On Wednesday, 17 August 2022 at 14:15:39 UTC+2, richar...@gmail.com wrote:
> And the model you choose affects what operations are defined on all values.
And the operations you choose affects the extensibility of your system!

> Nope, that your problem, You can have a comparative order without a metric.
That's not "my" problem. That's YOUR problem. You want a decidable < operator on elements of R.

Shame.

On Wednesday, 17 August 2022 at 20:17:44 UTC+8, richar...@gmail.com wrote:
> On 8/17/22 2:34 AM, wij wrote:
> >> ...
> > It already mentioned: Your math. cannot say "x+1 is closer than x to infinity".
> > You often say infinity is a (your?) "concept", what kind of concept?
> > Where is your books say infinity is a Special "vaue"?
> > Where in your books Infinity/Closer/Approaching is defined?
> >
> > You cannot use "-∞ <------ 0 ------> +∞" to debate anything.
> > You keep dream talking.
> I am just following the standard model of the Real Numbers.
>
> I don't have the time to teach you that.

I know, DEFINITION man. The problem is that you don't really know what the
standard model and the Real Numbers and DEFINITION you are talking about.
It is time to stop dream-talk and accept my proposal.

On Wednesday, 17 August 2022 at 14:17:44 UTC+2, richar...@gmail.com wrote:
> I am just following the standard model of the Real Numbers.
>
> I don't have the time to teach you that.
I believe you have mis-identified the problem. 🤣😂🤣😂🤣😂🤣😂

What you don't have is enough TIME to decide x<y for infinite-precision real numbers.
You know... because your Turing Machines don't halt on infinite data streams.

On Wednesday, 17 August 2022 at 13:44:14 UTC+1, Skep Dick wrote:
> On Wednesday, 17 August 2022 at 14:17:44 UTC+2, richar...@gmail.com wrote:
> > I am just following the standard model of the Real Numbers.
> >
> > I don't have the time to teach you that.
> I believe you have mis-identified the problem. 🤣😂🤣😂🤣😂🤣😂
>
> What you don't have is enough TIME to decide x<y for infinite-precision real numbers.
> You know... because your Turing Machines don't halt on infinite data streams.
>
No. So in the model, the infinite tape has to have a finite number of non-blank
symbols on a finite area of tape. Note that the numebr can be unbounded. For
a simple machine like "parity unary", you could have millions of 1s concatenated
on the tape, and it would eventually give the correct answer. But not if we created
a tape with an infinite number of ones. Or declared the blank symbol to represent
one in unary notation.

On Wednesday, 17 August 2022 at 14:52:36 UTC+2, malcolm.ar...@gmail.com wrote:
> No. So in the model, the infinite tape has to have a finite number of non-blank
> symbols on a finite area of tape.
You can't encode infinite-precision real numbers under those constraints.

On Wednesday, 17 August 2022 at 13:58:53 UTC+1, Skep Dick wrote:
> On Wednesday, 17 August 2022 at 14:52:36 UTC+2, malcolm.ar...@gmail.com wrote:
> > No. So in the model, the infinite tape has to have a finite number of non-blank
> > symbols on a finite area of tape.
> You can't encode infinite-precision real numbers under those constraints.
>
No. If we have two non-blank symbols, and the encoding is a sequence of non-
blank symbols followed by a blank, then we can encode 2^N different reals, where
N is the number of non-blank symbols.
So if we say that we are encoding numbers on 0-1, and the symbols represent binary
digits after the binary point, we can only encode rationals, and we can't encode 1.0
itself. However there are an infinite number of reals between 0 and 1.

On Wednesday, 17 August 2022 at 15:14:49 UTC+2, malcolm.ar...@gmail.com wrote:
> No. If we have two non-blank symbols, and the encoding is a sequence of non-
> blank symbols followed by a blank, then we can encode 2^N different reals, where
> N is the number of non-blank symbols.
> So if we say that we are encoding numbers on 0-1, and the symbols represent binary
> digits after the binary point, we can only encode rationals, and we can't encode 1.0
> itself. However there are an infinite number of reals between 0 and 1.
OK, but at the end of this exercise you have only encoded one real number.

<(x,y) is a binary operator.

On 17/08/2022 01:18, Alan Mackenzie wrote:
[Skep:]
>> It's no more or less "rigid" than the arbitrary choices made to
>> construct it.
> There is no such choice. If there were, there would be radically
> different mathematics in different cultures, much as there are radically
> differnt religions.

