Ben Bacarisse wrote:
> FredJeffries <fredjeffries@gmail.com> writes:
>
>> You have stumbled onto the difference between a 'potentially infinite'
>> process and an 'actually infinite' set. Contrary to popular opinion
>> around here, there is room for both tools.
>
> Who here does not accept the validity of potentially infinite processes?
>
> There is one poster who talks about "potentially infinite sets" but they
> have crazy theorems (well, conjectures actually since no proofs have
> been offered) like WMath's sets E and P with E in P and P \ {E} = P.
>
> Do you think there is also room for potentially infinite /sets/ here?
> If so, can you give an example of a theorem about them (with proof) that
> shows how they are different to the usual infinite sets?

Given the increasing dementia in Wolfgang Mückenheim's mind this days
("first unit fraction", sigh...) my guess is that the next breakthrough
in Mückenmath (aka "patamathematics") will be to distinguish actual
finite sets and potential finite sets :-))

Python <python@invalid.org> writes:

> Ben Bacarisse wrote:
>> FredJeffries <fredjeffries@gmail.com> writes:
>>
>>> You have stumbled onto the difference between a 'potentially infinite'
>>> process and an 'actually infinite' set. Contrary to popular opinion
>>> around here, there is room for both tools.
>>
>> Who here does not accept the validity of potentially infinite processes?
>> There is one poster who talks about "potentially infinite sets" but they
>> have crazy theorems (well, conjectures actually since no proofs have
>> been offered) like WMath's sets E and P with E in P and P \ {E} = P.
>> Do you think there is also room for potentially infinite /sets/ here?
>> If so, can you give an example of a theorem about them (with proof) that
>> shows how they are different to the usual infinite sets?
>
> Given the increasing dementia in Wolfgang Mückenheim's mind this days
> ("first unit fraction", sigh...) my guess is that the next breakthrough
> in Mückenmath (aka "patamathematics") will be to distinguish actual
> finite sets and potential finite sets :-))

Sadly he backed down at the last minute from asserting a third great
conjecture of WMaths. Having expressed incredulity that, in proper
mathematics, SUF(x) = {1/n < x} could "go from" being empty at 0 to
infinite for any x > 0, I suggested that "∃x > 0 such that SUF(x) is
finite" might be true in WMaths. He equivocated, but would not say yes.

(I had to be fussy and say ∃x in R and n in N such that x > 0 and SUF(x)
= n because WM is nothing if not vague about what a non-finite set is.)

--
Ben.

Le mercredi 24 mai 2023 à 21:18:00 UTC+2, Chris M. Thomasson a écrit :
> On 5/24/2023 8:45 AM, PengKuan Em wrote:
> > Hilbert thinks that infinity is the vessel but he is wrong. Infinity can be full.
> Saying its full implies that it cannot hold any more. Therefore, we are
> back at finite. Contradiction? An infinite pool can never be full...

So, Hilbert's paradox was blatantly wrong from the beginning. His hotel is infinite, but full.
While infinite hotel can never be full...

KP

Le mercredi 24 mai 2023 à 21:21:15 UTC+2, Chris M. Thomasson a écrit :
> On 5/24/2023 8:41 AM, PengKuan Em wrote:
> When you say "after infinity", that implies that infinity is finite.
> Whats wrong with this picture?
In fact, you have introduced a new notion: infinity cannot be full. So, Hilbert's hotel is not full, Natural numbers are not full, real numbers are not full.

Interesting. When I say "after infinity", there was not this notion.

KP

Le mercredi 24 mai 2023 à 21:26:19 UTC+2, Chris M. Thomasson a écrit :
> On 5/24/2023 8:37 AM, PengKuan Em wrote:
> There is no "after infinity". Infinity goes on forever. There is no last
> term.

Right.

