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Re: Déjà vu

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  Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-09 12:34 -0500
    Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-09 10:08 -0800
      Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-09 17:18 -0500
        Re: Déjà vu genmailus@gmail.com - 2017-01-09 14:28 -0800
          Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-09 17:55 -0500
            Re: Déjà vu Me <franz.fritschee.ff@gmail.com> - 2017-01-09 17:12 -0800
              Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-09 20:55 -0500
                Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-09 18:08 -0800
                  Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-09 23:08 -0800
                    Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 10:55 -0500
            Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-10 05:13 -0800
              Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 09:49 -0500
                Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-10 07:44 -0800
                  Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 11:45 -0500
                    Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-10 09:51 -0800
                    Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-10 10:33 -0800
                      Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-10 10:35 -0800
                        Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-10 13:00 -0800
                Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-10 11:05 -0800
                  Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 18:08 -0500
                    Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-10 17:25 -0800
                      Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 01:48 -0500
                        Re: Déjà vu "Ross A. Finlayson" <ross.finlayson@gmail.com> - 2017-01-10 23:24 -0800
                        Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 00:34 -0800
                          Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 11:26 -0500
                            Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 08:55 -0800
                              Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 12:13 -0500
                                Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 09:20 -0800
                                Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 09:22 -0800
                                  Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 09:23 -0800
                                  Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 13:11 -0500
                                    Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 10:26 -0800
                                      Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-11 13:42 -0500
                                        Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 10:50 -0800
                                        Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 10:58 -0800
                                        Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 11:01 -0800
                                          Re: Déjà vu John Gabriel <thenewcalculus@gmail.com> - 2017-01-11 11:03 -0800
                                            Re: Déjà vu bursejan@gmail.com - 2017-01-11 11:11 -0800
        Re: Déjà vu WM <wolfgang.mueckenheim@hs-augsburg.de> - 2017-01-10 00:03 -0800
          Re: Déjà vu Jim Burns <james.g.burns@att.net> - 2017-01-10 10:21 -0500

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#360290 — Re: Déjà vu

On 1/7/2017 12:22 PM, WM wrote:
> Am Samstag, 7. Januar 2017 16:20:22 UTC+1
>  schrieb Jim Burns:

>> You replace the limit set of a sequence of sets with
>> with a set with endpoints equal to the limits of
>> of the sequences of endpoints of the set sequence.
>
> Yes. lim [a_, b_n] = [lim a_, lim b_n].

You struggle over what a particular letter-sequence means,
"limit". But if I use a different letter-sequence, meaning
what I meant before, your argument has no footing. It falls
off the pier and flops around in the bay, irrelevant.

Define the _least upper bound_ of an increasing sequence
of sets A_n as a set B such that,
    for all n, A_n sub B
and
    for all sets C, if, for all n, A_n sub C, B sub C

LUB [ 0, 1 - 1/n ] = [ 0, 1 )

Is that "wrong"?

>>
>> Having changed the definition, you find that you disagree
>> with everyone else about the limit of that sequence of sets.
>
> The reason is that set theory puts union = limit,
>  which is wrong.

The least upper bound of an increasing sequence of sets
is the union of those sets.

>  But most have not comprehended that, in particular for
>  the sequence of FISONs or finite rows
> 1
> 1, 2
> 1, 2, 3
> ...
> The union is not a transfinite set, i.e.,
>  a set larger than all FISONs.

Every FISON is a proper subset of the union of all FISONs.
"Limit" or "least upper bound" is irrelevant to this.

Also, there is a mapping from the union to a
proper subset of the union, what we commonly consider
proof that the union is infinite.

Also also, this is not true of any FISON. They are
each finite.

>> Do you invent a new definition of limit for each question?
>
> No, I invented this limit only for the very special case
>  of inclusion monotonic seqeuences of real intervals,
>  in order to have another, simpler example where union
>  and limit differ in set theory.

So, you invent a new meaning for a word and any other meaning
becomes "wrong". Can anyone play this game?

Guess what "instructor" means when I refer to you, WM.

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#360300

Am Montag, 9. Januar 2017 18:34:41 UTC+1 schrieb Jim Burns:
> On 1/7/2017 12:22 PM, WM wrote:
> > Am Samstag, 7. Januar 2017 16:20:22 UTC+1
> >  schrieb Jim Burns:
> 
> >> You replace the limit set of a sequence of sets with
> >> with a set with endpoints equal to the limits of
> >> of the sequences of endpoints of the set sequence.
> >
> > Yes. lim [a_, b_n] = [lim a_, lim b_n].
> 
> You struggle over what a particular letter-sequence means,

No, I show that two different meanings are confused in set theory.
> 
> Define the _least upper bound_ of an increasing sequence
> of sets A_n as a set B such that,
>     for all n, A_n sub B
> and
>     for all sets C, if, for all n, A_n sub C, B sub C
> 
> LUB [ 0, 1 - 1/n ] = [ 0, 1 )
> 
> Is that "wrong"?

I do not say that one of the two limits is "wrong". I only show that there are two different limits. I do this in order to have an example easy enough to understand that the union and the analytical limit of the sequence of FISOns differ too.
> 
> > The reason is that set theory puts union = limit,
> >  which is wrong.
> 
> The least upper bound of an increasing sequence of sets
> is the union of those sets.

According to set theory the least upper bound of the sequence of FISONs is a set with cardinality aleph_0. The union of FISONs however has not cardinality aleph_0. Everybody can see this here
1
1, 2
1, 2, 3
...
The claim that all finite sets when written together in a single row yield a set that is larger than all, is simply foolish.
> 
> Every FISON is a proper subset of the union of all FISONs.

No. Since the union consists only of FISONs, it cannot surpass all.
> 
> Also, there is a mapping from the union to a
> proper subset of the union, what we commonly consider
> proof that the union is infinite.

The union is infinite, but not transfinite, i.e., not greater than all unioned sets.
> 
> Also also, this is not true of any FISON. They are
> each finite.

Of course. Their whole sequence can be defined recursively by F(n) = F(n-1) U {n}.
If there are infinitely many FISONs, then we have infinitely many such equantions and hence infinitely many unions already producing all infinitely many FISONs. None has card aleph_0. A further union, number omega + 1, without any further ingerdient, cannot surpass the all existing sets, because to surpass in cardinality means to have more elements.

Regards, WM

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#360377

On 1/9/2017 1:08 PM, WM wrote:
> Am Montag, 9. Januar 2017 18:34:41 UTC+1 schrieb Jim Burns:
>> On 1/7/2017 12:22 PM, WM wrote:
>>> Am Samstag, 7. Januar 2017 16:20:22 UTC+1
>>>  schrieb Jim Burns:

>>>> You replace the limit set of a sequence of sets with
>>>> with a set with endpoints equal to the limits of
>>>> of the sequences of endpoints of the set sequence.
>>>
>>> Yes. lim [a_, b_n] = [lim a_, lim b_n].
>>
>> You struggle over what a particular letter-sequence means,
>
> No, I show that two different meanings are confused
> in set theory.

"are confused". Passive voice. "Are confused" by whom?
You introduced another definition for "limit" and
now you have confused yourself.