Not convinced this is entirely true. It's only slightly perverse
to imagine a culture that had developed [combinatorial] games before the
formal construction of the reals. From there, the natural way to go is
to the surreals rather than to the reals. The only real [!] use then for
reals or hyperreals is that they are easier for the purposes of calculus.
I don't know where such a culture would have ended up; calculus is so
important for engineering/physics/... that perhaps the reals would have
come to dominate anyway, with the surreals studied only by academics.
[Or perhaps your claim is that the surreals are not radically different
from the reals? But I think the philosophy is very different even if
basic arithmetic eventually -- but only with a lot of effort -- works
out the same.]

May also be worth noting that religions develop at the tribal
level, and their radical differences were entrenched by the time it
became possible to compare world-wide, whereas the maths that was
available at tribal level almost has to be essentially the same world-
wide, and by [say] the 16thC, when much more advanced maths started to
develop, mathematicians across the civilised world were communicating
with each other, leading to a common[-ish] theory.

[...]
> The first mathematician likely grasped that 1 + 1 = 2 was an abstract
> truth. Or something like that.

Perhaps worth noting that animals have at least a rudimentary
grasp of that. Whether it goes much further than that is doubtful.

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Goodban

Andy Walker <anw@cuboid.co.uk> wrote:
> On 17/08/2022 01:18, Alan Mackenzie wrote:
> [Skep:]
>>> It's no more or less "rigid" than the arbitrary choices made to
>>> construct it.
>> There is no such choice. If there were, there would be radically
>> different mathematics in different cultures, much as there are radically
>> differnt religions.

> Not convinced this is entirely true. It's only slightly perverse
> to imagine a culture that had developed [combinatorial] games before the
> formal construction of the reals. From there, the natural way to go is
> to the surreals rather than to the reals. The only real [!] use then for
> reals or hyperreals is that they are easier for the purposes of calculus.
> I don't know where such a culture would have ended up; calculus is so
> important for engineering/physics/... that perhaps the reals would have
> come to dominate anyway, with the surreals studied only by academics.
> [Or perhaps your claim is that the surreals are not radically different
> from the reals? But I think the philosophy is very different even if
> basic arithmetic eventually -- but only with a lot of effort -- works
> out the same.]

But the real numbers and the hyperreals and the surreals are all part of
standard mathematics. They're all fields, for example.

> May also be worth noting that religions develop at the tribal
> level, and their radical differences were entrenched by the time it
> became possible to compare world-wide, whereas the maths that was
> available at tribal level almost has to be essentially the same world-
> wide, ....

I think that that's my point.

> .... and by [say] the 16thC, when much more advanced maths started to
> develop, mathematicians across the civilised world were communicating
> with each other, leading to a common[-ish] theory.

My other point is that there is no other theory possible. Otherwise,
why did it not develop?

> [...]
>> The first mathematician likely grasped that 1 + 1 = 2 was an abstract
>> truth. Or something like that.

> Perhaps worth noting that animals have at least a rudimentary
> grasp of that. Whether it goes much further than that is doubtful.

> --
> Andy Walker, Nottingham.
> Andy's music pages: www.cuboid.me.uk/andy/Music
> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Goodban

--
Alan Mackenzie (Nuremberg, Germany).

Skep Dick <skepdick22@gmail.com> wrote:
> On Wednesday, 17 August 2022 at 02:19:00 UTC+2, Alan Mackenzie wrote:
>> That 1 + 1 = 2, for example.
> That's not true. It depends on what + means.

>> There is no such choice. If there were, there would be radically
>> different mathematics in different cultures, much as there are radically
>> differnt religions.
> Which is precisely what you are observing. Have you bothered to read
> any history?

> Base 2 mathematics for computer scientists.
> Base 10 for the SI system.
> Base 60 for the Babylonians.
> Base 20 for the French

OK, 10 out of 10 for word play, but these ways of counting are not
radically different. They all use a base, for example.

>> Again, there is one system of maths and it is universal. If you were
>> right, there would be several distinct systems.
> There are several different systems!

> Natural numbers.
> Complex numbers.
> Real numbers.
> Surreal numbers.
> p-adic numbers.

They're all part of standard mathematics, such as is taught in degree
level courses.