But I'm curious why mathematicians are facinated by infinity. They get excited each time they see one while no one has used infinity concretely.

kp

On Wednesday, 24 May 2023 at 23:46:44 UTC+2, FredJeffries wrote:
> On Tuesday, May 23, 2023 at 3:17:02 PM UTC-7, Chris M. Thomasson wrote:
> >
> > For some reason, the idea of an infinite pool being "full" makes my mind
> > hurt a little bit. Think of an infinite waterfall pouring into a vessel
> > that can hold an infinite amount of water. It is never full... Humm...
>
> Congratulations. You have stumbled onto the difference between a
> 'potentially infinite' process and an 'actually infinite' set.

That remains plain bullshit, as I have argued quite enough already:
either "full" does mean "full", or it's Orwellian newspeak at its best.

> Contrary to popular opinion around here, there is room for both tools.

Yeah man, but each in its own place, and standard mathematics
does conflate the two (listen to the parrot Bacarisse, he is right
about *that*), and remains plain broken bullshit for the masses...

Julio

Le mercredi 24 mai 2023 à 21:27:07 UTC+2, Valeri Njuhan a écrit :
> Do You know, that You have been nominated for the nobel price in math. ?

There is not Nobel price in math. But anyway thanks for what you said.

kp

Le mercredi 24 mai 2023 à 23:28:38 UTC+2, FredJeffries a écrit :
> Congratulations. You have now stumbled onto the differences between/among the order types
> omega
> omega*
> omega* + omega.
>
> What would be the result if the rooms were arranged with the order type omega + omega*, i.e. if the positive integers came first and then the negative integers?
>
> Suppose the rooms were arranged in the order of the rational numbers?

One stumbles on all sort of thing when walking. So, keep walking.

kp

Le mercredi 24 mai 2023 à 23:30:51 UTC+2, FredJeffries a écrit :
> On Wednesday, May 24, 2023 at 8:37:09 AM UTC-7, PengKuan Em wrote:
> > I do not understand for what purpose Hilbert designed his paradox. Just to say that after infinity is a great void?
> To demonstrate that the ordinals omega and 1+omega are equal:
> 1+omega = omega
OK. Thanks.

Ben Bacarisse pretended :
> FredJeffries <fredjeffries@gmail.com> writes:
>
>> You have stumbled onto the difference between a 'potentially infinite'
>> process and an 'actually infinite' set. Contrary to popular opinion
>> around here, there is room for both tools.
>
> Who here does not accept the validity of potentially infinite processes?
>
> There is one poster who talks about "potentially infinite sets" but they
> have crazy theorems (well, conjectures actually since no proofs have
> been offered) like WMath's sets E and P with E in P and P \ {E} = P.
>
> Do you think there is also room for potentially infinite /sets/ here?
> If so, can you give an example of a theorem about them (with proof) that
> shows how they are different to the usual infinite sets?

I have mentioned previously something very close.

Studying the mathematical (set theoretical) structure of computer
viruses with the instruction set, the set of states, the set of
symbols, the number of heads, number of tapes, on a suitable stored
program random access machine with attached background storage. All
sets are finite except for the tape. I was introduced to the concept of
a 'semi-infinite' tape which was explained to be "as big as it needs to
be" and not to worry too much about it.

Le jeudi 25 mai 2023 à 09:39:59 UTC+2, Chris M. Thomasson a écrit :
> Here is a hotel that grows infinitely large, it always has room. Here is
> breaking down the process into steps, for a little while:
>
> https://youtu.be/AJlgxwdtHp4
>
> A DLA experiment.

