On 8/14/2021 9:20 AM, WM wrote:
> Jim Burns schrieb am Donnerstag, 12. August 2021 um 20:16:47 UTC+2:
>> On 8/12/2021 6:44 AM, Greg Cunt wrote:
>>> On Thursday, August 12, 2021 at 6:04:06 AM UTC+2,
>>> Jim Burns wrote:
>
>>> Hint: INTERSECTION M is what it is, ***completely***
>>> independent of any order defined on M.
>> The definition of the intersection of M does not mention
>> any order on M.
>
> Therefore all endsegments satisfying
> ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo

you have a set equal to a count.

> can be combined to
> |∩{E(k) : k ∈ ℕ_def}| = ℵo .
>
> Regards, WM
>

ℕ_def is meaningless

On 8/14/2021 9:29 AM, WM wrote:
> Jim Burns schrieb am Freitag, 13. August 2021 um 20:23:30 UTC+2:
>> On 8/12/2021 9:36 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Donnerstag, 12. August 2021 um 06:04:06 UTC+2:
>>
>>>> No last end segment, intersection is empty.
>>>
>>> Every definable endsegment is a last endsegment.
>> You have changed what "last"means so that there are two kinds
>> of "last", those that change and those that do not change.
>
> The endsegments satifying
> ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo
> dco not change. But their being last does change if we add another endsegment.

there is no "last" endsegment.

>>
>> For you, WM, I will say it this way:
>>
>> No [unchanging] last end segment, intersection is empty.
>
> This can only be realized by endsegments which are never last.
>>>>
>>>> WM rejects No Last End Segment.
>>>
>>> No, but I reject that last endsegments
>> ...[changing] last end segments..
>>> can yield a set with no last endsegment
>
>> ...no [unchanging] last end segment...
>>> - other than potentially infinite.
>>> But that has always a last one.
>> ...a [changing] last one...
>>
>> Once we make clear that you have changing and unchanging
>> end segments, we see that you are saying:
>>
>> | Every definable end segment is a [changing] last end segment.
>>
>> | I (WM) reject that [changing] last end segments can yield
>> | a set with no [unchanging] last end segments.
>> | - other than potentially infinite.
>> | But that has always a [changing] last one.
>>
>> How you manage to hold both these propositions in your
>> head at the same time without your head imploding is by
>> obscuring the difference between changing and unchanging.
>> This is a textbook case of equivocation, a fallacy.
>
> Every definable endsegment is the last one of its Finite initial segment. It can never make a set larger than finite.

nope, endsegments are infinite, did you change that ?

>>
>>> Every actually infinite set is larger than any finite set.
>>> But definable endsegments belong to finite sets by definition.
>> How I read
>> | E(k) belongs to a finite set
>> is
>> | The set {E(1),...,E(k)} of definable end segments between
>> | E(1) and definable E(k) is finite.
>>
>> That's true. This is also true:
>> | No set {E(1),...,E(k)} of definable end segments between
>> | E(1) and definable E(k) contains each definable end segment.
>
> That is true because the collection is potentially infinite.
>>
>> Because
>> Every definable end segment is a [changing] last end segment.
>> No definable end segment is an [unchanging] last end segment.
>> Each {E(1),...,E(k)} can be expanded to include more
>> definable end segments.
>
> Of course. What is difficult with this notion? There is only one important point: You must not confuse pot. and act. as set theorists deplorably deliberately do.
>
> Regards, WM
>

On Saturday, August 14, 2021 at 10:16:15 AM UTC-4, WM wrote:
> William schrieb am Donnerstag, 12. August 2021 um 15:54:10 UTC+2:
> > On Thursday, August 12, 2021 at 9:48:33 AM UTC-4, WM wrote:
> > > William schrieb am Donnerstag, 12. August 2021 um 15:45:34 UTC+2:
> > >
> > > > Every element of |N_F is used in E= (E(1), E(2), E(3),...)
> > > > For every element i of |N_F E(i) has cardinality aleph_0
> > > Important however is that all elements are used.
> > He said staring at an example where every element of |N_F is used,
> Every is not enough

Piffle. If every element of |N_F is used then no element of |N_F is not used;
i.e. "all elements are used".