Here, I have an idea: use the standard definition for
the limit of an increasing sequence of sets: union.
'Bye with the confusion.

>> Define the _least upper bound_ of an increasing sequence
>> of sets A_n as a set B such that,
>>     for all n, A_n sub B
>> and
>>     for all sets C, if, for all n, A_n sub C, B sub C
>>
>> LUB [ 0, 1 - 1/n ] = [ 0, 1 )
>>
>> Is that "wrong"?
>
> I do not say that one of the two limits is "wrong".

Ah. Therefore, when you say below:
   "The reason is that set theory puts union = limit,
    which is wrong",
it looks as though what you mean by saying something and
not saying something "has been confused". Of course,
with the use of passive voice, we have no idea
who did the confusing.

>  I only show that there are two different limits.
>  I do this in order to have an example easy enough
>  to understand that the union and the analytical limit
>  of the sequence of FISOns differ too.

I see no good reason for you to have introduced your
analytical limit.

You want some contradiction to appear so that you can
argue against complete infinity. But giving some
different definition of "limit" and getting different results
is not a contradiction -- any more than pointing out
the fact that there is no x such that 2*x = 3 _in the integers_
while at the same time there is such an x _in the rationals_.

That's not to say that I can't think of less-than-good
reasons.

>>> The reason is that set theory puts union = limit,
>>>  which is wrong.
>>
>> The least upper bound of an increasing sequence of sets
>> is the union of those sets.
>
> According to set theory the least upper bound of the
>  sequence of FISONs is a set with cardinality aleph_0.
>  The union of FISONs however has not cardinality aleph_0.

The least upper bound and the union are the same set.
Apparently, it "has been confused" that the same set both
has and doesn't have the cardinality aleph_0 .

>  Everybody can see this here
> 1
> 1, 2
> 1, 2, 3
> ...
> The claim that all finite sets when written together
> in a single row yield a set that is larger than all,
> is simply foolish.

No set in an _infinite_ increasing sequence of sets
can be as large as or larger then all the sets in the sequence.
If it were, it would have to be the last set, but there is no
last set.

>> Every FISON is a proper subset of the union of all FISONs.
>
> No. Since the union consists only of FISONs,
> it cannot surpass all.

This is as false as the last time you said this.
There is no last FISON, so the union _must_
surpass all.

>> Also, there is a mapping from the union to a
>> proper subset of the union, what we commonly consider
>> proof that the union is infinite.
>
> The union is infinite, but not transfinite,
>> i.e., not greater than all unioned sets.

Why not use the same meaning for "transfinite" as
everyone else?

>> Also also, this is not true of any FISON. They are
>> each finite.
>
> Of course. Their whole sequence can be defined recursively
> by F(n) = F(n-1) U {n}.

I would agree with the "of course", but not because of
that recursion relation -- which looks remarkably similar
to the recursive definition for von Neumann ordinals.
Von Neumann ordinals have no problem with being infinite.

> If there are infinitely many FISONs,
>  then we have infinitely many such equantions and hence
>  infinitely many unions already producing all infinitely
>  many FISONs. None has card aleph_0.

I'm hoping that you don't mean "of course" all FISONs
are finite because none has card aleph_0. That isn't
a reason, it's a re-statement.

>  A further union,
>  number omega + 1, without any further ingerdient, cannot
>  surpass the all existing sets, because to surpass in
>  cardinality means to have more elements.

The cardinality of omega + 1 is the same as the cardinality
of omega.

By an interesting non-coincidence, omega + 1 has a bijection
with omega which is a proper subset of omega + 1 and this
is the definition of an infinite set. Funny how that
works out.

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#360383

On Monday, 9 January 2017 14:18:18 UTC-8, Jim Burns  wrote:
> On 1/9/2017 1:08 PM, WM wrote:
> > Am Montag, 9. Januar 2017 18:34:41 UTC+1 schrieb Jim Burns:
> >> On 1/7/2017 12:22 PM, WM wrote:
> >>> Am Samstag, 7. Januar 2017 16:20:22 UTC+1
> >>>  schrieb Jim Burns:
> 
> >>>> You replace the limit set of a sequence of sets with
> >>>> with a set with endpoints equal to the limits of
> >>>> of the sequences of endpoints of the set sequence.
> >>>
> >>> Yes. lim [a_, b_n] = [lim a_, lim b_n].
> >>
> >> You struggle over what a particular letter-sequence means,
> >
> > No, I show that two different meanings are confused
> > in set theory.
> 
> "are confused". Passive voice. "Are confused" by whom?
> You introduced another definition for "limit" and
> now you have confused yourself.
> 
> Here, I have an idea: use the standard definition for
> the limit of an increasing sequence of sets: union.
> 'Bye with the confusion.
> 
> >> Define the _least upper bound_ of an increasing sequence
> >> of sets A_n as a set B such that,
> >>     for all n, A_n sub B
> >> and
> >>     for all sets C, if, for all n, A_n sub C, B sub C
> >>
> >> LUB [ 0, 1 - 1/n ] = [ 0, 1 )
> >>
> >> Is that "wrong"?
> >
> > I do not say that one of the two limits is "wrong".
> 
> Ah. Therefore, when you say below:
>    "The reason is that set theory puts union = limit,
>     which is wrong",
> it looks as though what you mean by saying something and
> not saying something "has been confused". Of course,
> with the use of passive voice, we have no idea
> who did the confusing.
> 
> >  I only show that there are two different limits.
> >  I do this in order to have an example easy enough
> >  to understand that the union and the analytical limit
> >  of the sequence of FISOns differ too.
> 
> I see no good reason for you to have introduced your
> analytical limit.
> 
> You want some contradiction to appear so that you can
> argue against complete infinity. 

I don't think any argument is needed here because there is no such thing as complete infinity. That is an oxymoron. WM wins each time because his arguments are sound.