Thanks

On Wednesday, May 24, 2023 at 9:12:13 PM UTC-7, Chris M. Thomasson wrote:
> On 5/24/2023 2:42 PM, FredJeffries wrote:
> > On Wednesday, May 24, 2023 at 12:16:25 PM UTC-7, Chris M. Thomasson wrote:
> >> On 5/23/2023 10:13 PM, FromTheRafters wrote:
> >>> Chris M. Thomasson was thinking very hard :
> >>>> I have to go into fictional magic land to create an infinite pool:
> >>>>
> >>>> https://oeis.org/A179759
> >>>>
> >>>> The land of creativity. The magic sack that can hold an infinite
> >>>> number of coins... ;^)
> >>>
> >>> But alas, it can only hold a countably infinite number of coins.
> >> The sack knows how many coins it is holding... Yes, it can count... ;^)
> >
> > More important than merely 'how many coins it is holding' is 'what is their ordering?'
> Say each coin is given an index number. Say this infinite sack is
> empty... I drop a coin in. It is given an index number of:
>
> sack_index = 0; // empty state
>
> drop a coin in:
> coin = sack_index;
> sack_index = sack_index + 1;
>
> So,
>
> sack contains nothing
>
> Drop some coins in:
>
> sack contains coin[0]
> sack contains coin[1]
> sack contains coin[2]
> sack contains coin[3]
> ...
>
> This sack can hold an infinite amount of coins. It can never get full.

Yes. Your sack CAN hold infinitely many coins (actual infinity). But (if you drop coins in one at a time) it never DOES contain infinitely many coins (potential infinity).

On Thursday, 25 May 2023 at 16:49:30 UTC+2, FredJeffries wrote:
> On Wednesday, May 24, 2023 at 9:12:13 PM UTC-7, Chris M. Thomasson wrote:

> > This sack can hold an infinite amount of coins. It can never get full.
>
> Yes. Your sack CAN hold infinitely many coins (actual infinity). But (if
> you drop coins in one at a time) it never DOES contain infinitely many
> coins (potential infinity).

Another piece of utter and totally crankish bullshit: and then I wonder how
can even the moderately sane and intelligent still be so stupidly destructive.

You guys are a fucking lost cause: get extinguished already...

*Plonk*

Julio

On Thursday, May 25, 2023 at 7:49:30 AM UTC-7, FredJeffries wrote:
> On Wednesday, May 24, 2023 at 9:12:13 PM UTC-7, Chris M. Thomasson wrote:
> > On 5/24/2023 2:42 PM, FredJeffries wrote:
> > > On Wednesday, May 24, 2023 at 12:16:25 PM UTC-7, Chris M. Thomasson wrote:
> > >> On 5/23/2023 10:13 PM, FromTheRafters wrote:
> > >>> Chris M. Thomasson was thinking very hard :
> > >>>> I have to go into fictional magic land to create an infinite pool:
> > >>>>
> > >>>> https://oeis.org/A179759
> > >>>>
> > >>>> The land of creativity. The magic sack that can hold an infinite
> > >>>> number of coins... ;^)
> > >>>
> > >>> But alas, it can only hold a countably infinite number of coins.
> > >> The sack knows how many coins it is holding... Yes, it can count... ;^)
> > >
> > > More important than merely 'how many coins it is holding' is 'what is their ordering?'
> > Say each coin is given an index number. Say this infinite sack is
> > empty... I drop a coin in. It is given an index number of:
> >
> > sack_index = 0; // empty state
> >
> > drop a coin in:
> > coin = sack_index;
> > sack_index = sack_index + 1;
> >
> > So,
> >
> > sack contains nothing
> >
> > Drop some coins in:
> >
> > sack contains coin[0]
> > sack contains coin[1]
> > sack contains coin[2]
> > sack contains coin[3]
> > ...
> >
> > This sack can hold an infinite amount of coins. It can never get full.
> Yes. Your sack CAN hold infinitely many coins (actual infinity). But (if you drop coins in one at a time) it never DOES contain infinitely many coins (potential infinity).

The notions of infinitesimals as iota-values or "bullets" has an infinitude of
them according to an external measure, or an "Aristotle's continuum".

I.e., written like "{ ... infinitely-many ...}".

The supertask is an idea about what happens "at infinity".

The differential equation about the hypergeometric has
three regular singular points: 0, 1, and infinity.

Drawing 3-D in 2-D perspective has a "point at infinity".