--
William Hughes

On Saturday, 14 August 2021 at 11:20:39 UTC-3, WM wrote:
[...]
> Therefore all endsegments satisfying
> ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo
> can be combined to
> |∩{E(k) : k ∈ ℕ_def}| = ℵo .

Of course not! I'd say that this inference is invalid by definition:

Definition: If WM makes a statement about an infinity of natural numbers, it is a wrong statement.

On 8/14/2021 10:26 AM, Sergio wrote:
> On 8/13/2021 1:23 PM, Jim Burns wrote:
>> On 8/12/2021 9:36 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Donnerstag, 12. August 2021 um 06:04:06 UTC+2:

>>>> No last end segment, intersection is empty.
>>>
>>> Every definable endsegment is a last endsegment.
>>
>> You have changed what "last"means so that there are two kinds
>> of "last", those that change and those that do not change.
>>
>> For you, WM, I will say it this way:
>>
>> No [unchanging] last end segment, intersection is empty.

> how about using cLast and uLast ?

This is a matter of taste.

When I am deciding on what notation to use for some
property or object, I like to expand on the page what I
want the reader to pay close attention to. If I and my
reader were at the same table, I would be pointing to it
with my finger. Making its presence physically larger
is a sort of disembodied finger-pointing.

Also, I like to be as explicit as possible, given other
constraints like space and time. "[changing] last" avoids
one layer of translation from "cLast" to "[changing] last".
No, it doesn't take a lot of mental resources to do that,
but there are other things I would prefer my reader to spend
their mental resources on.

Sometimes, maybe I take this a little far. I recently
mentioned "an able-to-be-stepped-through-one-by-one collection".
I'm not completely sure, but I think that sometimes I do
things like that just for the hell of it.

On Saturday, August 14, 2021 at 7:25:03 AM UTC-7, Sergio wrote:
> Every Ants
>
> All Ants
>
> Almost Ants
>
> Successor Ants
>
> Not Enough Ants
>
> Every Ant has a successor
>
> Almost Every Ant has a successor
>
> All Successor Ants
>
> .
> .
> .
> .

Ants go through some harrowing adventures
in their quest for mathematics.

On 8/14/2021 10:29 AM, WM wrote:
> Jim Burns schrieb
> am Freitag, 13. August 2021 um 20:23:30 UTC+2:

>> Once we make clear that you have changing and unchanging
>> [last] end segments, we see that you are saying:
>>
>> | Every definable end segment is a [changing] last end segment.
>>
>> | I (WM) reject that [changing] last end segments can yield
>> | a set with no [unchanging] last end segments.
>> | - other than potentially infinite.
>> | But that has always a [changing] last one.
>>
>> How you manage to hold both these propositions in your
>> head at the same time without your head imploding is by
>> obscuring the difference between changing and unchanging.
>> This is a textbook case of equivocation, a fallacy.
>
> Every definable endsegment is the last one of its Finite
> initial segment. It can never make a set larger than finite.

A set with only [changing] last end segments does not have
[unchanging] last end segments. However, you reject this.

>> Because
>> Every definable end segment is a [changing] last end segment.
>> No definable end segment is an [unchanging] last end segment.
>> Each {E(1),...,E(k)} can be expanded to include more
>> definable end segments.
>
> Of course. What is difficult with this notion?

<WM<JB>>
>>
>> WM rejects No Last End Segment.
>
> No, but I reject that last endsegments can yield
> a set with no last endsegment -
> other than potentially infinite.
> But that has always a last one.
>
</WM<JB>>

You refer to [changing] last end segments,
then [unchanging] last end segments,
then [changing] last end segments.

> There is only one important point:
> You must not confuse pot. and act.
> as set theorists deplorably deliberately do.

Set theorists do not confuse [changing] last endsegments
with [unchanging] last end segments.

