). Let me try and ask you a direct question: take the simple compactification of N with one point at infinity, where is the finite cover for that?

> > My *domain* of labels certainly cannot be associated with anything
> > effective or countable: that is the whole point, of an infinite
> > ordinal and of actual infinity.
>
> I'm all in favour of finding other ways of doing maths.
> (I do think that some ordinals do fit within the potentially infinite
> story, but let's leave that for the moment.)

Interesting...

> > That the standard is invalid re
> > infinities remains immediately at issue here.
>
> I'm not in the business of defending the established status quo.

OK, duly noted.

Julio

[Ross A. Finlayson]

Often it will be "omega-many" or "n-many"
over the usual range or index of iteration,
for example range of summation or otherwise
in the definition of cases of induction
for unbounded terms. There are n-many
or oo-many (infinity-many, omega-many).

That is, or these sets are called "countable".

It would be very interesting to many if
you could find an "application" of the
transfinite cardinals. This is of course
where much of higher mathematics is
"application" of induction over (1, .., oo).

Some people perceive the lack of
application of the "transfinite cardinals"
as a deficiency, and others that the
numbers themselves are more fundamentally
as of numbering, and iteration and induction,
before counting.

[Virgil]

In article <aa3008af-77e5-4646...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> You use that "domain" for the range but not for the domain of the sequence,
> then we are back to square one and what I have tagged incongruent.

A sequence is a function whose domain is always an initial subset of the
naturals, either finite or infinite.

[Ross A. Finlayson]

Infinite sequences (or series)
have infinite domains.


Virgil, some have "series" as the
sum and not just the "sequence".


I think we might each do well to
read (or re-read) Berkeley's treatise
on the infinitesimal "The Analyst".
Through the scope of time, you might
note that he wasn't so negative in
his outlook as one might suppose from
the one phrase "what are we to consider
these but ghosts of a departed quantity".

http://maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.pdf


Here what is this infinity in itself but
always the extra-ordinary. Where it is
only the bounded, how is that not trivial
and mundane to the sublime of what it is.

Also: put infinity to work. You'll find
that throughout mathematics, oo as the
lemniscate means the derivation of the case
of the limit that exists, as the value is
un-bounded and so diverges in the asymptotic
that its determined result yet maintain a
simple direct perfect clarity of purpose.

[Jim Burns]

The values of a sequence in w+1 = {0,1,...w} _can be_ anything
in w+1 but they don't _need to be_ any particular value in order
for that to be a sequence.

For example, the sequence in the rationals 1, 1/2, 1/3, ...
does not have all the rationals as values. Likewise, the
sequence in w+1 (n)_{n in |N} does not have all the values
in w+1 , missing w as it does, but it is a sequence in w+1 .

>> Using the discrete topology, this sequence
>> does not have a limit, since it does not become eventually constant
>> (as above):
>
> As said, what you write here is along the lines of what I am
> objecting to with my proof: in particular having to do with the fact
> that I rather *start* from an extended domain, and that the limit
> *does* exist, that is a given as we are defining it, not proving
> it...

It is your definition in your original post, w = lim_{n<w} Fn ,
that keeps the topology from being (completely) discrete.

If w is an isolated point, then every sequence converging to w
eventually becomes constant. You define a sequence that does not
eventually become constant as converging to w , so w is not
an isolated point. If not all points are isolated, then the
topology is not discrete.
<https://en.wikipedia.org/wiki/Isolated_point>

(Of course, apart from w , all the points in w+1 are discrete in
the usual topology.)

I may be beating a dead horse, but a topology for a set is a choice
of which subsets are designated as "open" (or alternately which
are "closed").
<https://en.wikipedia.org/wiki/General_topology#A_topology_on_a_set>

Which sets are open determines which sequences converge
because one version of the definition of convergence (the
topology-friendly version) is that the sequence b_n converges
to b if and only if _every open set containing b_ contains
all of the sequence b_n except for a finite number of terms
(that is, the exceptional n are bounded by some M).

