On 8/19/22 8:12 AM, Skep Dick wrote:
> On Friday, 19 August 2022 at 13:50:47 UTC+2, richar...@gmail.com wrote:
>>> Without explaininng the choice.
>> Then you aren't communicating. You need to use the one that is
>> applicable, as would be agreed upon.
> Applicable to WHAT? There is zero context!
>
> Definition/function APPLICATION is applied to concrete objects.
> Not an abstract one.

That is the problem. You want a definition of something without
specifying what it works on.

>
>> You are just proving you are anti-social and unable to effectively work
>> with others.
> What WORK, dickhead? There is no task on the table.

So, you aren't trying to acheve anything with your ranting? You admit it
is just meaningless?

>
> Do you even understand how synchronization works? You've made EVERY attempt to hinder communication/synchronization with your abstract sophistry.
>
> https://en.wikipedia.org/wiki/Synchronization_(computer_science)

>
>> Yes, but a GIVEN Mathematics is GIVEN.
> So GIVE it! GIVE me the Mathematics for the Real numbers. Where is it ?!?!?!
>
> ENCODE it.
> TRANSMIT it.
> COMMUNICATE it.

it is well defined in the literature. You are the one that seems to be
unable to process the definition because it is too abstract for you.

>
>> Thus things like "The Real Numbers" have precise definitions.
> You can't even precisely define "precise", you sophist!
>

Neither can you, so even though I have, you don't understand it.

>> Thus, we need to be clear WHICH Mathematics we are talking in, and why
>> there are "defaults" when not specified.
> It has helped you NOTHING. You have made it abundantly clear that whatever numbers you are talking about, it can't possibly be the standard Real numbers.
>

WHy not?

>>> Only YOU know. But you can't tell me.
>> Because it has been well defined for a long time but you can't seem to
>> understand it.
> The standard definition is incomprehensible.
>
> Maybe your understanding is much better. It's a shame you can't communicate it.

Since many people understand it, the problem isn't in the definition,
but in the receiver.

It has become clear that YOU are unable to handle the concept.

>
>> And that is where you are wrong. "Numbers" exist, they have been defined
> I don't see how that's possible. "define" is undefined.

No, it is well defined, maybe you just don't understand it.

>
> I mean, I can define you out of existence, but you are still there...
> You must be missing something about HOW definitions work.

You seem to be missing the part about definitions being a community
activity,

>
>>> Absolutely everything you are DOING can be re-interpreted from the lens of homotopy type theory/Univalent foundations.
>> Apparently not, since you seem to be unable to handle some of the concepts.
> Like you can't handle the concept of understanding understanding?
>
> Yeah, function self-application is tricky to idiots who restrict comprehension.

I understand that, but you are misapplying it.

>
>>> Without the need to cling onto the self-deception about your system being "consistent".
>> We assume consistentcy until proven wrong (but accept that it is a
>> possibility)
> It's an obvious fact to any software engineer. Your system represents, references and de-references NULL-types.
>
> Any non-idiot knows that's a runtime error.

Nope.

>
>> Can you prove it isn't? NOTE, the proof needs to follow the rules of or it isn't applicable, and so far it seems that is beyond your mental ability.
>> the system
> Bullshit. There's the system and there's the meta-system.
>
> Your foundations. The fact that your brain can't handle the null-pointer exceptions is just another clusterfuck in the system.

Why do youy say I can't handle the null-pointer exception?

>
>> knowledge-synchronization is based on having inadequate tools for the Job?
> What Job? You haven't specified a task.

Whatever you were trying to accomplish with it.

>
>> YOU are the one that made the fight, as you want to disagree with how
>> things are.
> Bullshit. I am pointing out that your description/definition of the Real Numbers fails to capture how things are!

Then you are doing a very bad job at it, since you start by not knowing
what you are talking about.

>
>> You seem to be doing a pretty good job at beating yourself up though, so
>> GOOD JOB. Keep it up and you will totally defeat yourself.
> There's no such thing as self-defeat. Let me demonstrate.
>
> This sentence defeats it self.
>
> Look! It's still there. You've been confusing self-affirmation for self-defeat all along. You dumb Mathematician!
>

Word Games. Shows you ran out of logic.

On Friday, 19 August 2022 at 14:22:51 UTC+2, richar...@gmail.com wrote:
> But that just shows you don't understand the definition, Yes -1 and 1
> are bounds.
Oh, you need not worry about my understanding! I subscribe to the axiom of unrestricted comprehension (despite its short-commings).

You said that a number is finite if it has finite bounds. And your argument/justification for this claim is that 0 is finite because it's bounded by -1 and 1.

But you haven't justified the finiteness/boundedness of -1 and 1 ?!?!

On Friday, 19 August 2022 at 14:22:51 UTC+2, richar...@gmail.com wrote:
> One way starts with 0 DEFINED as bounded, as a First Principle.
I really don't know what language I need to speak to get through to you.

I am not interested in definitional/judgmental properties! I am not interested in axioms DEFINING the boundedness of 0!
I am interested in propositional properties! I am interested in theorems PROVING the boundedness of 0!

I want to see your WORK.

On Friday, 19 August 2022 at 14:41:41 UTC+2, richar...@gmail.com wrote:
> On 8/19/22 8:12 AM, Skep Dick wrote:
> That is the problem. You want a definition of something without
> specifying what it works on.
Bullshit! The I have given you the domain of the functon: forall a. [a] × [a] -> [a]

Straight off the wikipedia page for Parametric polymorphism: https://en.wikipedia.org/wiki/Parametric_polymorphism

It works on ALL types.

Why is this confusing you?!?

> > What WORK, dickhead? There is no task on the table.
> So, you aren't trying to acheve anything with your ranting? You admit it
> is just meaningless?
It's pretty obvious what I am trying to do - point out your errors about the Real numbers.

You are immune to reason. A lot like Olcott.

> it is well defined in the literature.
That's a lie. Well-defined has a precise meaning. It may be defined, but it's definitely not well-defined.

>You are the one that seems to be unable to process the definition because it is too abstract for you.
I am perfectly capable of processing well-defined defintions.

You just haven't produced one of those yet.

> Neither can you, so even though I have, you don't understand it.
Nor do you understand understanding.

> WHy not?
For all the reasons already given which you fail to understand.

A bit like how Olcott refuses to understand why his decider is broken.

> Since many people understand it, the problem isn't in the definition,
> but in the receiver.
Many people understood the Earth is flat also. Argumentum ad populum is a fallacy.