Gus Gassmann schrieb am Samstag, 14. August 2021 um 17:26:31 UTC+2:
> On Saturday, 14 August 2021 at 11:20:39 UTC-3, WM wrote:
> [...]
> > Therefore all endsegments satisfying
> > ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo
> > can be combined to
> > |∩{E(k) : k ∈ ℕ_def}| = ℵo .
> Of course not!

Why not?

Regards, WM

Jim Burns schrieb am Samstag, 14. August 2021 um 19:02:48 UTC+2:
> On 8/14/2021 10:29 AM, WM wrote:

> > There is only one important point:
> > You must not confuse pot. and act.
> > as set theorists deplorably deliberately do.
> Set theorists do not confuse [changing] last endsegments
> with [unchanging] last end segments.

They do. Every endsegment which can be last is insufficient for an actual infinity. Set theorists however claim that there are no others and that actual infinit exists. That is wrong.

Regards, WM

William schrieb am Samstag, 14. August 2021 um 17:09:13 UTC+2:
> On Saturday, August 14, 2021 at 10:16:15 AM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 12. August 2021 um 15:54:10 UTC+2:
> > > On Thursday, August 12, 2021 at 9:48:33 AM UTC-4, WM wrote:
> > > > William schrieb am Donnerstag, 12. August 2021 um 15:45:34 UTC+2:
> > > >
> > > > > Every element of |N_F is used in E= (E(1), E(2), E(3),...)
> > > > > For every element i of |N_F E(i) has cardinality aleph_0
> > > > Important however is that all elements are used.
> > > He said staring at an example where every element of |N_F is used,
> > Every is not enough
> Piffle. If every element of |N_F is used then no element of |N_F is not used;

Wrong. But if you say that no element is not used, then
> i.e. "all elements are used".

and aleph_0 indexes are used. None remains. The requirement of two consecutive infinite sets is a nice idea, similar to this problem:

Up to every natural number n, Cantor's enumeration of the positive rational numbers is incomplete in every unit interval (k, k+1]. In every interval k > 1 much less rational numbers are enumerated than in interval 1. For an estimation see https://mathoverflow.net/questions/362791/what-fraction-of-fractions-does-cantors-famous-sequence-enumerate .

Assume that all ℵo incompletely enumerated unit intervals could be completely enumerated. Of course there is no last rational to be enumerated, but in order to complete an interval at least one natural number appears to be necessary. Then for ℵo intervals also ℵo natural numbers are required - in addition to the natural numbers applied up to this step. Since in the natural order of |N, used by Cantor, never two consecutive sets of cardinality ℵo can exist, this means that up to this step only finitely many natural numbers can have been applied. - For ℵo intervals! Therefore it is impossible that Cantor's enumeration of the positive rationals is complete.

Regards, WM

On Saturday, 14 August 2021 at 14:20:40 UTC-3, WM wrote:
> Gus Gassmann schrieb am Samstag, 14. August 2021 um 17:26:31 UTC+2:
> > On Saturday, 14 August 2021 at 11:20:39 UTC-3, WM wrote:
> > [...]
> > > Therefore all endsegments satisfying
> > > ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo (1)
> > > can be combined to
> > > |∩{E(k) : k ∈ ℕ_def}| = ℵo . (2)
> > Of course not!
> Why not?

If for no other reason than that you stated it. Your track record on these things is remarkable: 16+ years, and not a single true statement generated.

But here again for the umpteenth time: the intersections in (1) are *FINITE* intersections, and the intersection in (2) is over the *INFINITE* set ℕ_def, which has been shown to be equal to ℕ. You even accepted it at one point, but of course you now (claim that you) do not remember that any more.