The usual natural language notion of "infinity" is as of the
numbers and arbitrarily large. It's sometimes related to the abstraction
in the temporal what results constant redefinition of change in time,
like "infinity plus one" or "infinity times infinity" and so on,
reflecting for an arbitrarily large an arbitrarily larger an arbitrarily large.

The behavior of dynamical systems in equilibrium is often
in terms of "at infinity" (at t = oo, and, about x^dot and x^dot(t)).

The infinitely-divisible is a common concept in continuous quantities,
and more than less defines them.

"Transfinite cardinals" aren't the only "real infinities".

On 5/25/2023 7:24 AM, PengKuan Em wrote:
> Le mercredi 24 mai 2023 à 21:21:15 UTC+2, Chris M. Thomasson a écrit :
>> On 5/24/2023 8:41 AM, PengKuan Em wrote:
>> When you say "after infinity", that implies that infinity is finite.
>> Whats wrong with this picture?
> In fact, you have introduced a new notion: infinity cannot be full. So, Hilbert's hotel is not full, Natural numbers are not full, real numbers are not full.
>
> Interesting. When I say "after infinity", there was not this notion.

I can create a symbol, call it xxx[0].

I can say xxx[0] represents all of the infinity between 0 and 1.

I can say xxx[1] represents all of the infinity between 1 and 2.

Well, xxx[n] is infinity in and of itself. It is never full.

FromTheRafters <FTR@nomail.afraid.org> writes:

> Ben Bacarisse pretended :
>> FredJeffries <fredjeffries@gmail.com> writes:
>>
>>> You have stumbled onto the difference between a 'potentially infinite'
>>> process and an 'actually infinite' set. Contrary to popular opinion
>>> around here, there is room for both tools.
>>
>> Who here does not accept the validity of potentially infinite
>> processes?

Do you know of anyone? You may be seeing disagreement where there is
none.

>> There is one poster who talks about "potentially infinite sets" but they
>> have crazy theorems (well, conjectures actually since no proofs have
>> been offered) like WMath's sets E and P with E in P and P \ {E} = P.
>>
>> Do you think there is also room for potentially infinite /sets/ here?
>> If so, can you give an example of a theorem about them (with proof) that
>> shows how they are different to the usual infinite sets?
>
> I have mentioned previously something very close.

If it's a theorem (with proof) relating to potentially infinite sets I'd
really like to see it. Thanks. If you don't know one, that's fine.
Just say so.

--
Ben.

On Thursday, May 25, 2023 at 7:44:37 AM UTC-7, FromTheRafters wrote:
> Ben Bacarisse pretended :

> > Do you think there is also room for potentially infinite /sets/ here?
> > If so, can you give an example of a theorem about them (with proof) that
> > shows how they are different to the usual infinite sets?
> I have mentioned previously something very close.
>
> Studying the mathematical (set theoretical) structure of computer
> viruses with the instruction set, the set of states, the set of
> symbols, the number of heads, number of tapes, on a suitable stored
> program random access machine with attached background storage. All
> sets are finite except for the tape. I was introduced to the concept of
> a 'semi-infinite' tape which was explained to be "as big as it needs to
> be" and not to worry too much about it.

There is a similar phenomenon when considering the difference between polynomials and power series. Well, maybe it's not all that similar and maybe no one else finds it interesting.

For a power series we have (usually) infinitely many non-zero coefficients. For a polynomial, there are only finitely many non-zero coefficients, but there is no limit on how many of them there are.

Of course, we can see the difference when comparing the vector spaces of polynomials over a field vs power series over that field -- the former has a countably infinite basis whereas the latter doesn't.

Disclaimer: I AM NOT claiming that non-zero coefficients of polynomials are in any way a 'potentially infinite set'.

Just that there is a qualitative difference between the two. I have never come across a mathematical term for this difference.