> It has become clear that YOU are unable to handle the concept.
Not possible. You haven't even attempted to communicate the concept.

You keep pointing me to books. Like theists keep pointing me to books in order to understand their God.

> > I don't see how that's possible. "define" is undefined.
> No, it is well defined, maybe you just don't understand it.
You don't even know what well-defined means...

> You seem to be missing the part about definitions being a community
> activity,
They are a community activity. One programmer + One computer a community makes.

When you TEACH your language to your computer (you write the compiler) then your computer understands your language!
You seem to be unable to do that. Always blaming the computer for your inability to define your own language.

> > Yeah, function self-application is tricky to idiots who restrict comprehension.
> I understand that, but you are misapplying it.
No, I am not. It's stratified and it type-checks.

> > Any non-idiot knows that's a runtime error.
> Nope.
Hahahah. Yeah. And Olcott solved the halting problem.

> > Your foundations. The fact that your brain can't handle the null-pointer exceptions is just another clusterfuck in the system.
> Why do youy say I can't handle the null-pointer exception?
Because you have presented no algebraic effect handler.

> > What Job? You haven't specified a task.
> Whatever you were trying to accomplish with it.
That task is not possible given your hardware bug.

> > Bullshit. I am pointing out that your description/definition of the Real Numbers fails to capture how things are!
> Then you are doing a very bad job at it, since you start by not knowing
> what you are talking about.
I know precisely what I am talking about. Parametric polymorphism.

It's too bad you completely avoided the topic when I asked you to parametrize "lim(x -> y) z"

> > Look! It's still there. You've been confusing self-affirmation for self-defeat all along. You dumb Mathematician!
> Word Games. Shows you ran out of logic.
Shame! Looks like your logic-system can't handle reflection.

On 8/19/22 8:42 AM, Skep Dick wrote:
> On Friday, 19 August 2022 at 14:22:51 UTC+2, richar...@gmail.com wrote:
>> But that just shows you don't understand the definition, Yes -1 and 1
>> are bounds.
> Oh, you need not worry about my understanding! I subscribe to the axiom of unrestricted comprehension (despite its short-commings).
>
> You said that a number is finite if it has finite bounds. And your argument/justification for this claim is that 0 is finite because it's bounded by -1 and 1.
>
> But you haven't justified the finiteness/boundedness of -1 and 1 ?!?!
>

ONE justifcation for 0 being bounded, is that it is bounded by -1 and 1.

Another is just First Principles, that 0 is the start of the numbers and
is defined to be finite, but you don't seem to like First Principles and
special cases.

A second "First Principle" is the definition that if N if finite, then
N+1 is.

This makes ALL the Natural Numbers finite by induction, but NOT "w" as
we never get to "w-1" (whatever that mean) to show that is finite.

On 8/19/22 8:45 AM, Skep Dick wrote:
> On Friday, 19 August 2022 at 14:22:51 UTC+2, richar...@gmail.com wrote:
>> One way starts with 0 DEFINED as bounded, as a First Principle.
> I really don't know what language I need to speak to get through to you.
>
> I am not interested in definitional/judgmental properties! I am not interested in axioms DEFINING the boundedness of 0!
> I am interested in propositional properties! I am interested in theorems PROVING the boundedness of 0!
>
> I want to see your WORK.
>
>
>
>
>

So your system is just broken, as without First Principles, you have
nothing to prove with.

What can you actually prove without starting with a definition or axioms
(or an assumed proof)?

On 8/19/22 8:37 AM, Skep Dick wrote:
> On Friday, 19 August 2022 at 14:23:53 UTC+2, richar...@gmail.com wrote:
>> On 8/19/22 7:49 AM, Skep Dick wrote:
>>> On Friday, 19 August 2022 at 13:23:47 UTC+2, richar...@gmail.com wrote:
>>>>> That is you (according to you).
>>>> Then what is it?
>>> I have NO fucking idea.
>> Then why to you put down others that say it doesn't exist?
>
> Which part of ASKING are you misinterpreting as "put-down" you fucking cuntwaffle?
>
> If you tellme that you can see "past the end of infinities" I believe you at face value! This is as charitable as I can possibly get about such claims!
>
> It is precisely because YOU can see past the end of infinities and I can't is WHY I am asking YOU to tell me what the last digit of pi is!
>
>

You don't understand that your question is meaningless.

What is at the bottom of a bottomless pit.

What is at the end of an unending line.

What is the last word of an endless sequence of words.

They just don't exist.

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

> On 19/08/2022 01:30, Ben Bacarisse wrote:
>>>> If RL(LR) = 1/3, why is R(L) not 0?
>>> Because, viewed as a Hackenbush string, the player of R wins with
>>> or without the first move, so R(L) is > 0. It's a surreal number, but not
>>> a real number.
>> Right (I happen to know this) but I'm not sure you gave enough
>> information for anyone who does /not/ know the rules to work it out.
>
> I didn't. But you introduced Hackenbush!

Only because that's the interpretation I needed to square what you
wrote. If they are a not Hackenbush numbers, then why do you say that
R(L) > 0? The problem, for the reader, is why "Walker" numbers have
RL(LR) = 1/3 but R(L) > 0.

Given the name you used ("balanced binary"), the most natural
interpretation is of a binary representation who infinite strings are
the limits of infinite digit sums which works for RL(LR) but not,
apparently, for R(L).

And if they /are/ Hackenbush numbers, then saying that RL(LR) = 1/3 is
really throwing a spanner in the works (for the learner), because (but
here I am out of my confort zone) RL(LR) might be "identified" with 1/3
because there is no closer Hackenbush number to 1/3. That's analogous
to a Dedekind cut, but it isn't one because we are not constructing the
reals. Writing RL(LR) = 1/3 in this context is, in my option, not
helpful.

> [I'm not proposing to write a textbook here; anyone with a
> mathematical background and who is not already familiar with the
> surreals can look it up. I just wanted to give some pointers.]

Then I think you need to give a keyword to help people look it up. Did
you try searching for "balanced binary"?

--
Ben.

On Friday, 19 August 2022 at 15:46:22 UTC+2, richar...@gmail.com wrote:
> ONE justifcation for 0 being bounded, is that it is bounded by -1 and 1.
Being bounded by unbounded numbers... Gotcha!

> Another is just First Principles, that 0 is the start of the numbers and
> is defined to be finite, but you don't seem to like First Principles and
> special cases.
No shit! We are doing Mathematics! We are trying to solve general classes of problems!