[ .... ]

>> No, it [mathematics] is absolute, as demonstrated above. Bear in mind,
>> I've studied it, you haven't.
> You studied other people's invented notations.

Yes, since mathematics is far more difficult without appropriate
notations. But mathematics consists of a great deal more than mere
notations.

> I've invented notations that other people study.

> As a creator of formal languages I think I understand it much better
> than you.

This seems not to be the case.

--
Alan Mackenzie (Nuremberg, Germany).

On 17/08/2022 18:42, Alan Mackenzie wrote:
> But the real numbers and the hyperreals and the surreals are all part of
> standard mathematics. They're all fields, for example.

Yes, but .... I think if undergraduate curriculums started with
the hyperreals or surreals, the maths that would be the common ground of
"all" mathematicians, inc physicists, engineers and others, would overall
look quite different. It would look quite different again if we had had
computers a lot earlier and so had developed finite maths, NA and so on
further than analysis. If would look radically different again if stats
had developed earlier; likewise non-Euclidean geometry; or various
branches of mathematics. Who knows what maths on Alpha Centauri looks
like? What bits would essentially /have/ to be the same? Counting, but
what else?

>> May also be worth noting that religions develop at the tribal
>> level, and their radical differences were entrenched by the time it
>> became possible to compare world-wide, whereas the maths that was
>> available at tribal level almost has to be essentially the same world-
>> wide, ....
> I think that that's my point.

Yes, but that's only counting, and perhaps fractions. Tribes
don't discover calculus or fluid mechanics or other "modern" concepts
[ie, post-mediaeval].

>> .... and by [say] the 16thC, when much more advanced maths started to
>> develop, mathematicians across the civilised world were communicating
>> with each other, leading to a common[-ish] theory.
> My other point is that there is no other theory possible. Otherwise,
> why did it not develop?

Because, perhaps, the maths we have was developed by academics
who were in sufficient contact across the world? I don't know whether
there are important insights that no-one has yet spotted. FTAOD, I
don't expect our resident cranks to produce any.

[I'm not disagreeing with you very strongly, BTW. Mostly it's
a matter of emphasis.]

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Goodban

Andy Walker <anw@cuboid.co.uk> wrote:
> On 17/08/2022 18:42, Alan Mackenzie wrote:
>> But the real numbers and the hyperreals and the surreals are all part
>> of standard mathematics. They're all fields, for example.

> Yes, but .... I think if undergraduate curriculums started with
> the hyperreals or surreals, the maths that would be the common ground
> of "all" mathematicians, inc physicists, engineers and others, would
> overall look quite different.

That's an intriguing thought. I was at Cambridge at the time that Dr.
Conway's On Numbers and Games was published, and I have a copy of the
first edition.

> It would look quite different again if we had had computers a lot
> earlier and so had developed finite maths, NA and so on further than
> analysis. If would look radically different again if stats had
> developed earlier; likewise non-Euclidean geometry; or various
> branches of mathematics. Who knows what maths on Alpha Centauri looks
> like? What bits would essentially /have/ to be the same? Counting,
> but what else?

Maybe things would look different for a hundred or two hundred years, and
then "converge" again.

>>> May also be worth noting that religions develop at the tribal level,
>>> and their radical differences were entrenched by the time it became
>>> possible to compare world-wide, whereas the maths that was available
>>> at tribal level almost has to be essentially the same world- wide,
>>> ....
>> I think that that's my point.

> Yes, but that's only counting, and perhaps fractions. Tribes
> don't discover calculus or fluid mechanics or other "modern" concepts
> [ie, post-mediaeval].

>>> .... and by [say] the 16thC, when much more advanced maths started to
>>> develop, mathematicians across the civilised world were communicating
>>> with each other, leading to a common[-ish] theory.
>> My other point is that there is no other theory possible. Otherwise,
>> why did it not develop?

> Because, perhaps, the maths we have was developed by academics
> who were in sufficient contact across the world?

Maybe you're right. But contacts with (to us) "distant" cultures, such
as in China and Japan, are relatively recent. I don't think western
mathematicians in the last few hundred years have been fluent in Asian
languages, and I'm not sure Asian mathematicians ever learnt Latin that
the European academics used until comparatively recently. Maybe they
did, I just don't know.

> I don't know whether there are important insights that no-one has yet
> spotted.

Something of the significance of set theory, perhaps? Maybe there are.