On Wednesday, May 24, 2023 at 9:08:10 PM UTC-7, Chris M. Thomasson wrote:
> On 5/24/2023 2:34 PM, FredJeffries wrote:
> > On Wednesday, May 24, 2023 at 12:18:00 PM UTC-7, Chris M. Thomasson wrote:
> >> On 5/24/2023 8:45 AM, PengKuan Em wrote:
> >>> Le mercredi 24 mai 2023 à 01:15:39 UTC+2, FromTheRafters a écrit :
> >>>> Chris M. Thomasson used his keyboard to write :
> >>>> The vessel would have to be so large that it takes forever to build it.
> >>>>
> >>>> Stop wracking your brain trying to make mathematical objects act like
> >>>> real objects.
> >>> Hilbert thinks that infinity is the vessel but he is wrong. Infinity can be full.
> >> Saying its full implies that it cannot hold any more. Therefore, we are
> >> back at finite. Contradiction? An infinite pool can never be full...
> >
> > You are merely exploiting a difference in terminology: Does 'full' mean that each and every room is occupied or does it mean (as you suggest) that it cannot hold any more.
> >
> > Those are two very different properties. Do not confound them.
> Strange to me. Full as in each and every room is occupied makes me want
> to say this hotel is full and cannot hold any more.

The point of Hilbert's story is that those two conditions ARE NOT EQUIVALENT for infinite sets.

On 5/24/2023 5:46 PM, FredJeffries wrote:
> On Tuesday, May 23, 2023 at 3:17:02 PM UTC-7,
> Chris M. Thomasson wrote:

>> For some reason,
>> the idea of an infinite pool being "full"
>> makes my mind hurt a little bit.
>> Think of an infinite waterfall
>> pouring into a vessel that can hold
>> an infinite amount of water.
>> It is never full... Humm...
>
> Congratulations.
> You have stumbled onto
> the difference between
> a 'potentially infinite' process and
> an 'actually infinite' set.
> Contrary to popular opinion around here,
> there is room for both tools.

It's possible that my extensive remarks
about potential things and actual things
have been mis-read as referring to
_what is generally called_ "potential"
or "actual".

For the record, I have been focusing on
what Wolfgang Mückenheim of Hochshule
Augsburg calls "potential" and "actual",
so far as that can be discovered.

I'm skeptical about there being much
overlap between the two.

Keeping in mind that WM's claims are open
to revision moment by moment, depending
upon whatever he last wanted to be true,
I think that the keystone concept is Dedekind
(in)finity, AKA Bob-conservation.

WM-potentially-infinite sets do not have
Bob-conservation (are Dedekind infinite).

WM-actually-infinite sets do have Bob-
-conservation but have subsets which
do not have Bob-conservation.

The existence of a WM-actually-infinite
set requires elements that are not equal
to themselves. WM was made aware of this,
and he seemed to feel this was an advantage of
his system.

It would not be precise to say that
I think there is no room for WM-actuality
around here. It never occurred to me to
imagine a possible world in which it wasn't
around here. But I have been less than
completely welcoming, for example, to
things not equal to themselves.

Do you have advice with regard to those?

FredJeffries was thinking very hard :
> On Thursday, May 25, 2023 at 7:44:37 AM UTC-7, FromTheRafters wrote:
>> Ben Bacarisse pretended :
>
>>> Do you think there is also room for potentially infinite /sets/ here?
>>> If so, can you give an example of a theorem about them (with proof) that
>>> shows how they are different to the usual infinite sets?
>> I have mentioned previously something very close.
>>
>> Studying the mathematical (set theoretical) structure of computer
>> viruses with the instruction set, the set of states, the set of
>> symbols, the number of heads, number of tapes, on a suitable stored
>> program random access machine with attached background storage. All
>> sets are finite except for the tape. I was introduced to the concept of
>> a 'semi-infinite' tape which was explained to be "as big as it needs to
>> be" and not to worry too much about it.
>
> There is a similar phenomenon when considering the difference between
> polynomials and power series. Well, maybe it's not all that similar and maybe
> no one else finds it interesting.
>
> For a power series we have (usually) infinitely many non-zero coefficients.
> For a polynomial, there are only finitely many non-zero coefficients, but
> there is no limit on how many of them there are.