Nobody gives a shit if Olcott axiomatically solves a special case of the Halting problem.
Why should I give a shit if you axiomatically solve a special case of bounding?

> A second "First Principle" is the definition that if N if finite, then
> N+1 is.
That "IF" sure seems to be doing a lot of work for you...

> This makes ALL the Natural Numbers finite by induction, but NOT "w" as
> we never get to "w-1" (whatever that mean) to show that is finite.
Hah! Look how quickly you forgot the "IF"...

On Friday, 19 August 2022 at 15:48:25 UTC+2, richar...@gmail.com wrote:
> So your system is just broken, as without First Principles, you have
> nothing to prove with.
Then I'll use ∞-th principles instead!

> What can you actually prove without starting with a definition or axioms
> (or an assumed proof)?
I can prove the theorem I start with.

On Friday, 19 August 2022 at 15:50:27 UTC+2, richar...@gmail.com wrote:
> >
> You don't understand that your question is meaningless.
You don't understand what meaning is!

> What is at the bottom of a bottomless pit.
https://en.wikipedia.org/wiki/Bottom_type

> What is at the end of an unending line.
https://en.wikipedia.org/wiki/Bottom_type

> What is the last word of an endless sequence of words.
https://en.wikipedia.org/wiki/Bottom_type

> They just don't exist.
They exist. But they are uninhabited.

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

On 8/19/22 9:13 AM, Skep Dick wrote:
> On Friday, 19 August 2022 at 14:41:41 UTC+2, richar...@gmail.com wrote:
>> On 8/19/22 8:12 AM, Skep Dick wrote:
>> That is the problem. You want a definition of something without
>> specifying what it works on.
> Bullshit! The I have given you the domain of the functon: forall a. [a] × [a] -> [a]
>
> Straight off the wikipedia page for Parametric polymorphism: https://en.wikipedia.org/wiki/Parametric_polymorphism
>
> It works on ALL types.
>
> Why is this confusing you?!?

So, how do you USE that without knowing the ACTUAL type of a?

You talk about use being king.

>
>>> What WORK, dickhead? There is no task on the table.
>> So, you aren't trying to acheve anything with your ranting? You admit it
>> is just meaningless?
> It's pretty obvious what I am trying to do - point out your errors about the Real numbers.

So, what is my error, remember, to use the definitions of how things
actually work in the real number system.

If you "generic" number system breaks when trying to deal with actual
real numbers, that's your systems fault.

>
> You are immune to reason. A lot like Olcott.
>
>> it is well defined in the literature.
> That's a lie. Well-defined has a precise meaning. It may be defined, but it's definitely not well-defined.

Seems well enough for most people. What don't ou think is defined well
enough?

>
>
>> You are the one that seems to be unable to process the definition because it is too abstract for you.
> I am perfectly capable of processing well-defined defintions.
>
> You just haven't produced one of those yet.

Because I have, you just refuse to access it.

>
>> Neither can you, so even though I have, you don't understand it.
> Nor do you understand understanding.
>
>> WHy not?
> For all the reasons already given which you fail to understand.
>
> A bit like how Olcott refuses to understand why his decider is broken.

But WE point out the errors based on the definitions in the system.

You do not, as you refuse to be in the system. You try to point out
errors while being outside the system, just like Olcott.

>
>> Since many people understand it, the problem isn't in the definition,
>> but in the receiver.
> Many people understood the Earth is flat also. Argumentum ad populum is a fallacy.

Wrong logic.

>
>> It has become clear that YOU are unable to handle the concept.
> Not possible. You haven't even attempted to communicate the concept.
>
> You keep pointing me to books. Like theists keep pointing me to books in order to understand their God.

So, maybe you should look at the books. You might learn something.

>
>>> I don't see how that's possible. "define" is undefined.
>> No, it is well defined, maybe you just don't understand it.
> You don't even know what well-defined means...
>
>> You seem to be missing the part about definitions being a community
>> activity,
> They are a community activity. One programmer + One computer a community makes.

Lonely I guess.

>
> When you TEACH your language to your computer (you write the compiler) then your computer understands your language!
> You seem to be unable to do that. Always blaming the computer for your inability to define your own language.
>

And you get a system of 1, not the system that others talk of.

Just more of your anti-social understanding.

>>> Yeah, function self-application is tricky to idiots who restrict comprehension.
>> I understand that, but you are misapplying it.
> No, I am not. It's stratified and it type-checks.

Then why does your limit function try to do Real Number math on infinity?

It obviously didn't "type check" the input.

>
>>> Any non-idiot knows that's a runtime error.
>> Nope.
> Hahahah. Yeah. And Olcott solved the halting problem.
>
>>> Your foundations. The fact that your brain can't handle the null-pointer exceptions is just another clusterfuck in the system.
>> Why do youy say I can't handle the null-pointer exception?
> Because you have presented no algebraic effect handler.
>
>>> What Job? You haven't specified a task.
>> Whatever you were trying to accomplish with it.
> That task is not possible given your hardware bug.

I understand my hardware bugs, and seem to be adiquately working around
them. I don;t think you uderstand yours.

>
>>> Bullshit. I am pointing out that your description/definition of the Real Numbers fails to capture how things are!
>> Then you are doing a very bad job at it, since you start by not knowing
>> what you are talking about.
> I know precisely what I am talking about. Parametric polymorphism.

Then why don't you actually use it. "Lim(x->inf)f(x)" needs a different
rule then "Lim(x->y)f(x), y a Real Number".

Your polymorphism dispatch seems broken.

>
> It's too bad you completely avoided the topic when I asked you to parametrize "lim(x -> y) z"

Look again, you never asked that question, as your processor dropped the z.

lim(x -> y) z means:

if an L exist such that for ALL e > 0, |L - z| < e there exists an X
such that:

1) if y finite: | x - y | < | X - y |,
or
2) if y is infinity, x > X,
or
3) if y is -infinity, x < X,

So the shape of the domain that needs to within the error is
paramaterised on the type of y.