If w is an isolated point, the sequence Fn eventually (ignoring
the first M terms) must look like w, w, w, ... because
one of the open sets containing w is just {w} .
Amongst the reasons that this _cannot_ be is that it would mean
Fm = Fk for ~(m = k). So w is not an isolated point, and
the topology you chose with that definition is not discrete.

This assumes that lim_{n->w} n = w . If we don't assume that, then
w is just some random unconnected point, not a compactification.

Let O be an arbitrary collection of open sets U O = N+1
( N+1 = N U {w} , all n =< w )
Then w e A for some A e O .
Because lim_{n->w} n = w , and A is an open set containing q ,
A contains all but finitely many n (by the topology-friendly
definition of convergence, mentioned above).

Therefore A and _at most_ as many open sets from O as there
are n not in A will cover N+1. Thus, we know there is a
finite open cover.

[Ross A. Finlayson]

When you say "random", does that include
"random as a well-ordering of the reals"?

[Jim Burns]

On 3/24/2016 11:48 PM, Ross A. Finlayson wrote:
> On Thursday, March 24, 2016 at 8:30:45 AM UTC-7,
> Jim Burns wrote:
>> On 3/23/2016 7:42 PM, Julio Di Egidio wrote:


>>> For some reason I cannot get my head around the definition of compact
>>> (FYI, I am using Wikipedia). Let me try and ask you a direct
>>> question: take the simple compactification of N with one point at
>>> infinity, where is the finite cover for that?
>>
>> This assumes that lim_{n->w} n = w . If we don't assume that, then
>> w is just some random unconnected point, not a compactification.

> When you say "random", does that include
> "random as a well-ordering of the reals"?

It doesn't seem likely to me that I mean "random" in that way,
whatever way it is you are meaning it. But I'm not clear on
how you're using it. I don't think I would find it useful to
refer to a well-ordering of the reals as random.

It seems much more likely that I mean "random" in a way
approximately opposite to how you are using it. Here, I am
using "random" informally to mean something like "unconnected"
informally, as one might say "The twenty-dollar bill was
picked up by a random passer-by" to refer to a particular
someone who did not stand apart (metaphorically) from
other someones who could have done the same.

I find the mathematical/logical meaning of "random" interesting
and non-trivial, but here I was only throwing it in as a
(very tiny) bit of humor in a fairly dry exposition.

More seriously/less humorously, "random" seems to refer to
_lack_ of reasons for an outcome. For example, the Principle of
Indifference would take a symmetric six-sided die and assign
the same probability of 1/6 to it landing with any of its
faces up -- because symmetry allows us no physical reason to give
one face a higher probability than another.

(As I used it above, a point added to N that was just a point,
not the compactification of N, would have no better _reason_
to be added to N than any other point. Hence, random,
in my informal sense.)

<https://en.wikipedia.org/wiki/Principle_of_indifference>

Although I don't know what you mean by "random" that
seems to be opposite in sense from random as it might
be applied to a well-ordering of the reals.

[Ross A. Finlayson]

A well-ordering of the reals (that is not sweep
or the normal ordering as I usually and continuedly
promote) is so strongly "random" in a sense that it
must contain countably many elements that run off to
the bounds: before uncountably many elements are in
their normal order. There's no sense of "clustering"
possible in any distribution that follows, the points
in such a thing (that ZFC purports to exist) must
necessarily go off in their normal order. This is
different than a usual expectation, of independence
and identical distribution (i.i.d.) of samples. Of
"all the well-orderings of R" or R[0,1] that there
could be, then that there's some means to sample them
in some uniform manner where otherwise there's a way
to distinguish them, they are particularly scattered.

Then, how this applies to a topology of N is as of the
"Subsets Topology" example above, that a well-ordering
would converge to the compactification, or that otherwise
usual trivial results see reinforcement of the strong
property of continuity.

Then of course I point to sweep() as a function from
N to a continuous domain of [0,1] and also that it's
countable and also that it's well-ordered (for the
neat discrete topology of the unit segment courtesy
sweep).

[Ross A. Finlayson]

A well-ordering of a set is a well-ordering of any subset.