I had heard it stated as 'almost all' of a polynomial's coefficients
are zero. It says the same thing, when you know what 'almost all'
means.

> Of course, we can see the difference when comparing the vector spaces of
> polynomials over a field vs power series over that field -- the former has a
> countably infinite basis whereas the latter doesn't.
>
> Disclaimer: I AM NOT claiming that non-zero coefficients of polynomials are
> in any way a 'potentially infinite set'.

I can hear it now, the set of all possinomials is dark.

> Just that there is a qualitative difference between the two. I have never
> come across a mathematical term for this difference.

Me neither.

On 5/25/2023 4:06 PM, FredJeffries wrote:
> On Wednesday, May 24, 2023 at 9:08:10 PM UTC-7, Chris M. Thomasson wrote:
>> On 5/24/2023 2:34 PM, FredJeffries wrote:
>>> On Wednesday, May 24, 2023 at 12:18:00 PM UTC-7, Chris M. Thomasson wrote:
>>>> On 5/24/2023 8:45 AM, PengKuan Em wrote:
>>>>> Le mercredi 24 mai 2023 à 01:15:39 UTC+2, FromTheRafters a écrit :
>>>>>> Chris M. Thomasson used his keyboard to write :
>>>>>> The vessel would have to be so large that it takes forever to build it.
>>>>>>
>>>>>> Stop wracking your brain trying to make mathematical objects act like
>>>>>> real objects.
>>>>> Hilbert thinks that infinity is the vessel but he is wrong. Infinity can be full.
>>>> Saying its full implies that it cannot hold any more. Therefore, we are
>>>> back at finite. Contradiction? An infinite pool can never be full...
>>>
>>> You are merely exploiting a difference in terminology: Does 'full' mean that each and every room is occupied or does it mean (as you suggest) that it cannot hold any more.
>>>
>>> Those are two very different properties. Do not confound them.
>> Strange to me. Full as in each and every room is occupied makes me want
>> to say this hotel is full and cannot hold any more.
>
> The point of Hilbert's story is that those two conditions ARE NOT EQUIVALENT for infinite sets.
>

I can see a contrived scenario where is each room was instantly
occupied, then it would always be full, so to speak. So saying full,
means it has always has infinite occupants. I can create new rooms
forever, yet every new room is instantly occupied. So, its always full,
in a sense?

Make any sense? Or is this kook shit?

On Thursday, May 25, 2023 at 5:59:24 PM UTC-7, Jim Burns wrote:
> On 5/24/2023 5:46 PM, FredJeffries wrote:
> > On Tuesday, May 23, 2023 at 3:17:02 PM UTC-7,
> > Chris M. Thomasson wrote:
>
> >> For some reason,
> >> the idea of an infinite pool being "full"
> >> makes my mind hurt a little bit.
> >> Think of an infinite waterfall
> >> pouring into a vessel that can hold
> >> an infinite amount of water.
> >> It is never full... Humm...
> >
> > Congratulations.
> > You have stumbled onto
> > the difference between
> > a 'potentially infinite' process and
> > an 'actually infinite' set.
> > Contrary to popular opinion around here,
> > there is room for both tools.
> It's possible that my extensive remarks
> about potential things and actual things
> have been mis-read as referring to
> _what is generally called_ "potential"
> or "actual".
>
> For the record, I have been focusing on
> what Wolfgang Mückenheim of Hochshule
> Augsburg calls "potential" and "actual",
> so far as that can be discovered.
>
> I'm skeptical about there being much
> overlap between the two.
>
> Keeping in mind that WM's claims are open
> to revision moment by moment, depending
> upon whatever he last wanted to be true,
> I think that the keystone concept is Dedekind
> (in)finity, AKA Bob-conservation.
>
> WM-potentially-infinite sets do not have
> Bob-conservation (are Dedekind infinite).
>
> WM-actually-infinite sets do have Bob-
> -conservation but have subsets which
> do not have Bob-conservation.
>
> The existence of a WM-actually-infinite
> set requires elements that are not equal
> to themselves. WM was made aware of this,
> and he seemed to feel this was an advantage of
> his system.
>
> It would not be precise to say that
> I think there is no room for WM-actuality
> around here. It never occurred to me to
> imagine a possible world in which it wasn't
> around here. But I have been less than
> completely welcoming, for example, to
> things not equal to themselves.
>
> Do you have advice with regard to those?