>
>>> Look! It's still there. You've been confusing self-affirmation for self-defeat all along. You dumb Mathematician!
>> Word Games. Shows you ran out of logic.
> Shame! Looks like your logic-system can't handle reflection.
>

On Friday, 19 August 2022 at 19:59:31 UTC+8, richar...@gmail.com wrote:
> On 8/19/22 2:16 AM, wij wrote:
> > On Thursday, 18 August 2022 at 12:36:19 UTC+8, wij wrote:
> >> On Thursday, 18 August 2022 at 09:31:16 UTC+8, richar...@gmail.com wrote:
> >>> 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.
> >> It depends. The root reason is your notion of infinity is fuzzy.
> >> That leads to your infinity sometimes means unbounded large, sometimes a normal
> >> value, sometimes not, sometimes just an indication word. When really arguing,
> >> infinity does not exist. Like infinitesimal, at the beginning it is unbounded
> >> small non-zero VALUE, In the end, it is zero VALUE. When arguing, infinitesimal
> >> does not exist. What is the difference with POO Halt !!!
> >>
> >> How can we arguing this for a definite answer? No, we cannot.
> >> If we want a definite answer or a number containing infinity/infinitesimal in
> >> its expression definite, make the meaning unique first as I did.
> >
> > Let's say "-∞ <------ 0 ------> +∞" denotes the x-axis in textbooks.
> > The message the schematic conveys should be: -∞<0<+∞ (note that '∞' is now
> > 'legitimately' decorated with signs).
> > Assume "-∞<0<+∞" is accepted [note1]. And, to express the notion of 'close' or
> > 'arbitrarily approaching to', we assume there are special rules that '∞'
> > interacts with normal numbers. To express "x+1 closer than x to ∞":
> But that step presumes that we are ALLOWED to mix ∞ with other numbers
> in the mathematical operators.

Yes, allowed, like x<∞ (may not be precise) occurs in many textbooks.
You just showed your bias, not what your math. shows.

> Yes, we can define the "order" relationships on this "extended value" of
> infinity, but that doesn't mean that it has fully been broght into the
> family.

What kind of 'fully'. Did I show fully?

> In fact, we find that we can't even say that ∞ == ∞, as that is enough
> to cause problems. Infinities are ordered with the Reals, but only very
> limited with each other. -∞ < ∞ but that is about all we can say.

You admit 'ordered' now.
So you simply just don't know how algebra works based on definition, definition man.
Do you really understand how 'assumption' works in math.?

> >
> > 1) ∞-(x+1) < ∞-x
> > 2) ∞-x-1 < ∞-x
> > 3) ∞-1 < ∞
> > 4) ∞ < ∞+1
> >
> > Whatever the idea of '∞' is, as long as its meaning is unique, the notion of '∞'
> > must lead to the notion "∞ < ∞+1" being true (Is not x+1 closer than x by one
> > to infinity?).
> And you stated part of the problem, in a real sense ∞ is NOT "unique",
> there are MANY infinities that all have some common properties, but they
> are not all the same.
> >
> > [note1] You can refute "-∞<0<∞", but you would need to establish a 'theory' to
> > explain this assumed intermediate step. You need to explain lots more than simple
> > denial, explain enough to replace what is in the textbooks. Note that,
> > whatever you do is likely your own, not what the textbooks mean to say.
> > [note2] What does "arbitrarily close to" mean? Note the word 'distance' in
> > https://en.wikipedia.org/wiki/Limit_(mathematics)
> > Where is the approaching value of x from? Is it not the density property
> > guarantee you "For any two different numbers there exists a DIFFERENT
> > number", so you can find another x in between? What is wrong if we use
> > rational number x to approach like ancient Pythagorean did? Is the
> > result still valid by the limit argument? Why in the final, 'approach'
> > becomes 'equal', the x EQUALS to c, f(c)=L, the density property is
> > abandoned?
> > The limit theory has no concept of equality in its premise, it cannot
> > logically deduce the equality conclusion, except limit defines equality..
> > [note3] All decimals are real number, including infinitely long decimals. This is
> > general recognition, not any standard. Finite number of symbols cannot
> > define all of them. Actually, lots (technically almost all) of real numbers
> > are each not definable. What the Dedekind-cut like theory claims is false
> > (I don't need to look inside deep, circular arguments/definition should exist).

Why you repeatedly skip the other nonetheless important questions? If I won't get answer,
I will keep asking to remind you of your understanding is really like POO Halt.
Those questions are clear and simple. You just showed you evade in various ways
possible and keep talking about your definition/first principle/math (bias, really).

On 8/19/22 10:00 AM, Skep Dick wrote:
> On Friday, 19 August 2022 at 15:46:22 UTC+2, richar...@gmail.com wrote:
>> ONE justifcation for 0 being bounded, is that it is bounded by -1 and 1.
> Being bounded by unbounded numbers... Gotcha!

Nope, bounded by bounded numbers.

>
>> Another is just First Principles, that 0 is the start of the numbers and
>> is defined to be finite, but you don't seem to like First Principles and
>> special cases.
> No shit! We are doing Mathematics! We are trying to solve general classes of problems!
>
> Nobody gives a shit if Olcott axiomatically solves a special case of the Halting problem.

Except he doesn'tl, because he asserts things "axiomatically" that
aren't true, and thus can't be axioms in the system.

> Why should I give a shit if you axiomatically solve a special case of bounding?
>
>> A second "First Principle" is the definition that if N if finite, then
>> N+1 is.
> That "IF" sure seems to be doing a lot of work for you...

Yep, Induction is a powerful method when working with the Natural Numbers.

>
>> This makes ALL the Natural Numbers finite by induction, but NOT "w" as
>> we never get to "w-1" (whatever that mean) to show that is finite.
> Hah! Look how quickly you forgot the "IF"...

Because I don't need it, because I was stating a conclusion, not a rule.

It is a fact that we never got to w-1, just as we never got to w, in the
induciton.

Induction gets us to all the finite bounded and unbounded natural
numbers, it doesn't get us to infinity, just to unbounded numbers.

This makes them different.

On 8/19/22 10:04 AM, Skep Dick wrote:
> On Friday, 19 August 2022 at 15:48:25 UTC+2, richar...@gmail.com wrote:
>> So your system is just broken, as without First Principles, you have
>> nothing to prove with.
> Then I'll use ∞-th principles instead!
>
>> What can you actually prove without starting with a definition or axioms
>> (or an assumed proof)?
> I can prove the theorem I start with.
>

Nope, because if you ASSUME that, then you don't actually prove it.