Gus Gassmann schrieb am Samstag, 14. August 2021 um 19:31:37 UTC+2:
> On Saturday, 14 August 2021 at 14:20:40 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Samstag, 14. August 2021 um 17:26:31 UTC+2:
> > > On Saturday, 14 August 2021 at 11:20:39 UTC-3, WM wrote:
> > > [...]
> > > > Therefore all endsegments satisfying
> > > > ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo (1)
> > > > can be combined to
> > > > |∩{E(k) : k ∈ ℕ_def}| = ℵo . (2)
> > > Of course not!
> > Why not?

> the intersections in (1) are *FINITE* intersections, and the intersection in (2) is over the *INFINITE* set ℕ_def, which has been shown to be equal to ℕ.

Intersections are only depending on the sizes of the participating sets. Endsegments which in (1) have an infinite intersection have an infinite intersection in (2) too. When gathering the endsegments in (2) stop as soon as the intersection ceases to be infinite.

Regards, WM

On Saturday, 14 August 2021 at 14:37:41 UTC-3, WM wrote:
> Gus Gassmann schrieb am Samstag, 14. August 2021 um 19:31:37 UTC+2:
> > On Saturday, 14 August 2021 at 14:20:40 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Samstag, 14. August 2021 um 17:26:31 UTC+2:
> > > > On Saturday, 14 August 2021 at 11:20:39 UTC-3, WM wrote:
> > > > [...]
> > > > > Therefore all endsegments satisfying
> > > > > ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo (1)
> > > > > can be combined to
> > > > > |∩{E(k) : k ∈ ℕ_def}| = ℵo . (2)
> > > > Of course not!
> > > Why not?
> > the intersections in (1) are *FINITE* intersections, and the intersection in (2) is over the *INFINITE* set ℕ_def, which has been shown to be equal to ℕ.
> Intersections are only depending on the sizes of the participating sets. Endsegments which in (1) have an infinite intersection have an infinite intersection in (2) too. When gathering the endsegments in (2) stop as soon as the intersection ceases to be infinite.

First, oh exalted one, taking the intersection is not a stepwise process. Second, every end segment has cardinality aleph_0, so there is no finite point where "the intersection ceases to be infinite". How stupid can one be not to realize that after 16+ years of noodling, with zero results to show for it?

On Saturday, August 14, 2021 at 1:28:08 PM UTC-4, WM wrote:
> William schrieb am Samstag, 14. August 2021 um 17:09:13 UTC+2:

> > Piffle. If every element of |N_F is used then no element of |N_F is not used;
> Wrong.

?!

--
William Hughes

On 8/14/2021 12:28 PM, WM wrote:
> William schrieb am Samstag, 14. August 2021 um 17:09:13 UTC+2:
>> On Saturday, August 14, 2021 at 10:16:15 AM UTC-4, WM wrote:
>>> William schrieb am Donnerstag, 12. August 2021 um 15:54:10 UTC+2:
>>>> On Thursday, August 12, 2021 at 9:48:33 AM UTC-4, WM wrote:
>>>>> William schrieb am Donnerstag, 12. August 2021 um 15:45:34 UTC+2:
>>>>>
>>>>>> Every element of |N_F is used in E= (E(1), E(2), E(3),...)
>>>>>> For every element i of |N_F E(i) has cardinality aleph_0
>>>>> Important however is that all elements are used.
>>>> He said staring at an example where every element of |N_F is used,
>>> Every is not enough
>> Piffle. If every element of |N_F is used then no element of |N_F is not used;
>
> Wrong. But if you say that no element is not used, then

red herring again

1 every element is used
2 every element is not used
3 no element is used => #2
4 no element is not used => #1
5 all elements are used => #1
6 all elements are not used => #2

>
>> i.e. "all elements are used".
>
> and aleph_0 indexes are used.