> But giving some
> different definition of "limit" and getting different results
> is not a contradiction -- any more than pointing out
> the fact that there is no x such that 2*x = 3 _in the integers_
> while at the same time there is such an x _in the rationals_.
> 
> That's not to say that I can't think of less-than-good
> reasons.
> 
> >>> The reason is that set theory puts union = limit,
> >>>  which is wrong.
> >>
> >> The least upper bound of an increasing sequence of sets
> >> is the union of those sets.
> >
> > According to set theory the least upper bound of the
> >  sequence of FISONs is a set with cardinality aleph_0.
> >  The union of FISONs however has not cardinality aleph_0.
> 
> The least upper bound and the union are the same set.
> Apparently, it "has been confused" that the same set both
> has and doesn't have the cardinality aleph_0 .
> 
> >  Everybody can see this here
> > 1
> > 1, 2
> > 1, 2, 3
> > ...
> > The claim that all finite sets when written together
> > in a single row yield a set that is larger than all,
> > is simply foolish.
> 
> No set in an _infinite_ increasing sequence of sets
> can be as large as or larger then all the sets in the sequence.
> If it were, it would have to be the last set, but there is no
> last set.
> 
> >> Every FISON is a proper subset of the union of all FISONs.
> >
> > No. Since the union consists only of FISONs,
> > it cannot surpass all.
> 
> This is as false as the last time you said this.
> There is no last FISON, so the union _must_
> surpass all.
> 
> >> Also, there is a mapping from the union to a
> >> proper subset of the union, what we commonly consider
> >> proof that the union is infinite.
> >
> > The union is infinite, but not transfinite,
> >> i.e., not greater than all unioned sets.
> 
> Why not use the same meaning for "transfinite" as
> everyone else?
> 
> >> Also also, this is not true of any FISON. They are
> >> each finite.
> >
> > Of course. Their whole sequence can be defined recursively
> > by F(n) = F(n-1) U {n}.
> 
> I would agree with the "of course", but not because of
> that recursion relation -- which looks remarkably similar
> to the recursive definition for von Neumann ordinals.
> Von Neumann ordinals have no problem with being infinite.
> 
> > If there are infinitely many FISONs,
> >  then we have infinitely many such equantions and hence
> >  infinitely many unions already producing all infinitely
> >  many FISONs. None has card aleph_0.
> 
> I'm hoping that you don't mean "of course" all FISONs
> are finite because none has card aleph_0. That isn't
> a reason, it's a re-statement.
> 
> >  A further union,
> >  number omega + 1, without any further ingerdient, cannot
> >  surpass the all existing sets, because to surpass in
> >  cardinality means to have more elements.
> 
> The cardinality of omega + 1 is the same as the cardinality
> of omega.
> 
> By an interesting non-coincidence, omega + 1 has a bijection
> with omega which is a proper subset of omega + 1 and this
> is the definition of an infinite set. Funny how that
> works out.

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#360390

On 1/9/2017 5:28 PM, genmailus@gmail.com wrote:
> On Monday, 9 January 2017 14:18:18 UTC-8,
> Jim Burns  wrote:

>> You want some contradiction to appear so that you can
>> argue against complete infinity.
>
> I don't think any argument is needed here because
>  there is no such thing as complete infinity.

I haven't been able to get a clear description of
the distinction between complete infinity and
potential infinity.

Suppose I have a language L that contains the term 0
and, if x is a term in L, then Sx is also a term.

Does this language have a potential infinity of terms?
Does being a potential infinity keep me from doing
something that I would usually do?

(An example of something I might do that tends to give
anti-complete-infinitarians fits is to map each
term of L to to a set of terms of L. Is this an
oxymoron?)

Does language L have a complete infinity of terms?
Would that make L an oxymoron? Does that mean L
doesn't exist? If it does, why wouldn't L exist?

(I should point out that we use languages with
features very like what L has. Do these languages
"not exist"?)


>  That is an oxymoron. WM wins each time because
>  his arguments are sound.

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#360420

On Monday, January 9, 2017 at 11:56:01 PM UTC+1, Jim Burns wrote:
> On 1/9/2017 5:28 PM, genmailus@gmail.com wrote:
> > On Monday, 9 January 2017 14:18:18 UTC-8,
> > Jim Burns  wrote:
> 
> >> You want some contradiction to appear so that you can
> >> argue against complete infinity.
> >
> > I don't think any argument is needed here because
> >  there is no such thing as complete infinity.
> 
> I haven't been able to get a clear description of
> the distinction between complete infinity and
> potential infinity.

Well, a very short intro:

"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." [H.B. Enderton: "Elements of Set Theory", Academic Press, New York (1977) p. 14f] 

You might try this:

"Potential versus Completed Infinity" by  Eric Schechter
https://math.vanderbilt.edu/schectex/courses/thereals/potential.html

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#360428

On 1/9/2017 8:12 PM, Me wrote:
> On Monday, January 9, 2017 at 11:56:01 PM UTC+1,
> Jim Burns wrote:
>> On 1/9/2017 5:28 PM, genmailus@gmail.com wrote:
>>> On Monday, 9 January 2017 14:18:18 UTC-8,
>>> Jim Burns  wrote:

>>>> You want some contradiction to appear so that you can
>>>> argue against complete infinity.
>>>
>>> I don't think any argument is needed here because
>>>  there is no such thing as complete infinity.
>>
>> I haven't been able to get a clear description of
>> the distinction between complete infinity and
>> potential infinity.
>
> Well, a very short intro:

Thank you for your help. I will look at the web page.
I bet I could find something in the Stanford
Encyclopedia, too, and maybe I will.

I don't think all that will help with my actual
complaint, though -- which I have not been at all
clear about. I apologize if you feel misled by my
lack of clarity.

I believe that there are actual concepts of potential
and actual infinity, that there was once respectable
work done in that area, and there might even be
respectable work being done today.

However, the vast majority of references to potential
and complete infinity I see here don't seem clear at all,
and -- much worse -- don't agree with each other as
to what potential vs complete infinity is. I could
(conceivably) become a renowned expert on what is
called potential infinity in the literature without
coming a step closer to being able to hold a useful
conversation with WM about what _he_ calls potential
infinity. And even if I could do that with WM, it
wouldn't help me hold a useful conversation with anyone
else about potential infinity. Or so it seems to me.

Maybe genmailus@gmail.com has the same idea as WM
has as to what "potential infinity" means. My suspicion
is, though, that finding out what the idea is that
genmailus@gmail.com is defending won't help me understand
WM.



> "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." [H.B. Enderton: "Elements of
>  Set Theory", Academic Press, New York (1977) p. 14f]
>
>
> You might try this:
>
> "Potential versus Completed Infinity" by  Eric Schechter
> https://math.vanderbilt.edu/schectex/courses/thereals/potential.html
>

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#360431

On Monday, January 9, 2017 at 5:55:20 PM UTC-8, Jim Burns wrote:
> On 1/9/2017 8:12 PM, Me wrote:
> > On Monday, January 9, 2017 at 11:56:01 PM UTC+1,
> > Jim Burns wrote:
> >> On 1/9/2017 5:28 PM, genmailus@gmail.com wrote:
> >>> On Monday, 9 January 2017 14:18:18 UTC-8,
> >>> Jim Burns  wrote:
> 
> >>>> You want some contradiction to appear so that you can
> >>>> argue against complete infinity.
> >>>
> >>> I don't think any argument is needed here because
> >>>  there is no such thing as complete infinity.
> >>
> >> I haven't been able to get a clear description of
> >> the distinction between complete infinity and
> >> potential infinity.
> >
> > Well, a very short intro:
> 
> Thank you for your help. I will look at the web page.
> I bet I could find something in the Stanford
> Encyclopedia, too, and maybe I will.
> 
> I don't think all that will help with my actual
> complaint, though -- which I have not been at all
> clear about. I apologize if you feel misled by my
> lack of clarity.
> 
> I believe that there are actual concepts of potential
> and actual infinity, that there was once respectable
> work done in that area, and there might even be
> respectable work being done today.
> 
> However, the vast majority of references to potential
> and complete infinity I see here don't seem clear at all,
> and -- much worse -- don't agree with each other as
> to what potential vs complete infinity is. I could
> (conceivably) become a renowned expert on what is
> called potential infinity in the literature without
> coming a step closer to being able to hold a useful
> conversation with WM about what _he_ calls potential
> infinity. And even if I could do that with WM, it
> wouldn't help me hold a useful conversation with anyone
> else about potential infinity. Or so it seems to me.
> 
> Maybe genmailus@gmail.com has the same idea as WM
> has as to what "potential infinity" means. My suspicion
> is, though, that finding out what the idea is that
> genmailus@gmail.com is defending won't help me understand
> WM.
> 
> 
> 
> > "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." [H.B. Enderton: "Elements of
> >  Set Theory", Academic Press, New York (1977) p. 14f]
> >
> >
> > You might try this:
> >
> > "Potential versus Completed Infinity" by  Eric Schechter
> > https://math.vanderbilt.edu/schectex/courses/thereals/potential.html
> >