> FTAOD, I don't expect our resident cranks to produce any.

Sadly, I'm sure you're right, there.

> [I'm not disagreeing with you very strongly, BTW. Mostly it's
> a matter of emphasis.]

Yes, I understand. Thanks!

> --
> Andy Walker, Nottingham.
> Andy's music pages: www.cuboid.me.uk/andy/Music
> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Goodban

--
Alan Mackenzie (Nuremberg, Germany).

On Wednesday, August 17, 2022 at 12:08:11 AM UTC-7, Skep Dick wrote:
> On Wednesday, 17 August 2022 at 07:15:58 UTC+2, dklei...@gmail.com wrote:
> > Once chosen 0 does not change and the isomorphism
> > stops being of interest.
> It stops being of interest ??!?!?!
>
> -∞<----------------------0-> +∞
> is isomorphic to
> -∞<-0----------------------> +∞
>
> It gang-rapes my intuition that the isomorphism is being preserved despite my moving the "0" left and right!
>
There is a complete continuum of either side of 0. Seems obvious from the meaning of infinity
>
> It is incredibly interesting to me that Mathematics is so unintuitive!

On Wednesday, August 17, 2022 at 5:04:40 AM UTC-7, Skep Dick wrote:
> On Wednesday, 17 August 2022 at 13:54:24 UTC+2, richar...@gmail.com wrote:
> > Why is that a problem, many things that are True are unintuitive.
> Because Truth in Mathematics is ALWAYS relative to a model!
>
> And the model is semantic (intuition!) not syntactic (proofs).
> > Especially when you throw things like infinity into the mix, as we have
> > trouble really understanding it.
> I mean... your first sign of trouble is the ∃ operator. Which one is true?
>
> ∃∞ or ¬∃∞ ?
>
> And the answer is... whichever model you CHOOSE.
> > Maybe the biggest part of the issue is you are thinking of infinity
> > being AT the > symbol, when actually that is telling you it isn't there
> > but off a REAL FAR DISTANCE in that direction.
> Maybe your problem is that you think there is a DISTANCE metric to infinity?

Metrics are a poverty-stricken way to talk about limits. You need a topology.

wij <wyniijj2@gmail.com> wrote:
> On Wednesday, 17 August 2022 at 19:54:24 UTC+8, richar...@gmail.com wrote:
>> On 8/17/22 3:08 AM, Skep Dick wrote:
>> > On Wednesday, 17 August 2022 at 07:15:58 UTC+2, dklei...@gmail.com
>> > wrote:
>> >> Once chosen 0 does not change and the isomorphism stops being of
>> >> interest.
>> > It stops being of interest ??!?!?!

>> > -∞<----------------------0-> +∞
>> > is isomorphic to
>> > -∞<-0----------------------> +∞

>> > It gang-rapes my intuition that the isomorphism is being preserved
>> > despite my moving the "0" left and right!

>> > It is incredibly interesting to me that Mathematics is so
>> > unintuitive!

>> Why is that a problem, many things that are True are unintuitive.

>> Especially when you throw things like infinity into the mix, as we have
>> trouble really understanding it.

>> Maybe the biggest part of the issue is you are thinking of infinity
>> being AT the > symbol, when actually that is telling you it isn't there
>> but off a REAL FAR DISTANCE in that direction.

> What direction? Explain why x+1 is CLOSER than x to infinity.

That's really a troll question, which seems designed to embroil somebody
knowledgeable in stupid questions like how many angels can stand on the
head of a pin.

You want a real answer to your question? That is that "infinity" is not
a real number, and no real number has a distance to "infinity", distance
being something defined on the real numbers.

I think Richard used the phrase "REAL FAR DISTANCE in that direction" as
a thinking aid for those not versed in real analysis. It's false, as
children's fairy tales are "false", but it's a way of getting concepts
over in an easy to digest fashion.

--
Alan Mackenzie (Nuremberg, Germany).

On 17/08/2022 22:03, Alan Mackenzie wrote:
> [...] I was at Cambridge at the time that Dr.
> Conway's On Numbers and Games was published,

Ah. I was there doing Part III [now upgraded to MMath, yay!]
when he was appointed a Fellow of my college. I met him, but I was
specialising in astrophysics at the time, so although I was interested
in his work, I never really got to grips with it at the time. But a
few years later he gave a seminar in Nott'm, and we talked quite a lot,
after which I was hooked.