On 8/14/2021 12:37 PM, WM wrote:
> Gus Gassmann schrieb am Samstag, 14. August 2021 um 19:31:37 UTC+2:
>> On Saturday, 14 August 2021 at 14:20:40 UTC-3, WM wrote:
>>> Gus Gassmann schrieb am Samstag, 14. August 2021 um 17:26:31 UTC+2:
>>>> On Saturday, 14 August 2021 at 11:20:39 UTC-3, WM wrote:
>>>> [...]
>>>>> Therefore all endsegments satisfying
>>>>> ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo (1)
>>>>> can be combined to
>>>>> |∩{E(k) : k ∈ ℕ_def}| = ℵo . (2)
>>>> Of course not!
>>> Why not?
>
>> the intersections in (1) are *FINITE* intersections, and the intersection in (2) is over the *INFINITE* set ℕ_def, which has been shown to be equal to ℕ.
>
> Intersections are only depending on the sizes of the participating sets.

Wrong. Intersections are only depending on the elements of the
respective sets.

On 8/14/2021 10:20 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 12. August 2021 um 20:16:47 UTC+2:
>> On 8/12/2021 6:44 AM, Greg Cunt wrote:

>>> Hint: INTERSECTION M is what it is, ***completely***
>>> independent of any order defined on M.
>>
>> The definition of the intersection of M does not mention
>> any order on M.
>
> Therefore all endsegments satisfying
> ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo
> can be combined to
> |∩{E(k) : k ∈ ℕ_def}| = ℵo .

The intersection of a collection C of infinite end segments sub N_def
( meaning
( the set of elements in each end segment in C

is either an infinite end segment sub N_def or the empty set.

The intersection of all infinite end segments sub N_def
is a subset of each infinite end segment sub N_def.

For each infinite end segment sub N_def,
there is an infinite end segment it is NOT a subset of.

No infinite end segment is a subset of all infinite end
segments.

No infinite end segment is the intersection of all
infinite end segments.

The empty set is the intersection of all infinite end segments.

> |∩{E(k) : k ∈ ℕ_def}| = ℵo .

No.

On 8/14/2021 12:23 PM, WM wrote:
> Jim Burns schrieb am Samstag, 14. August 2021 um 19:02:48 UTC+2:
>> On 8/14/2021 10:29 AM, WM wrote:
>
>>> There is only one important point:
>>> You must not confuse pot. and act.
>>> as set theorists deplorably deliberately do.
>> Set theorists do not confuse [changing] last endsegments
>> with [unchanging] last end segments.
>
> They do. Every endsegment which can be last is insufficient for an actual infinity. Set theorists however claim that there are no others and that actual infinit exists. That is wrong.
>
> Regards, WM
>

"They" and "set theorists" are bogus.

Last Ant
Last Ants
Lasting Ants

On Saturday, 14 August 2021 at 15:25:42 UTC-3, William wrote:
> On Saturday, August 14, 2021 at 1:28:08 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 14. August 2021 um 17:09:13 UTC+2:
>
> > > Piffle. If every element of |N_F is used then no element of |N_F is not used;
> > Wrong.
> ?!

Dementia affecting reading abilities? (Can't plausibly be anything else.)

On Wednesday, August 11, 2021 at 9:04:06 PM UTC-7, Jim Burns wrote:
> On 8/11/2021 10:24 PM, Greg Cunt wrote:
> > On Wednesday, August 11, 2021 at 9:43:15 PM UTC+2,
> > WM wrote:
>
> >> intersections are indepedent of the order of <bla>.
> >
> > Indeed!
> >
> > Hint: INTERSECTION M is what it is, ***completely*** independent
> > of any order defined on M, you silly crank!
> > Hint: It only depends on the CONTENT of M.
> In this case, the order of the end segments is by inclusion.
> So, in this case, it's not possible to speak of their order
> without at the same time speaking of their content.
>
> ----
> Because of the order/content of the end segments,
> the intersection of a collection of end segments
> equals the last end segment in that collection
> -- but not if no last end segment exists.
>
> Last end segment, intersection is the last end segment.
>
> Because of the order/content of the end segments and
> because all {0,...,m} are finite,
> no number m is in the intersection of a collection of
> end segments unless there is a last end segment and
> m is in it.
>
> No last end segment, intersection is empty.
>
> ----
> I'm thinking about WM's appendix and...
> Last end segment, intersection is the last end segment.
> No last end segment, intersection is empty.
>
> WM rejects No Last End Segment.
> Instead, he defines into existence an appendix with the
> property that it makes the intersection empty.
>
> That's nonsense, I know.
> But this shines a light on how Wolfgang Mückenheim thinks
> mathematics should work -- one defines into existence things
> with the required properties -- _any_ required properties.
>
> This explains why he refers to unique prime factorization as
> being a _defined_ property of the natural numbers.
> As he sees it, defining into existence things with the
> unique-prime-factorization property is the only way to get
> the unique-prime-factorization property true without
> exception over some infinite domain.
>
> I think it also explains why he keeps adding dark numbers
> despite our repeated insistence that they be left out.
> He thinks it's standard operating procedure to just *poof*
> more things into the domain if you need them or want them.