As self-proclaimed retro-finitists, 
those closed-minded goons have 
pretty much nothing to teach 
or learn from you.  Their time 
is gone.  

(There are many a variety of 
numerical methods that provide 
closed-form solutions for various 
constrained systems that don't 
need closed-minded goon trolls, 
at all.)

Burns, you're amusing and insightful, 
you'd as well entertain your notions 
with the idea of speculating about 
the nature of the infinite (its 
nature, it's infinite).

Instead of wasting time trying to 
"understand" WM, maybe you should 
explain yourself.

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#360451

On Monday, January 9, 2017 at 6:08:55 PM UTC-8, Ross A. Finlayson wrote:
> On Monday, January 9, 2017 at 5:55:20 PM UTC-8, Jim Burns wrote:
> > On 1/9/2017 8:12 PM, Me wrote:
> > > On Monday, January 9, 2017 at 11:56:01 PM UTC+1,
> > > Jim Burns wrote:
> > >> On 1/9/2017 5:28 PM, genmailus@gmail.com wrote:
> > >>> On Monday, 9 January 2017 14:18:18 UTC-8,
> > >>> Jim Burns  wrote:
> > 
> > >>>> You want some contradiction to appear so that you can
> > >>>> argue against complete infinity.
> > >>>
> > >>> I don't think any argument is needed here because
> > >>>  there is no such thing as complete infinity.
> > >>
> > >> I haven't been able to get a clear description of
> > >> the distinction between complete infinity and
> > >> potential infinity.
> > >
> > > Well, a very short intro:
> > 
> > Thank you for your help. I will look at the web page.
> > I bet I could find something in the Stanford
> > Encyclopedia, too, and maybe I will.
> > 
> > I don't think all that will help with my actual
> > complaint, though -- which I have not been at all
> > clear about. I apologize if you feel misled by my
> > lack of clarity.
> > 
> > I believe that there are actual concepts of potential
> > and actual infinity, that there was once respectable
> > work done in that area, and there might even be
> > respectable work being done today.
> > 
> > However, the vast majority of references to potential
> > and complete infinity I see here don't seem clear at all,
> > and -- much worse -- don't agree with each other as
> > to what potential vs complete infinity is. I could
> > (conceivably) become a renowned expert on what is
> > called potential infinity in the literature without
> > coming a step closer to being able to hold a useful
> > conversation with WM about what _he_ calls potential
> > infinity. And even if I could do that with WM, it
> > wouldn't help me hold a useful conversation with anyone
> > else about potential infinity. Or so it seems to me.
> > 
> > Maybe genmailus@gmail.com has the same idea as WM
> > has as to what "potential infinity" means. My suspicion
> > is, though, that finding out what the idea is that
> > genmailus@gmail.com is defending won't help me understand
> > WM.
> > 
> > 
> > 
> > > "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." [H.B. Enderton: "Elements of
> > >  Set Theory", Academic Press, New York (1977) p. 14f]
> > >
> > >
> > > You might try this:
> > >
> > > "Potential versus Completed Infinity" by  Eric Schechter
> > > https://math.vanderbilt.edu/schectex/courses/thereals/potential.html
> > >
> 
> As self-proclaimed retro-finitists, 
> those closed-minded goons have 
> pretty much nothing to teach 
> or learn from you.  Their time 
> is gone.  
> 
> (There are many a variety of 
> numerical methods that provide 
> closed-form solutions for various 
> constrained systems that don't 
> need closed-minded goon trolls, 
> at all.)
> 
> Burns, you're amusing and insightful, 
> you'd as well entertain your notions 
> with the idea of speculating about 
> the nature of the infinite (its 
> nature, it's infinite).
> 
> Instead of wasting time trying to 
> "understand" WM, maybe you should 
> explain yourself.


Burns, I hope you'll excuse me.
It's not my place to be off-putting 
(though, by now I vouch that the 
local retro-finitists are trolls, 
and don't feed the trolls, 
and starve the trolls).

I just would hope that you would 
educate us instead of wasting 
effort on the ineducable.

We might learn something.

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#360486

On 1/10/2017 2:08 AM, Ross A. Finlayson wrote:

> Burns, I hope you'll excuse me.
> It's not my place to be off-putting
> (though, by now I vouch that the
> local retro-finitists are trolls,
> and don't feed the trolls,
> and starve the trolls).
>
> I just would hope that you would
> educate us instead of wasting
> effort on the ineducable.
>
> We might learn something.

My goal in reading and posting here is no more noble than
my own amusement -- though that might be hard to see. What
amuses me might well look like hard work to many other people.

Sometimes, I respond because I _must_ . I made a comparison
recently to a piece of chalk squeaking across a blackboard.
Sometimes, I have to _stop that chalk_ .

If you learn something from my posts, then I am glad.
I sincerely thank you for your attention.

I know I learn from posting, even if I only learn what
I think about some topic. That's useful, too.

   "How can I tell what I think till I see what I say?"
   -- E.M. Forster

As for feeding the trolls, they don't seem to like
many of the dainties I offer them. I'm not going to worry
about them getting fat on what I serve them.

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#360468

On Monday, 9 January 2017 14:56:01 UTC-8, Jim Burns  wrote:
> On 1/9/2017 5:28 PM, genmailus@gmail.com wrote:
> > On Monday, 9 January 2017 14:18:18 UTC-8,
> > Jim Burns  wrote:
> 
> >> You want some contradiction to appear so that you can
> >> argue against complete infinity.
> >
> > I don't think any argument is needed here because
> >  there is no such thing as complete infinity.
> 
> I haven't been able to get a clear description of
> the distinction between complete infinity and
> potential infinity.

That is untrue. WM has given you "innumerable" explanations.

Infinity: Junk. It does not exist.

The fact that there is even discussion about potential and actual infinity just proves that infinity is such a dubious concept, it needs an adjective in an attempt to clarify its meaning.

If infinity were well defined, there would be no need for additional adjectives.

Let me make it very simple for you: there is NO definition of infinity. You asking for a definition is the same as someone telling you that you can buy bird's milk and then you asking which bird produced the milk.