On 8/19/22 10:14 AM, wij wrote:
> On Friday, 19 August 2022 at 19:59:31 UTC+8, richar...@gmail.com wrote:
>> On 8/19/22 2:16 AM, wij wrote:
>>> On Thursday, 18 August 2022 at 12:36:19 UTC+8, wij wrote:
>>>> On Thursday, 18 August 2022 at 09:31:16 UTC+8, richar...@gmail.com wrote:
>>>>> 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.
>>>> It depends. The root reason is your notion of infinity is fuzzy.
>>>> That leads to your infinity sometimes means unbounded large, sometimes a normal
>>>> value, sometimes not, sometimes just an indication word. When really arguing,
>>>> infinity does not exist. Like infinitesimal, at the beginning it is unbounded
>>>> small non-zero VALUE, In the end, it is zero VALUE. When arguing, infinitesimal
>>>> does not exist. What is the difference with POO Halt !!!
>>>>
>>>> How can we arguing this for a definite answer? No, we cannot.
>>>> If we want a definite answer or a number containing infinity/infinitesimal in
>>>> its expression definite, make the meaning unique first as I did.
>>>
>>> Let's say "-∞ <------ 0 ------> +∞" denotes the x-axis in textbooks.
>>> The message the schematic conveys should be: -∞<0<+∞ (note that '∞' is now
>>> 'legitimately' decorated with signs).
>>> Assume "-∞<0<+∞" is accepted [note1]. And, to express the notion of 'close' or
>>> 'arbitrarily approaching to', we assume there are special rules that '∞'
>>> interacts with normal numbers. To express "x+1 closer than x to ∞":
>> But that step presumes that we are ALLOWED to mix ∞ with other numbers
>> in the mathematical operators.
>
> Yes, allowed, like x<∞ (may not be precise) occurs in many textbooks.
> You just showed your bias, not what your math. shows.
>
>> Yes, we can define the "order" relationships on this "extended value" of
>> infinity, but that doesn't mean that it has fully been broght into the
>> family.
>
> What kind of 'fully'. Did I show fully?

Things like ∞ doesn't have all the properties of an actual Real Number.

For instance, given:

x < ∞

we can NOT say that:

x-1 < ∞-1

because ∞ doesn't have a defined Mathematics in the Real Number System.

∞ can't be involved in "Algerbra" in the Real Number system.

>
>> In fact, we find that we can't even say that ∞ == ∞, as that is enough
>> to cause problems. Infinities are ordered with the Reals, but only very
>> limited with each other. -∞ < ∞ but that is about all we can say.
>
> You admit 'ordered' now.
> So you simply just don't know how algebra works based on definition, definition man.
> Do you really understand how 'assumption' works in math.?

Do you?

Algebra deals with values in the system. ∞ is NOT a "value" in the Real
Number system. it is a special concept that has some limited properties
in Relationship to the system.

The key point you miss is that when you go into one of the Extended Real
Number Systems where we add some form of infinity, we need to change the
rules of the Algebra that we are using (You do understand that it is "An
Algerbra", not "The Algebra", unless you make it "The Algerbra of the
Real Number System")

I learned that in High School.

>
>>>
>>> 1) ∞-(x+1) < ∞-x
>>> 2) ∞-x-1 < ∞-x
>>> 3) ∞-1 < ∞
>>> 4) ∞ < ∞+1
>>>
>>> Whatever the idea of '∞' is, as long as its meaning is unique, the notion of '∞'
>>> must lead to the notion "∞ < ∞+1" being true (Is not x+1 closer than x by one
>>> to infinity?).
>> And you stated part of the problem, in a real sense ∞ is NOT "unique",
>> there are MANY infinities that all have some common properties, but they
>> are not all the same.
>>>
>>> [note1] You can refute "-∞<0<∞", but you would need to establish a 'theory' to
>>> explain this assumed intermediate step. You need to explain lots more than simple
>>> denial, explain enough to replace what is in the textbooks. Note that,
>>> whatever you do is likely your own, not what the textbooks mean to say.
>>> [note2] What does "arbitrarily close to" mean? Note the word 'distance' in
>>> https://en.wikipedia.org/wiki/Limit_(mathematics)
>>> Where is the approaching value of x from? Is it not the density property
>>> guarantee you "For any two different numbers there exists a DIFFERENT
>>> number", so you can find another x in between? What is wrong if we use
>>> rational number x to approach like ancient Pythagorean did? Is the
>>> result still valid by the limit argument? Why in the final, 'approach'
>>> becomes 'equal', the x EQUALS to c, f(c)=L, the density property is
>>> abandoned?
>>> The limit theory has no concept of equality in its premise, it cannot
>>> logically deduce the equality conclusion, except limit defines equality.
>>> [note3] All decimals are real number, including infinitely long decimals. This is
>>> general recognition, not any standard. Finite number of symbols cannot
>>> define all of them. Actually, lots (technically almost all) of real numbers
>>> are each not definable. What the Dedekind-cut like theory claims is false
>>> (I don't need to look inside deep, circular arguments/definition should exist).
>
> Why you repeatedly skip the other nonetheless important questions? If I won't get answer,
> I will keep asking to remind you of your understanding is really like POO Halt.
> Those questions are clear and simple. You just showed you evade in various ways
> possible and keep talking about your definition/first principle/math (bias, really).

On Friday, 19 August 2022 at 16:09:04 UTC+2, richar...@gmail.com wrote:
> So, how do you USE that without knowing the ACTUAL type of a?
> You talk about use being king.
The ACTUAL (concrete) type of a is immaterial!
Because that's how polymorphism works - it supports ANY type.

What kind of an idiot are you. When you use a general tool for a particular job don't you know what the particualr task at hand is?!?

> So, what is my error, remember, to use the definitions of how things
> actually work in the real number system.
That's problem Number 1.

The definitions of how things work in the real numbers system don't correspont to how things work in the number system!

> If you "generic" number system breaks when trying to deal with actual
> real numbers, that's your systems fault.
No it isn't. It just means your "Real Numbers" are malformed.

> > That's a lie. Well-defined has a precise meaning. It may be defined, but it's definitely not well-defined.
> Seems well enough for most people. What don't ou think is defined well
> enough?
Everything that lacks well-definition! Given the standard definition of "well-definition"

https://en.wikipedia.org/wiki/Well-defined_expression

> > You just haven't produced one of those yet.
> Because I have, you just refuse to access it.
You aren't even shamed of lying anymore.

> > A bit like how Olcott refuses to understand why his decider is broken.
> But WE point out the errors based on the definitions in the system.
But any first order definition has infinite models. Surely you know this !?!

> You do not, as you refuse to be in the system. You try to point out
> errors while being outside the system, just like Olcott.
Surely you understand that "error" and "correctness" cannot be defined INSIDE the system?!?

You can only do that OUTSIDE the system. With Operational Semantics.

> > Many people understood the Earth is flat also. Argumentum ad populum is a fallacy.
> Wrong logic.
You can't define "wrong" in the system.