Yet, here it seems, a well-ordering of a bounded segment
of R, is not an initial segment of a well-ordering of any
different bounded segment nor unbounded segment, _but it
should be, via translation_.

Well-orderings of the reals that aren't the usual ordering
are quite "random".

[Jim Burns]

Ross, you have given two different descriptions of sweep()
previously, (i) sweep: N+1 -> [0,1] with the image points
equally spaced (and order preserving?), and (ii)
sweep: N+1 -> [0,1] order preserving and dense in [0,1] .
You've been shown that (i) and (ii) don't exist.

You seem to have a third description here, (iii) the image
of sweep() is countable and well-ordered. Here, it looks
like such functions can be found -- and pretty easily.

I don't see what such functions have to do with
any of the other topics where you have raised to
prospect of something you call sweep(). Maybe they
have nothing to do with each other.

> A well-ordering of a set is a well-ordering of any subset.
>
> Yet, here it seems, a well-ordering of a bounded segment
> of R, is not an initial segment of a well-ordering of any
> different bounded segment nor unbounded segment, _but it
> should be, via translation_.

When you talk about bounded segments of R and translations
you are referring to the _usual_ field operations on R,
'+', '-', '*', '/', '<' , and so, the usual ordering '<' .

When you talk about a well-ordering of the reals, you are
talking about some other order, which we might call '<<' .
We know that these are different orderings because '<'
is not a well-ordering and '<<' is.

You make some statements about a well-ordering (using '<<')
of a bounded segment (using '<'). You say something
about that "should be". This has all the appearance of
you forgetting that you are talking about two different orders
at the same time.

There is no apparent reason for what you say "should be"
to be.

> Well-orderings of the reals that aren't the usual ordering
> are quite "random".

Well, you have made it a little clearer what you mean by
"random". It' doesn't seem to have much to do with what
I mean by random.

Keep in mind, when you refer to "well-orderings of the reals
that aren't the usual ordering" you refer to all the
well-orderings of the reals, since the usual ordering
is not a well-ordering.

[Julio Di Egidio]

On Thursday, March 24, 2016 at 3:30:45 PM UTC, Jim Burns wrote:
> On 3/23/2016 7:42 PM, Julio Di Egidio wrote:
> > On Tuesday, March 22, 2016 at 6:50:13 PM UTC,
> > Alan Smaill wrote:

> >> Julio Di Egidio wrote:
<snip>

> >>> I was hoping you would flesh out "the view of "limit" ordinals as
> >>> least upper bounds" (the "original intention") and, in particular,
> >>> why the domain in that case would not be discrete.
> >>
> >> Let's see, take the sequence (n)_{n in |N} taking values in the
> >> enlarged domain with extra element w.
> >
> > You use that "domain" for the range but not for the domain of the
> > sequence, then we are back to square one and what I have tagged
> > incongruent.
>
> The values of a sequence in w+1 = {0,1,...w} _can be_ anything
> in w+1 but they don't _need to be_ any particular value in order
> for that to be a sequence.

<snip>

You miss the point: Alan wrote n in |N and I was countering with an w+1 (would be just w in my theory).

> >> Using the discrete topology, this sequence
> >> does not have a limit, since it does not become eventually constant
> >> (as above):
> >
> > As said, what you write here is along the lines of what I am
> > objecting to with my proof: in particular having to do with the fact
> > that I rather *start* from an extended domain, and that the limit
> > *does* exist, that is a given as we are defining it, not proving
> > it...
>
> It is your definition in your original post, w = lim_{n<w} Fn ,
> that keeps the topology from being (completely) discrete.

No definition, just poor presentation: I was introducing symbols and notions, the actual proof starts from "We prove that..."

> If w is an isolated point, then every sequence converging to w
> eventually becomes constant.

But that is exactly my argument! Our w is isolated by construction, hence the topology, hence ... the thesis.

<snip>

> If w is an isolated point, the sequence Fn eventually (ignoring
> the first M terms) must look like w, w, w, ... because
> one of the open sets containing w is just {w} .
> Amongst the reasons that this _cannot_ be is that it would mean
> Fm = Fk for ~(m = k).