Get some common sense man. You can't be that stupid. Or can you?

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#360481

Thank you for your contribution, Mr Gabriel.
You have given me an excellent example of the
sort of thing I was talking about to
Me <franz.fritschee.ff@gmail.com>

You made no attempt to clarify _what_ you were talking
about, and it's a pretty good bet that whatever
you mean (if anything) it is different from what
WM means (if anything) by the same terms.

In the face of that, whatever any actual mathematician
has to say about infinity, potential or complete,
seems pointless.

On 1/10/2017 8:13 AM, John Gabriel wrote:
> On Monday, 9 January 2017 14:56:01 UTC-8,
> Jim Burns  wrote:
>> On 1/9/2017 5:28 PM, genmailus@gmail.com wrote:
>>> On Monday, 9 January 2017 14:18:18 UTC-8,
>>> Jim Burns  wrote:

>>>> You want some contradiction to appear so that you can
>>>> argue against complete infinity.
>>>
>>> I don't think any argument is needed here because
>>>  there is no such thing as complete infinity.
>>
>> I haven't been able to get a clear description of
>> the distinction between complete infinity and
>> potential infinity.
>
> That is untrue.
>  WM has given you "innumerable" explanations.

It's more accurate to say
WM has given us innumerable "explanations".

> Infinity: Junk. It does not exist.

Perfect circles don't exist either. Welcome to mathematics.

Consider the _labels_ 0, 1, 2, 3, ...
Never mind what is getting labeled.

Is there a last label?

Suppose that we have rules for writing the next label,
which next label is none of the previous labels.
So, no label can be last.

What should we call the collection-without-last-one
of these labels?

Well, "infinite" occurs to me. But you've got your
panties in a bunch over _that word_ .

A lovely property of things-like-those-labels
is that there is a matching, one-to-one both ways,
between the collection of things and and an incomplete
part of the collection of things.
    (AKA, a bijection between the collection and a
    proper subset of the collection).

For example,
    0 --> 1
    1 --> 2
    2 --> 3
    3 --> 4
    ...

Or, another example,
    0 --> 0
    1 --> 2
    2 --> 4
    3 --> 6
    ...

So, instead of "infinite", we could make up a word,
"autosubbijectible" and call things-like-those-labels
autosubbijectible.

So, you say infinity does not exist.
Does autosubbijectibility exist?
If not, why not?

> The fact that there is even discussion about
>  potential and actual infinity just proves that infinity
>  is such a dubious concept, it needs an adjective in
>  an attempt to clarify its meaning.

I like this argument. (It's why I'm bothering to
respond this time.) It's so _useful_ ! One can prove
_anything_ with it. Just start a discussion about
something and that proves that ... uh ... something.
Anyway:

Let's talk about time machines. Hey-Oh! Time travel
must be possible. Good to know.

> If infinity were well defined, there would be no need for
>  additional adjectives.

Just like right triangles are obviously nonsense.
Good of you to point that out.

> Let me make it very simple for you:
>  there is NO definition of infinity.

You would be the fellow who doesn't know what
a definition is. So, this is kind of underwhelming.

>  You asking for a definition is the same as someone
>  telling you that you can buy bird's milk and
>  then you asking which bird produced the milk.
>
> Get some common sense man. You can't be that stupid.
>  Or can you?

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#360485

Am Dienstag, 10. Januar 2017 15:49:08 UTC+1 schrieb Jim Burns:
> Thank you for your contribution, Mr Gabriel.

> >> I haven't been able to get a clear description of
> >> the distinction between complete infinity and
> >> potential infinity.
> >
> > That is untrue.
> >  WM has given you "innumerable" explanations.
> 
> It's more accurate to say
> WM has given us innumerable "explanations".

It is very simple: The sets |N and Q can be "proved" equinumerous only, when they are existing completely.
An infinite Cantor list can be diagonalized only when it is complete such that no entris are remaining.
An infinite digit sequence cannot define anything unless it is completely there.
All that is actual infinty.

In potential infinity it is impossible to do these things. Uo to every digit, the real number remains undefined.

On the other hand, actual infinity is shown nonsense by the simple example of Scrooge McDuck who gets bankrupt when receiving 10 $ per day and spending 1 $, if he spends the lowest numbers. In case he spends the highest numbers, he will get rich.

As I just two hours ago told my students: In science the label can never influence a result. Therefore actual infinity and set theory based upon it is unscientific to say the least.

By the way, you can find the "actual" complete lesson here:
https://www.hs-augsburg.de/~mueckenh/Transfinity/GU12.PPT 

Regards, WM

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#360491

On 1/10/2017 10:44 AM, WM wrote:
> Am Dienstag, 10. Januar 2017 15:49:08 UTC+1
> schrieb Jim Burns:

>> Thank you for your contribution, Mr Gabriel.
>> You have given me an excellent example of the
>> sort of thing I was talking about to
>> Me <franz.fritschee.ff@gmail.com>
>>
>> You made no attempt to clarify _what_ you were talking
>> about, and it's a pretty good bet that whatever
>> you mean (if anything) it is different from what
>> WM means (if anything) by the same terms.
>>
>> In the face of that, whatever any actual mathematician
>> has to say about infinity, potential or complete,
>> seems pointless.
>>
>>>> I haven't been able to get a clear description of
>>>> the distinction between complete infinity and
>>>> potential infinity.
>>>
>>> That is untrue.
>>>  WM has given you "innumerable" explanations.
>>
>> It's more accurate to say
>> WM has given us innumerable "explanations".
>
> It is very simple: The sets |N and Q can be "proved"
>  equinumerous only, when they are existing completely.

This notion of sets (or other mathematical objects) coming
existence into seems to be at the heart of what you are
potential calling infinity.
    (I make no comparisons here with what others call
    potential infinity.)

It seems to call for a more finely-grained notion of
truth than the more common notion of "exactly one of
true or false".

Let A be a set about which we know
-- {} is in A
-- if x is in A, then (x U {x}) is in A

Let B be the intersection of all sets C such that
-- C is a subset of A (and may be A)
-- {} is in C
-- if x is in C, then (x U {x}) is in C

This is a particular set B (sometimes called the
minimal inductive set). Also, it is infinite.

Given my indoctrination, I would say that, for a particular
set D, exactly one of "D is in B" and "D is not in B"
is true, and the other is false. Do you agree with that?

If you agree, then we are done here, and there doesn't
seem to be any difference (beyond vocabulary) between
what you call potential infinity and what I call infinity.

If you disagree, can you _clearly_ describe how that
is supposed to work?

Is there something other than in B or not in B that might
describe D?

Does the description of D (whatever it is) _change_
with time or with location? If it does change, does
it _change back_ , maybe like rewinding a tape, so
that _everyone_ (in all of space and time) has the
opportunity to "observe" D coming into B (or not)?

My suspicion is that the mathematicians of a certain
era didn't want to have to bother with these questions
(and more! Many more!). They're really just a distraction
from what are more commonly thought of as mathematical
questions. So, they scrapped potential infinity in
favor of "exactly one of true or false".