> > You keep pointing me to books. Like theists keep pointing me to books in order to understand their God.
> So, maybe you should look at the books. You might learn something.
I doubt it. I am an empiricist. Language (Mathematics included) is only for (self) expression.

> > They are a community activity. One programmer + One computer a community makes.
> Lonely I guess.
But coherent.

> >
> And you get a system of 1, not the system that others talk of.
You don't understand transpilers work?
You don't understand how dependent types/wire protocols/marshalling/unmarshalling works?

Shame! Almost like you've never integrated different systems over a network

> Just more of your anti-social understanding.
I mean. I can know how distributed consensus protocols work so...

This 2-week long discussion with you takes 2-5 minutes with anyone who understands computer science ;)

Talk about effective communication...

> > No, I am not. It's stratified and it type-checks.
> Then why does your limit function try to do Real Number math on infinity?
MY real function? It's YOUR real function.

lim(x -> y) z

let me rewrite it for you in a syntax that's clearer...

lim(approaches(x,∞)) z

approaches() is a polymorphic binary function. First argument is a Real number, Second argument is ∞. It sure sounds like the product type of R x ∞

> It obviously didn't "type check" the input.
Ah! So are you finally admitting that there is a type error in the expresion "lim(x: R -> ∞) z"

> > That task is not possible given your hardware bug.
> I understand my hardware bugs, and seem to be adiquately working around
> them. I don;t think you uderstand yours.
Shame! You are a piss-poor systems designer if you are fixing hardware bugs in software.

Great hackjob. Piss poor design.

> > I know precisely what I am talking about. Parametric polymorphism.
> Then why don't you actually use it. "Lim(x->inf)f(x)" needs a different
> rule then "Lim(x->y)f(x), y a Real Number".
No it doesn't. Generic functions are generic.

> Your polymorphism dispatch seems broken.
Let the compiler handle that. Implementation details!!!

> lim(x -> y) z means:
>
> if an L exist such that for ALL e > 0, |L - z| < e there exists an X
> such that:
>
> 1) if y finite: | x - y | < | X - y |,
> or
> 2) if y is infinity, x > X,
> or
> 3) if y is -infinity, x < X,
Details details details

The proof of the generic type handles ALL possible implementations.

> So the shape of the domain that needs to within the error is
> paramaterised on the type of y.
And the type of y is ???

You know... given that y is bound to infinity.

On 8/19/22 10:33 AM, Skep Dick wrote:
> On Friday, 19 August 2022 at 16:09:04 UTC+2, richar...@gmail.com wrote:
>> So, how do you USE that without knowing the ACTUAL type of a?
>> You talk about use being king.
> The ACTUAL (concrete) type of a is immaterial!
> Because that's how polymorphism works - it supports ANY type.
>
> What kind of an idiot are you. When you use a general tool for a particular job don't you know what the particualr task at hand is?!?
>

So, How do ou use your general limit property then? I thought that was
your problem?

>
>> So, what is my error, remember, to use the definitions of how things
>> actually work in the real number system.
> That's problem Number 1.
>
> The definitions of how things work in the real numbers system don't correspont to how things work in the number system!

Which "Number system" are you talking about, obviously not the Reals?

>
>> If you "generic" number system breaks when trying to deal with actual
>> real numbers, that's your systems fault.
> No it isn't. It just means your "Real Numbers" are malformed.

They work perfectly fine.

>
>>> That's a lie. Well-defined has a precise meaning. It may be defined, but it's definitely not well-defined.
>> Seems well enough for most people. What don't ou think is defined well
>> enough?
> Everything that lacks well-definition! Given the standard definition of "well-definition"
>
> https://en.wikipedia.org/wiki/Well-defined_expression

But that applies to only EXPRESSIONS, and FUNCTIONS, not systems

So your definition of well-definition doesn't seem to be well-defined,
since you want it to apply to a bigger domain that it was defined for.

>
>>> You just haven't produced one of those yet.
>> Because I have, you just refuse to access it.
> You aren't even shamed of lying anymore.

I state I use the appropriate standard definitions unless specified. If
you won't look them up, thats YOUR fault.

>
>>> A bit like how Olcott refuses to understand why his decider is broken.
>> But WE point out the errors based on the definitions in the system.
> But any first order definition has infinite models. Surely you know this !?!

So?

>
>> You do not, as you refuse to be in the system. You try to point out
>> errors while being outside the system, just like Olcott.
> Surely you understand that "error" and "correctness" cannot be defined INSIDE the system?!?
>
> You can only do that OUTSIDE the system. With Operational Semantics.

No, error and correctness OF A STATEMENT IN THE SYSTEM, are determined
in the system.

If you want to look at things like completeness or inconsistency, you
may need to go outside the system.
>
>>> Many people understood the Earth is flat also. Argumentum ad populum is a fallacy.
>> Wrong logic.
> You can't define "wrong" in the system.

In that context, sure I can, a logic not accepted as valid in the system.

>
>>> You keep pointing me to books. Like theists keep pointing me to books in order to understand their God.
>> So, maybe you should look at the books. You might learn something.
> I doubt it. I am an empiricist. Language (Mathematics included) is only for (self) expression.

Too bad, if you think you can't learn, that tends to be self-fullfilling.

>
>>> They are a community activity. One programmer + One computer a community makes.
>> Lonely I guess.
> But coherent.

You think.

>
>>>
>> And you get a system of 1, not the system that others talk of.
> You don't understand transpilers work?
> You don't understand how dependent types/wire protocols/marshalling/unmarshalling works?
>
> Shame! Almost like you've never integrated different systems over a network

I think you don't even know the type of "system" that we have been
talking about.

>
>> Just more of your anti-social understanding.
> I mean. I can know how distributed consensus protocols work so...
>
> This 2-week long discussion with you takes 2-5 minutes with anyone who understands computer science ;)
>
> Talk about effective communication...

Your the one that is making it hard.

>
>>> No, I am not. It's stratified and it type-checks.
>> Then why does your limit function try to do Real Number math on infinity?
> MY real function? It's YOUR real function.
>
> lim(x -> y) z
>
> let me rewrite it for you in a syntax that's clearer...
>
> lim(approaches(x,∞)) z
>
> approaches() is a polymorphic binary function. First argument is a Real number, Second argument is ∞. It sure sounds like the product type of R x ∞
>
>> It obviously didn't "type check" the input.
> Ah! So are you finally admitting that there is a type error in the expresion "lim(x: R -> ∞) z"

No, there isn't. The expression is, in your words, polymorphic on the
type of the limit. It has one definition if the limit to approach is a
finite number, and a slightly different definition if the limit to
approach is an infinite.