Nope, as w is indeed isolated...

> So w is not an isolated point, and
> the topology you chose with that definition is not discrete.

<snip>

> This assumes that lim_{n->w} n = w . If we don't assume that, then
> w is just some random unconnected point, not a compactification.

The limit is *then* defined, *after* omega, i.e. it is the opposite of an assumption: again, my argument goes the other way round.

Indeed, in my construction w is an *unconnected* point, though not a "random" one, it comes with specific arithmetic properties: its order in the sequence, but I'd think at least also an extension of the successor/predecessor function, or, equivalently, an extension of the inductive definition of the set-sequence itself.

(But I cannot comment on compact (or the topology): I'll need a little bit more time to go through that part. Thanks for your help.)

Julio

[Virgil]

In article <07e75e56-3bc1-4c71...@googlegroups.com>,

Julio Di Egidio <ju...@diegidio.name> wrote:

> Julio

Connectedness is a topologically defined property, so without a specific
topology, or at least a specified set of topologies, in mind, w cannot
be either connected or unconnected.

[Ross A. Finlayson]

As you might not have noticed that before,
it seems somewhat odd that the reals are to
to be uncountable, but the set of pairs of
reals is somehow less (because each would
have a non-empty disjoint containing a
unique rational, of which there are less).


Then, though I might convince Newton or Leibniz
or anyone by drawing a line segment that a
line segment of points is only by drawing the
line, in fact I have that Cantor proves it also.

[Ross A. Finlayson]

Besides that "there can't be uncountably many
disjoint intervals" to begin with (because
each would contains a distinct rational and
those are less than uncountable), there can't
be uncountably many endpoints of intervals,
that's not quite so clear, because there is to
be established that the pairwise disjoint holds
for intervals that might intersect, but, it's
rather clear, because any of these bounded
intervals are again contained in as many more
intervals as there are, each with a non-empty
disjoint, and, any two non-identical intervals
do have a non-empty disjoint (of at least one
interval).

So, luckily calculus is built on a definition
of "measure 1.0" from measure theory and the
transfinite cardinals contribute nothing to
the analytical character of the reals. This
is where there is at least one interval for
each real number x, for example the interval
between zero and x.

This line of argument readily (and in a
constructivist manner) shows the dense
(but nowhere continuous) rationals and
their complement in the continuous reals
being equinumerous (equipollent, equivalent).

[Jim Burns]

On 3/26/2016 2:41 PM, Julio Di Egidio wrote:
> On Thursday, March 24, 2016 at 3:30:45 PM UTC,
> Jim Burns wrote:
>> On 3/23/2016 7:42 PM, Julio Di Egidio wrote:
>>> On Tuesday, March 22, 2016 at 6:50:13 PM UTC,
>>> Alan Smaill wrote:

>>>> Using the discrete topology, this sequence
>>>> does not have a limit, since it does not become eventually constant
>>>> (as above):
>>>
>>> As said, what you write here is along the lines of what I am
>>> objecting to with my proof: in particular having to do with the fact
>>> that I rather *start* from an extended domain, and that the limit
>>> *does* exist, that is a given as we are defining it, not proving
>>> it...

[...]

>> If w is an isolated point, then every sequence converging to w
>> eventually becomes constant.
>
> But that is exactly my argument! Our w is isolated by construction,
> hence the topology, hence ... the thesis.

I don't want to ask you what your argument has been, since you are
currently working out some aspects of it, and I don't want to
bump your elbow. I don't see enough of this on USEnet or the
internet and I certainly don't want to make it harder to do.
Unfortunately, I don't have a good enough idea of what your
argument is. I strongly suspect that this is because I have not
read all of the thread, so please do not consider this a
criticism.

The thesis still seems to be " w in w ".

The argument seems to involve a series, and w = omega is used
three ways, as the domain of the series, as the range of the
series (or maybe w+1 is, which is closely related), and as an
element in the range of the series. The overabundance of w's may be
part of my problem.