Yeah I'd advise to ignore WM and his misunderstanding at best
and otherwise his abuse of notation, and learn something about
the "running constants" in physics that explain discretization and
continuum limits and supertasks and superclassical models of flow
in what must be "real" scalar infinitesimals and their scalar infinities.
(Wholes.)

On 5/25/2023 2:36 PM, Chris M. Thomasson wrote:
> On 5/25/2023 7:24 AM, PengKuan Em wrote:
>> Le mercredi 24 mai 2023 à 21:21:15 UTC+2, Chris M. Thomasson a écrit :
>>> On 5/24/2023 8:41 AM, PengKuan Em wrote:
>>> When you say "after infinity", that implies that infinity is finite.
>>> Whats wrong with this picture?
>> In fact, you have introduced a new notion: infinity cannot be full.
>> So, Hilbert's hotel is not full, Natural numbers are not full, real
>> numbers are not full.
>>
>> Interesting. When I say "after infinity", there was not this notion.
>
>
> I can create a symbol, call it xxx[0].
>
> I can say xxx[0] represents all of the infinity between 0 and 1.
>
> I can say xxx[1] represents all of the infinity between 1 and 2.
>
> Well, xxx[n] is infinity in and of itself. It is never full.
>

We can define its current state to be full, but it can always hold more,
and more, ...

On 5/24/2023 10:11 PM, Chris M. Thomasson wrote:
> On 5/24/2023 9:00 PM, Chris M. Thomasson wrote:
>> On 5/24/2023 2:52 PM, FromTheRafters wrote:
>>> Chris M. Thomasson brought next idea :
>>>> On 5/24/2023 8:41 AM, PengKuan Em wrote:
>>>>> Le mercredi 24 mai 2023 à 00:17:02 UTC+2, Chris M. Thomasson a écrit :
>>>>>> On 5/22/2023 6:15 AM, PengKuan Em wrote:
>>>>>>
>>>>>> For some reason, the idea of an infinite pool being "full" makes
>>>>>> my mind
>>>>>> hurt a little bit. Think of an infinite waterfall pouring into a
>>>>>> vessel
>>>>>> that can hold an infinite amount of water. It is never full...
>>>>>> Humm...
>>>>>
>>>>> Yes, this is what I said above, Hilbert's paradox means that after
>>>>> infinity there is a huge void that can contain all things.
>>>>
>>>> When you say "after infinity", that implies that infinity is finite.
>>>> Whats wrong with this picture?
>>>
>>> Intuition fails. Learning about the surreals you find that a
>>> particular class of numbers have early birthdays. In his construction
>>> after infinitely many of these he goes on to the next infinitely many
>>> and then the next.
>>
>> If one tries to iterate all of the rationals between 0 and 1, it will
>> never be able to actually reach one. If one tries to iterate all of
>> the rationals between 1 and 2, it will never be able to actually reach
>> two.
>> On and on.
>
> I can think of a so-called hyper process that creates an individual
> process for each interval:
>
> 0 to 1
> 1 to 2
> 2 to 3
> on and on...

The problem is that what about the infinity between 0 to 1/2?

So that requires another hyper process, on and on....