Note that it is perfectly feasible (if that is your
pleasure) to describe the gradual coming-into-B of
potential infinity with the in-or-out of complete
infinity. Nothing seems to me to have been lost by
that choice. If I'm wrong about that, what, then,
has been lost?

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#360499

Am Dienstag, 10. Januar 2017 17:45:38 UTC+1 schrieb Jim Burns:

> > It is very simple: The sets |N and Q can be "proved"
> >  equinumerous only, when they are existing completely.
> 
> This notion of sets (or other mathematical objects) coming
> existence into seems to be at the heart of what you are
> potential calling infinity.
>     (I make no comparisons here with what others call
>     potential infinity.)

"Should we briefly characterize the new view of the infinite introduced by Cantor, we could certainly say: In analysis we have to deal only with the infinitely small and the infinitely large as a limit-notion, as something becoming, arising, being under construction, i.e., as we put it, with the potential infinite. But this is not the proper infinite. This we have for instance when we consider the entirety of the numbers 1, 2, 3, 4, ... itself as a completed unit, or the points of a length as an entirety of things which is completely available. This sort of infinity is named actual infinite." [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167]
> 
> It seems to call for a more finely-grained notion of
> truth than the more common notion of "exactly one of
> true or false".

Yes, like in intuitionism:

On an intuitionistic view, neither the truth-value of a statement nor any other mathematical entity can be given as the final result of an infinite process, since an infinite process is precisely one that does not have a final result: that is why, when the domain of quantification is infinite, an existentially quantified statement cannot be regarded in advance as determinately either true or false, and a universally quantified one cannot be thought of as being true accidentally, that is independently of there being a proof of it, a proof which must depend intrinsically upon our grasp of the process whereby the domain is generated.  [Michael Dummet: "Elements of intuitionism" 2nd ed., Clarendon Press, Oxford (2000) pp. 40ff]
> 
> Let A be a set about which we know
> -- {} is in A
> -- if x is in A, then (x U {x}) is in A
> 
> Let B be the intersection of all sets C such that
> -- C is a subset of A (and may be A)
> -- {} is in C
> -- if x is in C, then (x U {x}) is in C
> 
> This is a particular set B (sometimes called the
> minimal inductive set). Also, it is infinite.

It is not complete, hence not a set in the terms of set theory. It exists up to every desired x, but every such x belongs to a finite initial segment whereas infinitely many further x may come into being without changing this situation.
> 
> Given my indoctrination, I would say that, for a particular
> set D, exactly one of "D is in B" and "D is not in B"
> is true, and the other is false. Do you agree with that?

Every set D is also existing only up to every elemnt which belongs to a finite intial segment. I do not claim that potential infinity is completely explored. There are questions which can be answered for every natural number, independent of its "being existing" and other which resist any answer.
> 
> If you agree, then we are done here, and there doesn't
> seem to be any difference (beyond vocabulary) between
> what you call potential infinity and what I call infinity.

You have to claim that every natural number indexes a rational number and that no rationalnumber remains without index. That is in contradiction by the mathematical limit of the function of the cardinality of the rationals < n which are not indexed by a natural < n. This function is infinite for every n and should be zero in the limit? Not in mathematics.
> 
> If you disagree, can you _clearly_ describe how that
> is supposed to work?
> 
> Is there something other than in B or not in B that might
> describe D?

I am afraid, that in the end, in order to solve these questions, we have no choice but accept the constraints of reality. We have 10^100 bits available. Therefore we can use not more elements of mathematics simultaneously.
> 
> Does the description of D (whatever it is) _change_
> with time or with location? If it does change, does
> it _change back_ , maybe like rewinding a tape, so
> that _everyone_ (in all of space and time) has the
> opportunity to "observe" D coming into B (or not)?
> 
> My suspicion is that the mathematicians of a certain
> era didn't want to have to bother with these questions
> (and more! Many more!). They're really just a distraction
> from what are more commonly thought of as mathematical
> questions. So, they scrapped potential infinity in
> favor of "exactly one of true or false".

I don't think so. It was Cantor's approach, mainly based on his religious belief, and then the amazing, paradoxical results of equinumerousity of |N and Q and the "proof" of transcendental numbers.

But it leads to McDuck's becoming poor or rich, depending on the labels attached to his dollars. That is not acceptable in a scientific environment.

Regards, WM

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#360501

On Tuesday, January 10, 2017 at 8:45:38 AM UTC-8, Jim Burns wrote:
> On 1/10/2017 10:44 AM, WM wrote:
> > Am Dienstag, 10. Januar 2017 15:49:08 UTC+1
> > schrieb Jim Burns:
> 
> >> Thank you for your contribution, Mr Gabriel.
> >> You have given me an excellent example of the
> >> sort of thing I was talking about to
> >> Me <franz.fritschee.ff@gmail.com>
> >>
> >> You made no attempt to clarify _what_ you were talking
> >> about, and it's a pretty good bet that whatever
> >> you mean (if anything) it is different from what
> >> WM means (if anything) by the same terms.
> >>
> >> In the face of that, whatever any actual mathematician
> >> has to say about infinity, potential or complete,
> >> seems pointless.
> >>
> >>>> I haven't been able to get a clear description of
> >>>> the distinction between complete infinity and
> >>>> potential infinity.
> >>>
> >>> That is untrue.
> >>>  WM has given you "innumerable" explanations.
> >>
> >> It's more accurate to say
> >> WM has given us innumerable "explanations".
> >
> > It is very simple: The sets |N and Q can be "proved"
> >  equinumerous only, when they are existing completely.
> 
> This notion of sets (or other mathematical objects) coming
> existence into seems to be at the heart of what you are
> potential calling infinity.
>     (I make no comparisons here with what others call
>     potential infinity.)
> 
> It seems to call for a more finely-grained notion of
> truth than the more common notion of "exactly one of
> true or false".
> 
> Let A be a set about which we know
> -- {} is in A
> -- if x is in A, then (x U {x}) is in A
> 
> Let B be the intersection of all sets C such that
> -- C is a subset of A (and may be A)
> -- {} is in C
> -- if x is in C, then (x U {x}) is in C
> 
> This is a particular set B (sometimes called the
> minimal inductive set). Also, it is infinite.
> 
> Given my indoctrination, I would say that, for a particular
> set D, exactly one of "D is in B" and "D is not in B"
> is true, and the other is false. Do you agree with that?
> 
> If you agree, then we are done here, and there doesn't
> seem to be any difference (beyond vocabulary) between
> what you call potential infinity and what I call infinity.
> 
> If you disagree, can you _clearly_ describe how that
> is supposed to work?
> 
> Is there something other than in B or not in B that might
> describe D?
> 
> Does the description of D (whatever it is) _change_
> with time or with location? If it does change, does
> it _change back_ , maybe like rewinding a tape, so
> that _everyone_ (in all of space and time) has the
> opportunity to "observe" D coming into B (or not)?
> 
> My suspicion is that the mathematicians of a certain
> era didn't want to have to bother with these questions
> (and more! Many more!). They're really just a distraction
> from what are more commonly thought of as mathematical
> questions. So, they scrapped potential infinity in
> favor of "exactly one of true or false".
> 
> Note that it is perfectly feasible (if that is your
> pleasure) to describe the gradual coming-into-B of
> potential infinity with the in-or-out of complete
> infinity. Nothing seems to me to have been lost by
> that choice. If I'm wrong about that, what, then,
> has been lost?