I think you mentioned that you want to avoid our presuppositions.
This seems to me like a worthy goal. As part of that, though,
I think that you will need to be unusually explicit about
what you mean by each of the w's . I could get a way with
saying lim_{n->w} n = w and letting others assume I mean the
usual sort of things -- but that is where presuppositions
creep in.

[Jim Burns]

On 3/27/2016 7:19 PM, Jim Burns wrote:
[ to Julio Di Egidio ]

> I think you mentioned that you want to avoid our presuppositions.
> This seems to me like a worthy goal. As part of that, though,
> I think that you will need to be unusually explicit about
> what you mean by each of the w's . I could get a way with
> saying lim_{n->w} n = w and letting others assume I mean the
> usual sort of things -- but that is where presuppositions
> creep in.

Whether w in W is true or false depends upon what we mean by
(the set) W and (the element) w . I'm going to give a proof of
w not in W for what I consider very reasonable interpretations
of (the set) W and (the element) w . It's possible that you will
find some presuppositions of mine that you think should be rejected.
I don't think so, but I wouldn't think so, whether or not.

By (the set) W , I mean some set with the same order as the
familiar W = omega = { {}, {{}}, {{},{{}}}, ... } . This set,
with 0 = {} and Sx = x U {x} , satisfies the second-order
Peano axioms, and is order-isomorphic to any other set W'
with 0' and S' satisfying the second-order Peano axioms.

One thing that makes this particular order interesting and
important is that every infinite set contains a subset with
this order. Therefore, it would be reasonable to call W
the first infinite ordinal, whatever its contents and
whatever its label.

Let a finite element x of W be such that the subset of W
less than x, {y| y<x }, is finite.

By (the element) w , I mean a member of (the set) W which
is larger than all and only the finite elements of (the set) W ,
which is a set satisfying the second-order Peano axioms.

I intend to show that a set satisfying 2OPA ("W") contains
infinitely many finite elements but it does not contain
any infinite elements (such as "w").
Since there are infinitely many finite elements that
(the element) w would need to be greater than if it were
in (the set) W , (the element) w would necessarily be an
infinite element.
In this sense, w not in W .

----
*Define* the set A to be infinite if and only if there is
a bijection between A and a proper subset of A,
and define a set to be finite if it is not infinite.

Infinite(A) <->
exist f: A -> A/Z &
f a bijection & Z a non-empty subset of A

Finite(A) <-> ~Infinite(A)

*Define*
Inductive(A,z,f) <-> z e A & f: A -> A

*Define*
Peano(W,z,f) <->
P1) Inductive(W,z,f)
P2) all x, y, f(x) = f(y) <-> x = y
P3) all x, ~( f(x) = z )
P4) ALL C sub W, Inductive(C,z,f) -> C = W

*Theorem*
Infinite(A) ->
exists W sub A, z e W, f: W -> W,
Peano(W,z,f)

Proof sketch:
Infinite(A).
Let f: A -> A/Z and z e Z .
Define W = intersect{ C sub A | Inductive(C,z,f) }
...
Peano(W,z,f)

(I include this theorem because this is the step from
which all these consequences flow. It is this intersection
that makes induction (P4) valid for W.
By making sure there are no _proper_ subsets of W which
are inductive (or else there would be "extra" elements in W)
it is enough to show that C is inductive (in P4) to show
that C is W.
It is this intersection that excludes any "extra" elements
from W -- and w is an extra element, as we'll see.)


Peano(W,z,f)

*Define* an order '<' on B such that
x < f(y) <-> x < y V x = y

Let Before(x) be the subset of elements y of W less than z
Before(x) = { y | y < x }

*Define* inductively Before: W -> P(W)

Before(z) = {}

Before(f(x)) = Before(x) U {x}

(Note that, because induction is valid for W,z,f ,
Before(x) and < are well-defined here.)