> and I have a copy of the
> first edition.

Likewise. Also "Winning Ways".

> [...] But contacts with (to us) "distant" cultures, such
> as in China and Japan, are relatively recent. I don't think western
> mathematicians in the last few hundred years have been fluent in Asian
> languages, and I'm not sure Asian mathematicians ever learnt Latin that
> the European academics used until comparatively recently. Maybe they
> did, I just don't know.

I think you're right that direct contact was rare. But there
were India and the Arabs between, and, I believe, more contact than
people recognised until recently between the Mongol empire and Europe.
It seems to be only in recent years that the contributions of non-
European mathematicians has been [more] fully recognised, I suspect
because of growth in "History of Maths" as a proper academic subject.

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Goodban

Andy Walker <anw@cuboid.co.uk> writes:

> On 17/08/2022 22:03, Alan Mackenzie wrote:
>> [...] I was at Cambridge at the time that Dr.
>> Conway's On Numbers and Games was published,
>
> Ah. I was there doing Part III [now upgraded to MMath, yay!]
> when he was appointed a Fellow of my college. I met him, but I was
> specialising in astrophysics at the time, so although I was interested
> in his work, I never really got to grips with it at the time. But a
> few years later he gave a seminar in Nott'm, and we talked quite a lot,
> after which I was hooked.

Seems there are a few of us! I knew him at Cambridge, but a few years
later. He was my undergraduate dissertation adviser. He was an
extraordinarily clear communicator, and very generous with his time. I
was sad to hear of his death due to Covid.

--
Ben.

On Thursday, 18 August 2022 at 05:18:26 UTC+8, Alan Mackenzie wrote:
> wij <wyni...@gmail.com> wrote:
> > On Wednesday, 17 August 2022 at 19:54:24 UTC+8, richar...@gmail.com wrote:
> >> On 8/17/22 3:08 AM, Skep Dick wrote:
> >> > On Wednesday, 17 August 2022 at 07:15:58 UTC+2, dklei...@gmail.com
> >> > wrote:
> >> >> Once chosen 0 does not change and the isomorphism stops being of
> >> >> interest.
> >> > It stops being of interest ??!?!?!
>
> >> > -∞<----------------------0-> +∞
> >> > is isomorphic to
> >> > -∞<-0----------------------> +∞
>
> >> > It gang-rapes my intuition that the isomorphism is being preserved
> >> > despite my moving the "0" left and right!
> >> > It is incredibly interesting to me that Mathematics is so
> >> > unintuitive!
> >> Why is that a problem, many things that are True are unintuitive.
> >> Especially when you throw things like infinity into the mix, as we have
> >> trouble really understanding it.
> >> Maybe the biggest part of the issue is you are thinking of infinity
> >> being AT the > symbol, when actually that is telling you it isn't there
> >> but off a REAL FAR DISTANCE in that direction.
> > What direction? Explain why x+1 is CLOSER than x to infinity.
>
> That's really a troll question, which seems designed to embroil somebody
> knowledgeable in stupid questions like how many angels can stand on the
> head of a pin.
>
> You want a real answer to your question? That is that "infinity" is not
> a real number, and no real number has a distance to "infinity", distance
> being something defined on the real numbers.
>
> I think Richard used the phrase "REAL FAR DISTANCE in that direction" as
> a thinking aid for those not versed in real analysis. It's false, as
> children's fairy tales are "false", but it's a way of getting concepts
> over in an easy to digest fashion.
> --
> Alan Mackenzie (Nuremberg, Germany).

Do you understand, as you say so, you are stating YOUR OWN math.

Do you ever see the X-Axis "-∞ <------ 0 ------> +∞" printed on textbooks?

What message does it try to convey as clearly as possible? Is any of your vague 'concept' clearer?
If x+1 is not closer than x to infinity, what does close/approach mean and
demonstrated like in the wiki?
https://en.wikipedia.org/wiki/Limit_(mathematics)

I or YOU, a troll trying "to embroil somebody knowledgeable in stupid questions
like how many angels can stand on the head of a pin."?

Don't pretend you say what the textbooks say and mean.
If you don't have proper answer, please shut up, lie and troll.