Thanks and it seems interesting to consider the compactification,
of the naturals, what results variously the fixed point at infinity
does or doesn't (that, it's independent) have the property of a
"prime at infinity" or "composite at infinity" for what factors
represent roots, for which we can thank the laws of logarithms
and the products in powers of additive terms.

The non-logical axioms of ZF set theory are that a) sets are
well-founded, and b) the infinity in ZF is well-founded.
These of course are, "restrictions of comprehension",
that unrestricted comprehension sees fail without qualifications.

I.e. there's as much in a sense "wishing away".

On Thursday, August 12, 2021 at 7:22:19 AM UTC-7, Sergio wrote:
> On 8/12/2021 8:39 AM, WM wrote:
> > zelos...@gmail.com schrieb am Donnerstag, 12. August 2021 um 06:51:48 UTC+2:
> >> tisdag 10 augusti 2021 kl. 13:36:58 UTC+2 skrev WM:
> >
> >>> Among fools of matheology this may be a majority opinion.
> >>>
> >> mathematics, stop with this "matheology" thing,
> >
> > No! Never!! Unless you withdraw the belief that more than countably many real numbers could be well-ordered, i.e., distinguished as individuals.
> >
> > Regards, WM
> >
> "distinguished as individuals" is not a math term, and requires an
> External Observer Dependent Math, which is a crock.

Euh, the individua of continua, are, largely explored as
points or atoms.

They're defined either way as one in terms of the other:
wholes and parts.

Distinguishability is actually a thing, also. (Eg "discernibles".)

Crank-bashers must take care they don't bash themselves.

Ciao

On Saturday, August 14, 2021 at 12:12:38 PM UTC-7, Ross A. Finlayson wrote:
> On Thursday, August 12, 2021 at 7:22:19 AM UTC-7, Sergio wrote:
> > On 8/12/2021 8:39 AM, WM wrote:
> > > zelos...@gmail.com schrieb am Donnerstag, 12. August 2021 um 06:51:48 UTC+2:
> > >> tisdag 10 augusti 2021 kl. 13:36:58 UTC+2 skrev WM:
> > >
> > >>> Among fools of matheology this may be a majority opinion.
> > >>>
> > >> mathematics, stop with this "matheology" thing,
> > >
> > > No! Never!! Unless you withdraw the belief that more than countably many real numbers could be well-ordered, i.e., distinguished as individuals.
> > >
> > > Regards, WM
> > >
> > "distinguished as individuals" is not a math term, and requires an
> > External Observer Dependent Math, which is a crock.
>
> Euh, the individua of continua, are, largely explored as
> points or atoms.
>
> They're defined either way as one in terms of the other:
> wholes and parts.
>
> Distinguishability is actually a thing, also. (Eg "discernibles".)
>
> Crank-bashers must take care they don't bash themselves.
>
> Ciao

Anyways for a real challenge I've put up a
paper tiger what's quite laudable and strong.

For example after a 10,000 post "On Well-Ordering the Reals, Infinity",
there's some neat consideration that line-drawing is the only way.

Of course though that's "well-ordering _these_ reals".

.... What I would expect a comprehensive mathematics as
a foundation must be to include.