On Thursday, May 25, 2023 at 8:08:05 PM UTC-7, Chris M. Thomasson wrote:
> On 5/24/2023 10:11 PM, Chris M. Thomasson wrote:
> > On 5/24/2023 9:00 PM, Chris M. Thomasson wrote:
> >> On 5/24/2023 2:52 PM, FromTheRafters wrote:
> >>> Chris M. Thomasson brought next idea :
> >>>> On 5/24/2023 8:41 AM, PengKuan Em wrote:
> >>>>> Le mercredi 24 mai 2023 à 00:17:02 UTC+2, Chris M. Thomasson a écrit :
> >>>>>> On 5/22/2023 6:15 AM, PengKuan Em wrote:
> >>>>>>
> >>>>>> For some reason, the idea of an infinite pool being "full" makes
> >>>>>> my mind
> >>>>>> hurt a little bit. Think of an infinite waterfall pouring into a
> >>>>>> vessel
> >>>>>> that can hold an infinite amount of water. It is never full...
> >>>>>> Humm...
> >>>>>
> >>>>> Yes, this is what I said above, Hilbert's paradox means that after
> >>>>> infinity there is a huge void that can contain all things.
> >>>>
> >>>> When you say "after infinity", that implies that infinity is finite.
> >>>> Whats wrong with this picture?
> >>>
> >>> Intuition fails. Learning about the surreals you find that a
> >>> particular class of numbers have early birthdays. In his construction
> >>> after infinitely many of these he goes on to the next infinitely many
> >>> and then the next.
> >>
> >> If one tries to iterate all of the rationals between 0 and 1, it will
> >> never be able to actually reach one. If one tries to iterate all of
> >> the rationals between 1 and 2, it will never be able to actually reach
> >> two.
> >> On and on.
> >
> > I can think of a so-called hyper process that creates an individual
> > process for each interval:
> >
> > 0 to 1
> > 1 to 2
> > 2 to 3
> > on and on...
>
> The problem is that what about the infinity between 0 to 1/2?
>
> So that requires another hyper process, on and on....

My favorite is n/d, n-> d, d-> oo.

One great thing about it is book-ending the numbers zero and infinity,
calling that an integer continuum, then basically shrinking that to
[0,1], so it's another definition of continuity, than the usual complete
ordered field's, and extent and density and completeness and measure,
is established for it. (Constructively, ....)

Of course, if you also know transfinite set theory, then it gets into
why all the arguments for uncountability happen to exclude it.

Of course those 'real numbers' or "iota-values" aren't "standard",
where the complete ordered field's is "standard", in fact it's another
definition of continuity altogether, then that it suffices being
a continuous domain.

This sort of thing is also called an "Aristotle's continuum", or,
"standard infinitesimals", i.e. where infinitesimals are standard,
vis-a-vis that standard real numbers' infinitesimals are "zero".

This is basically the model of infinitesimals that pre-calculus and
establishing theorems about the limits and for delta-epsilonics has
that "this is the definition of a modular clock continuum and the
non-integer part of real numbers in growing magnitude or as
uniform continuous motion in time, as infinitely-divisiible and
infinitely-divded, we are going to _not_ use this model and instead
we will use delta-epsilonics so everybody can do mathematics without
infinity".

Getting into axiomatic set theory then of course, there's built up
how there's a regularity about it of course, then about how usual
set theory makes way for non-Cartesian functions, so the various
partsd don't conflict, about this being sort of unique as between integers
and the unit, while otherwise in Cartesian functions in Cantorian set theory,
a point-set of a unit's reals is uncountable. (There's a bit about ubiquitous
ordinals for numbering before counting and successor as order type as powerset.)

It's really great then how it's illustrated there are at least three definitions
of continuity, and at least four, which makes for lots of interesting ways
to stake out "potential" and "actual" vis-a-vis "effective" infinity,
which are real and true mathematics and help resolve most all
or any "paradoxes" the presence of which indicates incomplete theories.

Then, it sort of only requires a matter of deduction about the inductive
either way of integers incrementing and wholes being divded, and in the middle
of those, unavailable to inductive inference either way, that deductive inference
builds the bridge, through: "the middle of nowhere".

It's better and anybody who doesn't know that is lacking,
if only in an overall more complete (and consistent) sense.

So, continuity is already pretty central to foundations already.