Not sure if you "approaches" is actually a function, unless it is just
returning its inputs to the limit operator(the variable that the limit
is bound over and the "value" that variable is limiting over).

The x -> y is just a notation way of passing the two part parameter to
the limit operator that also takes the expression as a "parameter"

>
>>> That task is not possible given your hardware bug.
>> I understand my hardware bugs, and seem to be adiquately working around
>> them. I don;t think you uderstand yours.
> Shame! You are a piss-poor systems designer if you are fixing hardware bugs in software.
>
> Great hackjob. Piss poor design.

Seems to work well for me.

>
>>> I know precisely what I am talking about. Parametric polymorphism.
>> Then why don't you actually use it. "Lim(x->inf)f(x)" needs a different
>> rule then "Lim(x->y)f(x), y a Real Number".
> No it doesn't. Generic functions are generic.

And then can't actualy do anything.

How does YOUR limit operator handle the two different types of limit ranges?

>
>> Your polymorphism dispatch seems broken.
> Let the compiler handle that. Implementation details!!!
>
>> lim(x -> y) z means:
>>
>> if an L exist such that for ALL e > 0, |L - z| < e there exists an X
>> such that:
>>
>> 1) if y finite: | x - y | < | X - y |,
>> or
>> 2) if y is infinity, x > X,
>> or
>> 3) if y is -infinity, x < X,
> Details details details
>
> The proof of the generic type handles ALL possible implementations.

Does it? without forking definitions?

>
>> So the shape of the domain that needs to within the error is
>> paramaterised on the type of y.
> And the type of y is ???
>
> You know... given that y is bound to infinity.
>

When Y is bound to infinity, then its type is an infinite.

Limits to infinites use different rules than limits to finites.

Details matter.

On Friday, 19 August 2022 at 22:30:10 UTC+8, richar...@gmail.com wrote:
> On 8/19/22 10:14 AM, wij wrote:
> > On Friday, 19 August 2022 at 19:59:31 UTC+8, richar...@gmail.com wrote:
> >> On 8/19/22 2:16 AM, wij wrote:
> >>> On Thursday, 18 August 2022 at 12:36:19 UTC+8, wij wrote:
> >>>> On Thursday, 18 August 2022 at 09:31:16 UTC+8, richar...@gmail.com wrote:
> >>>>> 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.
> >>>> It depends. The root reason is your notion of infinity is fuzzy.
> >>>> That leads to your infinity sometimes means unbounded large, sometimes a normal
> >>>> value, sometimes not, sometimes just an indication word. When really arguing,
> >>>> infinity does not exist. Like infinitesimal, at the beginning it is unbounded
> >>>> small non-zero VALUE, In the end, it is zero VALUE. When arguing, infinitesimal
> >>>> does not exist. What is the difference with POO Halt !!!
> >>>>
> >>>> How can we arguing this for a definite answer? No, we cannot.
> >>>> If we want a definite answer or a number containing infinity/infinitesimal in
> >>>> its expression definite, make the meaning unique first as I did.
> >>>
> >>> Let's say "-∞ <------ 0 ------> +∞" denotes the x-axis in textbooks.
> >>> The message the schematic conveys should be: -∞<0<+∞ (note that '∞' is now
> >>> 'legitimately' decorated with signs).
> >>> Assume "-∞<0<+∞" is accepted [note1]. And, to express the notion of 'close' or
> >>> 'arbitrarily approaching to', we assume there are special rules that '∞'
> >>> interacts with normal numbers. To express "x+1 closer than x to ∞":
> >> But that step presumes that we are ALLOWED to mix ∞ with other numbers
> >> in the mathematical operators.
> >
> > Yes, allowed, like x<∞ (may not be precise) occurs in many textbooks.
> > You just showed your bias, not what your math. shows.
> >
> >> Yes, we can define the "order" relationships on this "extended value" of
> >> infinity, but that doesn't mean that it has fully been broght into the
> >> family.
> >
> > What kind of 'fully'. Did I show fully?
> Things like ∞ doesn't have all the properties of an actual Real Number.

You rejected it for precise meaning.
Not ∞ itself lacks property. You make ∞ to denote fuzzy notion.

> For instance, given:
>
> x < ∞
>
> we can NOT say that:
>
> x-1 < ∞-1
>
> because ∞ doesn't have a defined Mathematics in the Real Number System.
>
> ∞ can't be involved in "Algerbra" in the Real Number system.

Did you see the proposed definition? It works perfectly in algebra.

> >
> >> In fact, we find that we can't even say that ∞ == ∞, as that is enough
> >> to cause problems. Infinities are ordered with the Reals, but only very
> >> limited with each other. -∞ < ∞ but that is about all we can say.
> >
> > You admit 'ordered' now.
> > So you simply just don't know how algebra works based on definition, definition man.
> > Do you really understand how 'assumption' works in math.?
> Do you?
>
> Algebra deals with values in the system. ∞ is NOT a "value" in the Real
> Number system. it is a special concept that has some limited properties
> in Relationship to the system.
Really surprised, you don't really understand algebra.
Let me correct you. Algebra deals with SYMBOL. Your notion of 'value' is basically
used by engineers. Engineers can and often assume value too small is zero, math.(logic) cannot.

> The key point you miss is that when you go into one of the Extended Real
> Number Systems where we add some form of infinity, we need to change the
> rules of the Algebra that we are using (You do understand that it is "An
> Algerbra", not "The Algebra", unless you make it "The Algerbra of the
> Real Number System")

That is how definition works and why it is for, to define an ARBITRARY system.
People in academic environment HAS TO do their math. in the way like Extended
Real Number. They cannot deny existing math. or there are big, very real troubles.

> I learned that in High School.

I learned it by myself.