Peano(W,z,f) with order <
Peano(V,w,g) with order <<

Define inductively iso: W -> V

iso(z) = w

iso(x) = x' -> iso(f(x)) = g(x')

*Theorem*
iso: W -> V is a bijection

for x' = iso(x) and y' = iso(y)
x' << y' <-> x < y

W,z,f and V,w,g are order-isomorphic


Peano(W,z,f)

*Theorem*
Infinite(W)

Proof sketch
f: W -> W/{z} is a bijection.

*Theorem*
all x e W, Finite(Before(x))

Proof by induction

(to prove base)
Finite(Before(z)) <-> Finite({})

(to prove step)
Finite(Before(x)) -> Finite(Before(f(x)))

which is equivalent to
Infinite(Before(f(x))) -> Infinite(Before(x))

Assume Infinite(Before(f(x))) .

Let h: Before(f(x)) -> Before(f(x))/Z
be a bijection
with Z a non-empty subset of Before(f(x))

Note that Before(f(x)) = Before(x) U {x}

Case 1. x e Z

Define h': Before(x) -> Before(x)/Z'

by h'(y) = h(y) , all y

where Z' = Z - {x} + {f(x)}
so Z' is a non-empty subset of Before(x)
and h' is a bijection

Infinite(Before(x))

Case 2. x ~e Z

exists u, v, f(u) = x , f(x) = v

Define h": Before(x) -> Before(x)/Z

by h"(u) = v
h"(y) = h(y) , otherwise

where Z is the same non-empty subset of Before(f(x))
and a non-empty subset of Before(x)
and h" is a bijection

Infinite(Before(x))

Therefore,
Infinite(Before(f(x))) -> Infinite(Before(x))
and
Finite(Before(x)) -> Finite(Before(f(x)))

By induction,
all x e W, Finite(Before(x))


*In Summary*
All elements of W have finitely many elements before them.
There are infinitely many elements in W, and so there are
infinitely many elements with finitely many elements
before them.
If there were an element in W with all the finite elements
before it ("w"), there would be infinitely many such finite
elements before it.
But there isn't such an element ("w").
Therefore w not in W .

[Jim Burns]

On 3/29/2016 2:16 PM, Jim Burns wrote:
> On 3/27/2016 7:19 PM, Jim Burns wrote:
> [ to Julio Di Egidio ]

>> I think you mentioned that you want to avoid our presuppositions.
>> This seems to me like a worthy goal. As part of that, though,
>> I think that you will need to be unusually explicit about
>> what you mean by each of the w's . I could get a way with
>> saying lim_{n->w} n = w and letting others assume I mean the
>> usual sort of things -- but that is where presuppositions
>> creep in.
>
> Whether w in W is true or false depends upon what we mean by
> (the set) W and (the element) w . I'm going to give a proof of
> w not in W for what I consider very reasonable interpretations
> of (the set) W and (the element) w . It's possible that you will
> find some presuppositions of mine that you think should be rejected.
> I don't think so, but I wouldn't think so, whether or not.

I noticed that the central part of my argument for
w not in W is quite short, so I thought I'd give it a
little more emphasis. (My last post was maybe a bit long.)
Definitions and associated arguments are still there upthread.
This is just the Executive Summary.

*Theorem*
1) Infinite(W)

*Theorem*
2) all x e W, Finite(Before(x))

*Define*
3) Before(w) = { x e W | Finite(Before(x)) }

By (2) and (3),
{ x e W | Finite(Before(x)) } = W

Before(w) = W

By (1),
Infinite(Before(w))

However, by (2),
w e W -> Finite(Before(w))

Therefore,
w ~e W

[Ross A. Finlayson]

"Thus, for instance, the smallest of the
infinite integers is the limit of the finite
integers, though all finite integers are at
an infinite distance from it."

-- Bertrand Russell

[Me]

On Tuesday, March 29, 2016 at 8:16:51 PM UTC+2, Jim Burns wrote:
>
> Whether w in W is true or false depends upon what we mean by

> (the set) W and (the element) w. I'm going to give a proof of

> w not in W for what I consider very reasonable interpretations

> of (the set) W and (the element) w. [...]

So just let's consider w = W with

> the familiar W = omega = {{}, {{}}, {{}, {{}}}, ... }.