On 8/17/22 8:23 AM, Skep Dick wrote:
> On Wednesday, 17 August 2022 at 14:15:39 UTC+2, richar...@gmail.com wrote:
>> And the model you choose affects what operations are defined on all values.
> And the operations you choose affects the extensibility of your system!
>
>> Nope, that your problem, You can have a comparative order without a metric.
> That's not "my" problem. That's YOUR problem. You want a decidable < operator on elements of R.
>
> Shame.

There IS a decidable < operation on elements of R.

The fact that "closer to infinity", since infinity isn't a normal value
of the field is defined differently than "closer to x" where x IS a
member of the field isn't a problem.

That is just operator overloading.

x is closer to infinity than y if x > y, which holds for all x and y
that are actually members of the Real Number system.

On 8/17/22 8:58 AM, Skep Dick wrote:
> On Wednesday, 17 August 2022 at 14:52:36 UTC+2, malcolm.ar...@gmail.com wrote:
>> No. So in the model, the infinite tape has to have a finite number of non-blank
>> symbols on a finite area of tape.
> You can't encode infinite-precision real numbers under those constraints.
>

Right, and Turing Machines don't do arbitrary computations over the
Reals BECAUSE of that fact.

Turing Machies are DEFINED to work with things with a finite representation.

Yes, you might be able to work out some "Turing-Like" machine that could
store Real Numbers on its tape. It likely would need some "state"
register to hold a Real Number, and operations that mainipulated that
state register with the number on the Tape.

But that isn't actually a Turing Machine.

wij <wyniijj2@gmail.com> wrote:
> On Thursday, 18 August 2022 at 05:18:26 UTC+8, Alan Mackenzie wrote:
>> wij <wyni...@gmail.com> wrote:
>> > On Wednesday, 17 August 2022 at 19:54:24 UTC+8, richar...@gmail.com
>> > wrote:
>> >> On 8/17/22 3:08 AM, Skep Dick wrote:
>> >> > On Wednesday, 17 August 2022 at 07:15:58 UTC+2,
>> >> > dklei...@gmail.com wrote:
>> >> >> Once chosen 0 does not change and the isomorphism stops being of
>> >> >> interest.
>> >> > It stops being of interest ??!?!?!

>> >> > -∞<----------------------0-> +∞
>> >> > is isomorphic to
>> >> > -∞<-0----------------------> +∞

>> >> > It gang-rapes my intuition that the isomorphism is being
>> >> > preserved despite my moving the "0" left and right! It is
>> >> > incredibly interesting to me that Mathematics is so unintuitive!
>> >> Why is that a problem, many things that are True are unintuitive.
>> >> Especially when you throw things like infinity into the mix, as we
>> >> have trouble really understanding it. Maybe the biggest part of
>> >> the issue is you are thinking of infinity being AT the > symbol,
>> >> when actually that is telling you it isn't there but off a REAL FAR
>> >> DISTANCE in that direction.
>> > What direction? Explain why x+1 is CLOSER than x to infinity.

>> That's really a troll question, which seems designed to embroil somebody
>> knowledgeable in stupid questions like how many angels can stand on the
>> head of a pin.

>> You want a real answer to your question? That is that "infinity" is not
>> a real number, and no real number has a distance to "infinity", distance
>> being something defined on the real numbers.

>> I think Richard used the phrase "REAL FAR DISTANCE in that direction" as
>> a thinking aid for those not versed in real analysis. It's false, as
>> children's fairy tales are "false", but it's a way of getting concepts
>> over in an easy to digest fashion.
>> --
>> Alan Mackenzie (Nuremberg, Germany).

> Do you understand, as you say so, you are stating YOUR OWN math.

There is no such thing. There is only one "mathematics".

> Do you ever see the X-Axis "-∞ <------ 0 ------> +∞" printed on textbooks?

No.

> What message does it try to convey as clearly as possible?

I imagine that it is trying to say there is no limit on the X-Axis, and
that -oo and oo are not on it. Which is, in fact, the case.

> Is any of your vague 'concept' clearer?

I don't know what you're talking about.

> If x+1 is not closer than x to infinity, what does close/approach mean and
> demonstrated like in the wiki?
> https://en.wikipedia.org/wiki/Limit_(mathematics)

Quote me a sentence from it you find puzzling, and I'll try to explain
it.

> I or YOU, a troll trying "to embroil somebody knowledgeable in stupid
> questions like how many angels can stand on the head of a pin."?