You may have inferred that this local retro-finitist 
troll was having a "potential infinite" that was the 
inductive set, but rather he refers to that as being 
the "doesn't exist because it is beyond the bounded".

So, you volunteered for him a correct interpretation 
of your interpretation, but without the surrounding 
context of otherwise his rejection of your (or, any) 
interpretation.


Then, there's my interpretation, about that the formation 
of your set B or "minimal inductive set" has that the 
quantification over finite elements (when they are the 
only regular elements that are already there) arrives 
at a non-well-founded (inductive) set before then what 
you accredit as the minimal inductive set.

Here this is in the context that the "infinitary" version 
of the usual pair-wise union or intersection involves the 
quantification over the elements as so satisfy the predicate, 
where that in the pair-wise there's one for the next.

Here, D "was" and "was in" B+ (the non-well-founded set that 
quantification delivers), and, it's not in B (having restricted 
the comprehension or censored what was the result of the operation 
to its "well-founded" part of its value, by some pair-wise disjoint).

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#360502

On Tuesday, January 10, 2017 at 10:33:18 AM UTC-8, Ross A. Finlayson wrote:
> On Tuesday, January 10, 2017 at 8:45:38 AM UTC-8, Jim Burns wrote:
> > On 1/10/2017 10:44 AM, WM wrote:
> > > Am Dienstag, 10. Januar 2017 15:49:08 UTC+1
> > > schrieb Jim Burns:
> > 
> > >> Thank you for your contribution, Mr Gabriel.
> > >> You have given me an excellent example of the
> > >> sort of thing I was talking about to
> > >> Me <franz.fritschee.ff@gmail.com>
> > >>
> > >> You made no attempt to clarify _what_ you were talking
> > >> about, and it's a pretty good bet that whatever
> > >> you mean (if anything) it is different from what
> > >> WM means (if anything) by the same terms.
> > >>
> > >> In the face of that, whatever any actual mathematician
> > >> has to say about infinity, potential or complete,
> > >> seems pointless.
> > >>
> > >>>> I haven't been able to get a clear description of
> > >>>> the distinction between complete infinity and
> > >>>> potential infinity.
> > >>>
> > >>> That is untrue.
> > >>>  WM has given you "innumerable" explanations.
> > >>
> > >> It's more accurate to say
> > >> WM has given us innumerable "explanations".
> > >
> > > It is very simple: The sets |N and Q can be "proved"
> > >  equinumerous only, when they are existing completely.
> > 
> > This notion of sets (or other mathematical objects) coming
> > existence into seems to be at the heart of what you are
> > potential calling infinity.
> >     (I make no comparisons here with what others call
> >     potential infinity.)
> > 
> > It seems to call for a more finely-grained notion of
> > truth than the more common notion of "exactly one of
> > true or false".
> > 
> > Let A be a set about which we know
> > -- {} is in A
> > -- if x is in A, then (x U {x}) is in A
> > 
> > Let B be the intersection of all sets C such that
> > -- C is a subset of A (and may be A)
> > -- {} is in C
> > -- if x is in C, then (x U {x}) is in C
> > 
> > This is a particular set B (sometimes called the
> > minimal inductive set). Also, it is infinite.
> > 
> > Given my indoctrination, I would say that, for a particular
> > set D, exactly one of "D is in B" and "D is not in B"
> > is true, and the other is false. Do you agree with that?
> > 
> > If you agree, then we are done here, and there doesn't
> > seem to be any difference (beyond vocabulary) between
> > what you call potential infinity and what I call infinity.
> > 
> > If you disagree, can you _clearly_ describe how that
> > is supposed to work?
> > 
> > Is there something other than in B or not in B that might
> > describe D?
> > 
> > Does the description of D (whatever it is) _change_
> > with time or with location? If it does change, does
> > it _change back_ , maybe like rewinding a tape, so
> > that _everyone_ (in all of space and time) has the
> > opportunity to "observe" D coming into B (or not)?
> > 
> > My suspicion is that the mathematicians of a certain
> > era didn't want to have to bother with these questions
> > (and more! Many more!). They're really just a distraction
> > from what are more commonly thought of as mathematical
> > questions. So, they scrapped potential infinity in
> > favor of "exactly one of true or false".
> > 
> > Note that it is perfectly feasible (if that is your
> > pleasure) to describe the gradual coming-into-B of
> > potential infinity with the in-or-out of complete
> > infinity. Nothing seems to me to have been lost by
> > that choice. If I'm wrong about that, what, then,
> > has been lost?
> 
> You may have inferred that this local retro-finitist 
> troll was having a "potential infinite" that was the 
> inductive set, but rather he refers to that as being 
> the "doesn't exist because it is beyond the bounded".
> 
> So, you volunteered for him a correct interpretation 
> of your interpretation, but without the surrounding 
> context of otherwise his rejection of your (or, any) 
> interpretation.
> 
> 
> Then, there's my interpretation, about that the formation 
> of your set B or "minimal inductive set" has that the 
> quantification over finite elements (when they are the 
> only regular elements that are already there) arrives 
> at a non-well-founded (inductive) set before then what 
> you accredit as the minimal inductive set.
> 
> Here this is in the context that the "infinitary" version 
> of the usual pair-wise union or intersection involves the 
> quantification over the elements as so satisfy the predicate, 
> where that in the pair-wise there's one for the next.
> 
> Here, D "was" and "was in" B+ (the non-well-founded set that 
> quantification delivers), and, it's not in B (having restricted 
> the comprehension or censored what was the result of the operation 
> to its "well-founded" part of its value, by some pair-wise disjoint).

That is, it's so clear that B+ doesn't have any 
well-founded elements not in B and vice versa.