On 8/14/2021 1:23 PM, WM wrote:
> Jim Burns schrieb
> am Samstag, 14. August 2021 um 19:02:48 UTC+2:
>> On 8/14/2021 10:29 AM, WM wrote:

>>> There is only one important point:
>>> You must not confuse pot. and act.
>>> as set theorists deplorably deliberately do.
>>
>> Set theorists do not confuse [changing] last endsegments
>> with [unchanging] last end segments.
>
> They do.

You do.

<WM<JB>>
>>
>> WM rejects No Last End Segment.
>
> No, but I reject that last endsegments can yield
> a set with no last endsegment -
> other than potentially infinite.
> But that has always a last one.
>
</WM<JB>>

> Every endsegment which can be last is insufficient for
> an actual infinity.

Each end segment can be described with one description..
Having been described, it can be reasoned about.

Depending upon your mood, I suppose, you call this either
potential infinity or actual infinity.
Whatever you call it, we can reason about _all_ of
those infinitely many individuals.

> Set theorists however claim that there are no others and
> that actual infinit exists. That is wrong.

I claim that we can reason about each of some collection
of described individuals, and that we can do this whether
or not there is a last one of these individuals, or whether
or not it is possible to step to one-by-one each of these
individuals, and that we can reason about them WITHOUT our
claims applying to other individuals we have NOT described.

On Saturday, August 14, 2021 at 12:46:56 PM UTC-7, Jim Burns wrote:
> On 8/14/2021 1:23 PM, WM wrote:
> > Jim Burns schrieb
> > am Samstag, 14. August 2021 um 19:02:48 UTC+2:
> >> On 8/14/2021 10:29 AM, WM wrote:
>
> >>> There is only one important point:
> >>> You must not confuse pot. and act.
> >>> as set theorists deplorably deliberately do.
> >>
> >> Set theorists do not confuse [changing] last endsegments
> >> with [unchanging] last end segments.
> >
> > They do.
> You do.
> <WM<JB>>
> >>
> >> WM rejects No Last End Segment.
> >
> > No, but I reject that last endsegments can yield
> > a set with no last endsegment -
> > other than potentially infinite.
> > But that has always a last one.
> >
> </WM<JB>>
> > Every endsegment which can be last is insufficient for
> > an actual infinity.
> Each end segment can be described with one description..
> Having been described, it can be reasoned about.
>
> Depending upon your mood, I suppose, you call this either
> potential infinity or actual infinity.
> Whatever you call it, we can reason about _all_ of
> those infinitely many individuals.
> > Set theorists however claim that there are no others and
> > that actual infinit exists. That is wrong.
> I claim that we can reason about each of some collection
> of described individuals, and that we can do this whether
> or not there is a last one of these individuals, or whether
> or not it is possible to step to one-by-one each of these
> individuals, and that we can reason about them WITHOUT our
> claims applying to other individuals we have NOT described.

I like a theory where "ubiquitous ordinals" make for
order type is powerset is successor.

When we reason about 0..oo it's in a sense same
as oo..0 with the notion what "it's difference what
implements counting".

The usual "sputnik of quantification" like "Russell's
element or Burali-Forti's ordinal and so on", point to
why it's necessary for a rapprochement in foundations,
basically to include the "extra-standard".

Then as for whether a standard or ground model of
the integers even exists, is that bounded fragments
and extensions do.

I spent a lot of time studying foundations
mostly because it's illuminating. Also because
it's difficult in comprehension and strong exercise.

.... That then it becomes ingrained and free ...,
that foundations is foundations is foundations.

Mark my words: mark my words.

Line-drawing: marked by word.

Sergio schrieb am Samstag, 14. August 2021 um 20:56:13 UTC+2:

>
> 1 every element is used

nevertheless most elements are not used, because every element belongs to a finite initial segment followed by aleph_0 elements.

> 5 all elements are used => #1

Wrong. All elements are not followed by aleph_0 elements. If every element is used, then aleph_0 elements remain. If all elements are used, then none remains.

Regards, WM