> Don't pretend you say what the textbooks say and mean. If you don't
> have proper answer, please shut up, lie and troll.

Well, thanks for such a congenial and polite exchange.

--
Alan Mackenzie (Nuremberg, Germany).

On 8/17/22 8:25 AM, wij wrote:
> On Wednesday, 17 August 2022 at 20:17:44 UTC+8, richar...@gmail.com wrote:
>> On 8/17/22 2:34 AM, wij wrote:
>>>> ...
>>> It already mentioned: Your math. cannot say "x+1 is closer than x to infinity".
>>> You often say infinity is a (your?) "concept", what kind of concept?
>>> Where is your books say infinity is a Special "vaue"?
>>> Where in your books Infinity/Closer/Approaching is defined?
>>>
>>> You cannot use "-∞ <------ 0 ------> +∞" to debate anything.
>>> You keep dream talking.
>> I am just following the standard model of the Real Numbers.
>>
>> I don't have the time to teach you that.
>
> I know, DEFINITION man. The problem is that you don't really know what the
> standard model and the Real Numbers and DEFINITION you are talking about.
> It is time to stop dream-talk and accept my proposal.

The problem is that once you try to define that "infinity - x"
represents a number, you open the system to contradictions.

It has been shown that trying to make "infinity" act like a number
REQUIRES the remove of some other "standard" rule of the Real Number
system that we like to be able to assume.

Thus, there is a sharp line between the Real Number System that doesn't
include "Infinity" as a number, and the various Trans-Finite system that
do allow "Infinity" to be treated as a "Normal" member of the system.

One of the big problems is that there are many ways to get to "infinity"
and if you try to make it work as a normal value, and let "infinity" ==
"infinity" be true.

For instancd, the sum of the odd Natural numbers is infinity, as is the
sum of the even Natural numbers, as is the sum of the Natual numbers.

But clearly the sum of the Natural Numbers is the sum of the odd Natural
Numbers + the Sum of the Even Natural Numbers so

Infinity = Infinity + Infinity.

Subtract an Infinity from both sides and you get:

0 = Infinity.

Which seems crazy. This is what happens when you try to treat "Infinity"
as just a normal number. You need to remove certain operations, at least
when some of the operands are an infinite.

On 8/17/2022 6:54 PM, Richard Damon wrote:
> On 8/17/22 8:23 AM, Skep Dick wrote:
>> On Wednesday, 17 August 2022 at 14:15:39 UTC+2, richar...@gmail.com
>> wrote:
>>> And the model you choose affects what operations are defined on all
>>> values.
>> And the operations you choose affects the extensibility of your system!
>>
>>> Nope, that your problem, You can have a comparative order without a
>>> metric.
>> That's not "my" problem. That's YOUR problem. You want a decidable <
>> operator on elements of R.
>>
>> Shame.
>
> There IS a decidable < operation on elements of R.
>
> The fact that "closer to infinity", since infinity isn't a normal value
> of the field is defined differently than "closer to x" where x IS a
> member of the field isn't a problem.
>
> That is just operator overloading.
>
> x is closer to infinity than y if x > y, which holds for all x and y
> that are actually members of the Real Number system.

What on Earth do you mean closer to infinity?

Surely inf - x and inf - y are equal for finite x an y whether they are
equal or not. Actually both differences equal inf. If you don't like
that, you might say they are incomparable but that probably isn't true.
--
Jeff Barnett

On 8/17/22 10:28 PM, Jeff Barnett wrote:
> On 8/17/2022 6:54 PM, Richard Damon wrote:
>> On 8/17/22 8:23 AM, Skep Dick wrote:
>>> On Wednesday, 17 August 2022 at 14:15:39 UTC+2, richar...@gmail.com
>>> wrote:
>>>> And the model you choose affects what operations are defined on all
>>>> values.
>>> And the operations you choose affects the extensibility of your system!
>>>
>>>> Nope, that your problem, You can have a comparative order without a
>>>> metric.
>>> That's not "my" problem. That's YOUR problem. You want a decidable <
>>> operator on elements of R.
>>>
>>> Shame.
>>
>> There IS a decidable < operation on elements of R.
>>
>> The fact that "closer to infinity", since infinity isn't a normal
>> value of the field is defined differently than "closer to x" where x
>> IS a member of the field isn't a problem.
>>
>> That is just operator overloading.
>>
>> x is closer to infinity than y if x > y, which holds for all x and y
>> that are actually members of the Real Number system.
>
> What on Earth do you mean closer to infinity?
>
> Surely inf - x and inf - y are equal for finite x an y whether they are
> equal or not. Actually both differences equal inf. If you don't like
> that, you might say they are incomparable but that probably isn't true.