You certainly know that in ZF(C) we can prove the existence of such a set (due to the axiom of infinity + ...) and even without the axiom of foundation we get easily that W !e W (since every set in W is finite, but W is infinite).

Hence at least in the context of ZF(C) (even without "foundation" or "regularity") Julio's statement "W e W" is wrong. (Did he mention an alternative system he's working in/with?)

[Me]

On Wednesday, March 30, 2016 at 11:41:39 AM UTC+2, Me wrote:
>
> Hence at least in the context of ZF(C) (even without "foundation" or
> "regularity") Julio's statement "W e W" is wrong. (Did he mention an
> alternative system he's working in/with?)

@Julio: It's not good enough (in mathematics) just to claim "I am contending standard mathematics". Actually, you will have to STATE the axioms and/or rules of your "alernative" system, otherwise your "considerations" are just crap (even though rather elaborated crap, that is).

[Jim Burns]

If I understand correctly, Julio is trying to minimize his
assumptions about W (the set) and w (the element). So, what
alternative he would be using would be in play, would be
to be determined.

I may be over-interpreting the discussion, but I think
this discussion is an attack on "the context", regular mathematics,
so just saying w ~e w in ZF(C) would be beside the point.

If, for example, w ~e w in ZF(C) but other considerations
force us to conclude w e w , I think Julio would be very pleased
with this result.

I think the "other considerations" would have to be what we
mean by omega, its definition -- or maybe, how it _should_ be
defined, or something like that. Julio seems to have been treating
the topology of the ordinals as "other considerations", but
I'm not entirely clear what is going on there.

My contribution is what I consider the minimum restriction on
what W (the set) is and what w (the element) is. Keep in mind
that the set and the element are two different roles, and they
need not be the are mathematical object. (With foundation,
they obviously are not the same, as I'm sure Julio knows.)

Julio has focused on topology, I have focused on order.

I consider the bare minimum description of the set in " w e w "
to be something _with the order_ of the usual w .
Thus, w (the set) would satisfy the Peano axioms.

I consider the bare minimum description of the element in " w e w "
to be something _at the same place in the order_ of the set.
In analogy with 0 being the first in order of whatever
version of the set W we are looking at. 0 is {} in
the set of all finite von Neumann ordinals, for example.

I think that, if w e W, then w would have to be the first
after all of the finitely-positioned elements in W .
That seems to me to be the essence of what we mean by omega.
By finitely-positioned element, I mean an element
with only finitely many elements less than it.
There could be such an element. There are such elements
in other orders, such as w+1. But I can show there isn't
such a w in W .

If we assume _less_ about w (the element) and W (the set)
it is no longer clear to me that we are still talking
about w = omega in any relevant way.

[Julio Di Egidio]

On Wednesday, March 30, 2016 at 7:55:27 PM UTC+1, Jim Burns wrote:

> Julio has focused on topology, I have focused on order.

Nope, but typical projections: I had focused on order, it is you guys who have raised topological issues...

Never mind: thank you for the feedback, I'll look into your posts as soon as I manage.

Julio

[Jim Burns]

On 3/29/2016 9:29 PM, Ross A. Finlayson wrote:
> On Tuesday, March 29, 2016 at 5:28:49 PM UTC-7,
> Jim Burns wrote:


>> *Theorem*
>> 1) Infinite(W)
>>
>> *Theorem*
>> 2) all x e W, Finite(Before(x))
>>
>> *Define*
>> 3) Before(w) = { x e W | Finite(Before(x)) }

> "Thus, for instance, the smallest of the
> infinite integers is the limit of the finite
> integers, though all finite integers are at
> an infinite distance from it."
>
> -- Bertrand Russell

In set theory, an ordinal number, or ordinal, is one
generalization of the concept of a natural number that
is used to describe a way to arrange a collection of
objects in order, one after another.
-- Wikipedia, "Ordinal number"

According to modern usage, there are infinitely many
natural numbers, all of which are finite -- and there are
infinitely many finite ordinals, all of which (not surprisingly)
are before the first infinite ordinal. These are closely
related but not quite identical facts.