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#360519

On Tuesday, January 10, 2017 at 10:36:09 AM UTC-8, Ross A. Finlayson wrote:
> On Tuesday, January 10, 2017 at 10:33:18 AM UTC-8, Ross A. Finlayson wrote:
> > On Tuesday, January 10, 2017 at 8:45:38 AM UTC-8, Jim Burns wrote:
> > > On 1/10/2017 10:44 AM, WM wrote:
> > > > Am Dienstag, 10. Januar 2017 15:49:08 UTC+1
> > > > schrieb Jim Burns:
> > > 
> > > >> Thank you for your contribution, Mr Gabriel.
> > > >> You have given me an excellent example of the
> > > >> sort of thing I was talking about to
> > > >> Me <franz.fritschee.ff@gmail.com>
> > > >>
> > > >> You made no attempt to clarify _what_ you were talking
> > > >> about, and it's a pretty good bet that whatever
> > > >> you mean (if anything) it is different from what
> > > >> WM means (if anything) by the same terms.
> > > >>
> > > >> In the face of that, whatever any actual mathematician
> > > >> has to say about infinity, potential or complete,
> > > >> seems pointless.
> > > >>
> > > >>>> I haven't been able to get a clear description of
> > > >>>> the distinction between complete infinity and
> > > >>>> potential infinity.
> > > >>>
> > > >>> That is untrue.
> > > >>>  WM has given you "innumerable" explanations.
> > > >>
> > > >> It's more accurate to say
> > > >> WM has given us innumerable "explanations".
> > > >
> > > > It is very simple: The sets |N and Q can be "proved"
> > > >  equinumerous only, when they are existing completely.
> > > 
> > > This notion of sets (or other mathematical objects) coming
> > > existence into seems to be at the heart of what you are
> > > potential calling infinity.
> > >     (I make no comparisons here with what others call
> > >     potential infinity.)
> > > 
> > > It seems to call for a more finely-grained notion of
> > > truth than the more common notion of "exactly one of
> > > true or false".
> > > 
> > > Let A be a set about which we know
> > > -- {} is in A
> > > -- if x is in A, then (x U {x}) is in A
> > > 
> > > Let B be the intersection of all sets C such that
> > > -- C is a subset of A (and may be A)
> > > -- {} is in C
> > > -- if x is in C, then (x U {x}) is in C
> > > 
> > > This is a particular set B (sometimes called the
> > > minimal inductive set). Also, it is infinite.
> > > 
> > > Given my indoctrination, I would say that, for a particular
> > > set D, exactly one of "D is in B" and "D is not in B"
> > > is true, and the other is false. Do you agree with that?
> > > 
> > > If you agree, then we are done here, and there doesn't
> > > seem to be any difference (beyond vocabulary) between
> > > what you call potential infinity and what I call infinity.
> > > 
> > > If you disagree, can you _clearly_ describe how that
> > > is supposed to work?
> > > 
> > > Is there something other than in B or not in B that might
> > > describe D?
> > > 
> > > Does the description of D (whatever it is) _change_
> > > with time or with location? If it does change, does
> > > it _change back_ , maybe like rewinding a tape, so
> > > that _everyone_ (in all of space and time) has the
> > > opportunity to "observe" D coming into B (or not)?
> > > 
> > > My suspicion is that the mathematicians of a certain
> > > era didn't want to have to bother with these questions
> > > (and more! Many more!). They're really just a distraction
> > > from what are more commonly thought of as mathematical
> > > questions. So, they scrapped potential infinity in
> > > favor of "exactly one of true or false".
> > > 
> > > Note that it is perfectly feasible (if that is your
> > > pleasure) to describe the gradual coming-into-B of
> > > potential infinity with the in-or-out of complete
> > > infinity. Nothing seems to me to have been lost by
> > > that choice. If I'm wrong about that, what, then,
> > > has been lost?
> > 
> > You may have inferred that this local retro-finitist 
> > troll was having a "potential infinite" that was the 
> > inductive set, but rather he refers to that as being 
> > the "doesn't exist because it is beyond the bounded".
> > 
> > So, you volunteered for him a correct interpretation 
> > of your interpretation, but without the surrounding 
> > context of otherwise his rejection of your (or, any) 
> > interpretation.
> > 
> > 
> > Then, there's my interpretation, about that the formation 
> > of your set B or "minimal inductive set" has that the 
> > quantification over finite elements (when they are the 
> > only regular elements that are already there) arrives 
> > at a non-well-founded (inductive) set before then what 
> > you accredit as the minimal inductive set.
> > 
> > Here this is in the context that the "infinitary" version 
> > of the usual pair-wise union or intersection involves the 
> > quantification over the elements as so satisfy the predicate, 
> > where that in the pair-wise there's one for the next.
> > 
> > Here, D "was" and "was in" B+ (the non-well-founded set that 
> > quantification delivers), and, it's not in B (having restricted 
> > the comprehension or censored what was the result of the operation 
> > to its "well-founded" part of its value, by some pair-wise disjoint).
> 
> That is, it's so clear that B+ doesn't have any 
> well-founded elements not in B and vice versa.


If you ever noticed before when I would 
write N+ for N, as successor or order type 
of the inductive set, here writing then 
B+ for B as containing itself, that is 
also the same, that B+ contains B as an 
element, and N+ contains N as an element.

This might have escaped your notice 
as it was fifteen or more years ago.

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#360508

On Tuesday, 10 January 2017 06:49:08 UTC-8, Jim Burns  wrote:
> Thank you for your contribution, Mr Gabriel.
> You have given me an excellent example of the
> sort of thing I was talking about to
> Me <franz.fritschee.ff@gmail.com>

Sorry, no time to read nonsense.

> 
> You made no attempt to clarify _what_ you were talking
> about, and it's a pretty good bet that whatever
> you mean (if anything) it is different from what
> WM means (if anything) by the same terms.

Oh really? You mean that "There is no such thing as infinity." is not clarifying my standpoint? Chuckle.

> 
> In the face of that, whatever any actual mathematician
> has to say about infinity, potential or complete,
> seems pointless.

Yes! Your logic seems to be intact still.

> 
> On 1/10/2017 8:13 AM, John Gabriel wrote:
> > On Monday, 9 January 2017 14:56:01 UTC-8,
> > Jim Burns  wrote:
> >> On 1/9/2017 5:28 PM, genmailus@gmail.com wrote:
> >>> On Monday, 9 January 2017 14:18:18 UTC-8,
> >>> Jim Burns  wrote:
> 
> >>>> You want some contradiction to appear so that you can
> >>>> argue against complete infinity.
> >>>
> >>> I don't think any argument is needed here because
> >>>  there is no such thing as complete infinity.
> >>
> >> I haven't been able to get a clear description of
> >> the distinction between complete infinity and
> >> potential infinity.
> >
> > That is untrue.
> >  WM has given you "innumerable" explanations.
> 
> It's more accurate to say
> WM has given us innumerable "explanations".

No, because that would imply that infinity might be plausible, but it's obvious that only WM's explanations are plausible.

> 
> > Infinity: Junk. It does not exist.
> 
> Perfect circles don't exist either. 

The circle concept is perfect. No instantiation of it is ever perfect or even close.

> Welcome to mythmatics.

Indeed.

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#360535

On 1/10/2017 2:05 PM, John Gabriel wrote:
> On Tuesday, 10 January 2017 06:49:08 UTC-8,
> Jim Burns  wrote:

>> You made no attempt to clarify _what_ you were talking
>> about, and it's a pretty good bet that whatever
>> you mean (if anything) it is different from what
>> WM means (if anything) by the same terms.
>
> Oh really? You mean that
>    "There is no such thing as infinity."
> is not clarifying my standpoint? Chuckle

Chuckle. No, it's not clarifying your standpoint.
I don't expect you to understand that.

Here's a similar example:
    "There is such a thing as autosubbijectibility."

Do you feel that's all I need to say in order to
clarify my position? Or do you think that I should
explain what I mean better?

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