But x and y ARE comparable, and so we know which one if farther in the
direction towards that which is towards infinity.

We don't need to measure how close somethings are to "The Sky" to tell
which one is higher, i.e closer to the sky.

Closer to (positve) infinity thus becomes an equivalent expression to
larger.
Note, this definition, which is used in limit theory implies that there
are two different "Infinite" limits, one to the positive end, and one to
the negative end, +Inf and -Inf, and closer to +Inf is larger, and
closer to -Inf is the smaller (more negative, or less positive)

On Thursday, 18 August 2022 at 09:20:42 UTC+8, Alan Mackenzie wrote:
> wij <wyni...@gmail.com> wrote:
> > On Thursday, 18 August 2022 at 05:18:26 UTC+8, Alan Mackenzie wrote:
> >> wij <wyni...@gmail.com> wrote:
> >> > On Wednesday, 17 August 2022 at 19:54:24 UTC+8, richar...@gmail.com
> >> > wrote:
> >> >> On 8/17/22 3:08 AM, Skep Dick wrote:
> >> >> > On Wednesday, 17 August 2022 at 07:15:58 UTC+2,
> >> >> > dklei...@gmail.com wrote:
> >> >> >> Once chosen 0 does not change and the isomorphism stops being of
> >> >> >> interest.
> >> >> > It stops being of interest ??!?!?!
>
> >> >> > -∞<----------------------0-> +∞
> >> >> > is isomorphic to
> >> >> > -∞<-0----------------------> +∞
>
> >> >> > It gang-rapes my intuition that the isomorphism is being
> >> >> > preserved despite my moving the "0" left and right! It is
> >> >> > incredibly interesting to me that Mathematics is so unintuitive!
> >> >> Why is that a problem, many things that are True are unintuitive.
> >> >> Especially when you throw things like infinity into the mix, as we
> >> >> have trouble really understanding it. Maybe the biggest part of
> >> >> the issue is you are thinking of infinity being AT the > symbol,
> >> >> when actually that is telling you it isn't there but off a REAL FAR
> >> >> DISTANCE in that direction.
> >> > What direction? Explain why x+1 is CLOSER than x to infinity.
>
> >> That's really a troll question, which seems designed to embroil somebody
> >> knowledgeable in stupid questions like how many angels can stand on the
> >> head of a pin.
>
> >> You want a real answer to your question? That is that "infinity" is not
> >> a real number, and no real number has a distance to "infinity", distance
> >> being something defined on the real numbers.
>
> >> I think Richard used the phrase "REAL FAR DISTANCE in that direction" as
> >> a thinking aid for those not versed in real analysis. It's false, as
> >> children's fairy tales are "false", but it's a way of getting concepts
> >> over in an easy to digest fashion.
> >> --
> >> Alan Mackenzie (Nuremberg, Germany).
>
> > Do you understand, as you say so, you are stating YOUR OWN math.
> There is no such thing. There is only one "mathematics".
> > Do you ever see the X-Axis "-∞ <------ 0 ------> +∞" printed on textbooks?
> No.

Shame.

> > What message does it try to convey as clearly as possible?
> I imagine that it is trying to say there is no limit on the X-Axis, and
> that -oo and oo are not on it. Which is, in fact, the case.
> > Is any of your vague 'concept' clearer?
> I don't know what you're talking about.
> > If x+1 is not closer than x to infinity, what does close/approach mean and
> > demonstrated like in the wiki?
> > https://en.wikipedia.org/wiki/Limit_(mathematics)
> Quote me a sentence from it you find puzzling, and I'll try to explain
> it.
> > I or YOU, a troll trying "to embroil somebody knowledgeable in stupid
> > questions like how many angels can stand on the head of a pin."?
>
> > Don't pretend you say what the textbooks say and mean. If you don't
> > have proper answer, please shut up, lie and troll.
> Well, thanks for such a congenial and polite exchange.
> --
> Alan Mackenzie (Nuremberg, Germany).

--
Idiot is idiot, wherever s/he is, educated or not, civilized or not.