It looks to me as though Bertrand Russel is using "integer"
(especially "infinite integer") the way I would use "ordinal".
This is not criticism of Russell's usage, which was probably
perfectly fine for when he was writing. But it is a theorem
that each natural number has only finitely many natural numbers
before it == there are no infinite integers, given our current
usage.

[Jim Burns]

On 3/26/2016 7:40 PM, Ross A. Finlayson wrote:

> As you might not have noticed that before,
> it seems somewhat odd that the reals are to
> to be uncountable, but the set of pairs of
> reals is somehow less (because each would
> have a non-empty disjoint containing a
> unique rational, of which there are less).

There are at least as many pairs of reals as there are reals.
Consider just the pairs of the form {0,x} , for example.
So, the {0,x} and pairs and segments are uncountable.

But the segments in the set { (0,x) } are not disjoint.
It is the requirement that they be _disjoint_ that forces
each rational to be in _only one_ segment. There can be
no more _disjoint_ segments than there are rationals.

Note, that although this means each disjoint segment must have
_at least_ one rational only its own, _actually_ there are
a countable _infinity_ of rationals only its own in each segment .

This is the sort of thing that makes infinity fun.

[Ross A. Finlayson]

There are as many intervals with endpoint zero
as there are real numbers.


Each two of them are having a non-empty disjoint,
one way or the other, that is an interval (with
or without zero).

So, there aren't more of those than there are
rationals, because there's a distinct rational
in each disjoint.

There's (at least) one for each.

There are as many non-empty and distinct disjoints
as there are real numbers, between a given interval
starting at zero and all the others.


Russell used to joke that he was the Pope, here,
if you can't quantify over the intervals of real
numbers, which in a sense define magnitudes, then
you are very carefully treading on the ice, and,
we've seen somewhat a spate of hot summers these days.

You are claiming "foundational" support to be
voiding a constructive topological argument.

That seems the bit more egregrious than noting,
for example, that quantifying over the finite
ordinals, would be the Russell set (of Russell's
paradox), which would be omega and it contains
itself, and noting that "defining" the infinity
as so "ordinary" saw instead the conclusion that
it was so "well-founded". That would be rather
more egregious.

[Ross A. Finlayson]

This example of usage is a way to show (N e N), or
N = N+, w e w, etcetera, that the naturals are compact.

Russell's paradox as applied to just the finite ordinals
is another example.

This gets into usual reasoning why "restrictions" of
comprehension as the axioms of infinity and well-
foundedness are: are not logical in the sense of
not being independent axioms (as they are not
"expanding" comprehension).

[Jim Burns]

On 3/30/2016 11:13 PM, Ross A. Finlayson wrote:

>
> There are as many intervals with endpoint zero
> as there are real numbers.
>
>
> Each two of them are having a non-empty disjoint,
> one way or the other, that is an interval (with
> or without zero).

I think I know what you mean by "disjoint" here.
Let's say, for sets A and B , either A/B or B/A
(or both) are non-empty.

That is not what you need for your argument.
You need for _each_ set to have a rational
that _all_ the other sets do not have.
In that case, there cannot be more sets than rationals.

Suppose there is a rational x in A/B.
Is x not also in A/C and A/D and so on?
How do you know?



Consider the family of intervals { [0,x] | x > 0 } and
an individual member of that family [0,pi] .
What is it that distinguishes [0,pi] from the others?

Luckily, either [0,pi] is a proper subset or a proper superset
of very other set in that family.
_Having more_ than [0,pi] distinguishes each [0,x] in
{ [0,x] | x > pi } from [0,pi]
_Having more_ than each [0,x] in { [0,x] | 0 < x < pi }
distinguishes [0,pi] from each [0,x]

What sets [0,pi] apart from the rest of its family is just {pi} .

One way to look at this
U = intersect{ [0,x] | x > pi }
= [0,pi]

L = union{ [0,x] | 0 < x < pi }
= [0,pi)

{pi} = U/L


{pi} is not an interval, really, and it certainly does not
have a rational.