> >
> >>>
> >>> 1) ∞-(x+1) < ∞-x
> >>> 2) ∞-x-1 < ∞-x
> >>> 3) ∞-1 < ∞
> >>> 4) ∞ < ∞+1
> >>>
> >>> Whatever the idea of '∞' is, as long as its meaning is unique, the notion of '∞'
> >>> must lead to the notion "∞ < ∞+1" being true (Is not x+1 closer than x by one
> >>> to infinity?).
> >> And you stated part of the problem, in a real sense ∞ is NOT "unique",
> >> there are MANY infinities that all have some common properties, but they
> >> are not all the same.
> >>>
> >>> [note1] You can refute "-∞<0<∞", but you would need to establish a 'theory' to
> >>> explain this assumed intermediate step. You need to explain lots more than simple
> >>> denial, explain enough to replace what is in the textbooks. Note that,
> >>> whatever you do is likely your own, not what the textbooks mean to say.
> >>> [note2] What does "arbitrarily close to" mean? Note the word 'distance' in
> >>> https://en.wikipedia.org/wiki/Limit_(mathematics)
> >>> Where is the approaching value of x from? Is it not the density property
> >>> guarantee you "For any two different numbers there exists a DIFFERENT
> >>> number", so you can find another x in between? What is wrong if we use
> >>> rational number x to approach like ancient Pythagorean did? Is the
> >>> result still valid by the limit argument? Why in the final, 'approach'
> >>> becomes 'equal', the x EQUALS to c, f(c)=L, the density property is
> >>> abandoned?
> >>> The limit theory has no concept of equality in its premise, it cannot
> >>> logically deduce the equality conclusion, except limit defines equality.
> >>> [note3] All decimals are real number, including infinitely long decimals. This is
> >>> general recognition, not any standard. Finite number of symbols cannot
> >>> define all of them. Actually, lots (technically almost all) of real numbers
> >>> are each not definable. What the Dedekind-cut like theory claims is false
> >>> (I don't need to look inside deep, circular arguments/definition should exist).
> >
> > Why you repeatedly skip the other nonetheless important questions? If I won't get answer,
> > I will keep asking to remind you of your understanding is really like POO Halt.
> > Those questions are clear and simple. You just showed you evade in various ways
> > possible and keep talking about your definition/first principle/math (bias, really).
> When the later question are based on the error done first, they can't
> actually be answered.

On 8/19/22 11:17 AM, wij wrote:
> On Friday, 19 August 2022 at 22:30:10 UTC+8, richar...@gmail.com wrote:
>> On 8/19/22 10:14 AM, wij wrote:
>>> On Friday, 19 August 2022 at 19:59:31 UTC+8, richar...@gmail.com wrote:
>>>> On 8/19/22 2:16 AM, wij wrote:
>>>>> On Thursday, 18 August 2022 at 12:36:19 UTC+8, wij wrote:
>>>>>> On Thursday, 18 August 2022 at 09:31:16 UTC+8, richar...@gmail.com wrote:
>>>>>>> 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.
>>>>>> It depends. The root reason is your notion of infinity is fuzzy.
>>>>>> That leads to your infinity sometimes means unbounded large, sometimes a normal
>>>>>> value, sometimes not, sometimes just an indication word. When really arguing,
>>>>>> infinity does not exist. Like infinitesimal, at the beginning it is unbounded
>>>>>> small non-zero VALUE, In the end, it is zero VALUE. When arguing, infinitesimal
>>>>>> does not exist. What is the difference with POO Halt !!!
>>>>>>
>>>>>> How can we arguing this for a definite answer? No, we cannot.
>>>>>> If we want a definite answer or a number containing infinity/infinitesimal in
>>>>>> its expression definite, make the meaning unique first as I did.
>>>>>
>>>>> Let's say "-∞ <------ 0 ------> +∞" denotes the x-axis in textbooks.
>>>>> The message the schematic conveys should be: -∞<0<+∞ (note that '∞' is now
>>>>> 'legitimately' decorated with signs).
>>>>> Assume "-∞<0<+∞" is accepted [note1]. And, to express the notion of 'close' or
>>>>> 'arbitrarily approaching to', we assume there are special rules that '∞'
>>>>> interacts with normal numbers. To express "x+1 closer than x to ∞":
>>>> But that step presumes that we are ALLOWED to mix ∞ with other numbers
>>>> in the mathematical operators.
>>>
>>> Yes, allowed, like x<∞ (may not be precise) occurs in many textbooks.
>>> You just showed your bias, not what your math. shows.
>>>
>>>> Yes, we can define the "order" relationships on this "extended value" of
>>>> infinity, but that doesn't mean that it has fully been broght into the
>>>> family.
>>>
>>> What kind of 'fully'. Did I show fully?
>> Things like ∞ doesn't have all the properties of an actual Real Number.
>
> You rejected it for precise meaning.
> Not ∞ itself lacks property. You make ∞ to denote fuzzy notion.
>
>> For instance, given:
>>
>> x < ∞
>>
>> we can NOT say that:
>>
>> x-1 < ∞-1
>>
>> because ∞ doesn't have a defined Mathematics in the Real Number System.
>>
>> ∞ can't be involved in "Algerbra" in the Real Number system.
>
> Did you see the proposed definition? It works perfectly in algebra.

>
>>>
>>>> In fact, we find that we can't even say that ∞ == ∞, as that is enough
>>>> to cause problems. Infinities are ordered with the Reals, but only very
>>>> limited with each other. -∞ < ∞ but that is about all we can say.
>>>
>>> You admit 'ordered' now.
>>> So you simply just don't know how algebra works based on definition, definition man.
>>> Do you really understand how 'assumption' works in math.?
>> Do you?
>>
>> Algebra deals with values in the system. ∞ is NOT a "value" in the Real
>> Number system. it is a special concept that has some limited properties
>> in Relationship to the system.
>
> Really surprised, you don't really understand algebra.
> Let me correct you. Algebra deals with SYMBOL. Your notion of 'value' is basically
> used by engineers. Engineers can and often assume value too small is zero, math.(logic) cannot.
>
>> The key point you miss is that when you go into one of the Extended Real
>> Number Systems where we add some form of infinity, we need to change the
>> rules of the Algebra that we are using (You do understand that it is "An
>> Algerbra", not "The Algebra", unless you make it "The Algerbra of the
>> Real Number System")
>
> That is how definition works and why it is for, to define an ARBITRARY system.
> People in academic environment HAS TO do their math. in the way like Extended
> Real Number. They cannot deny existing math. or there are big, very real troubles.
>
>> I learned that in High School.
>
> I learned it by myself.

Maybe not so well.

>
>>>
>>>>>
>>>>> 1) ∞-(x+1) < ∞-x
>>>>> 2) ∞-x-1 < ∞-x
>>>>> 3) ∞-1 < ∞
>>>>> 4) ∞ < ∞+1
>>>>>
>>>>> Whatever the idea of '∞' is, as long as its meaning is unique, the notion of '∞'
>>>>> must lead to the notion "∞ < ∞+1" being true (Is not x+1 closer than x by one
>>>>> to infinity?).
>>>> And you stated part of the problem, in a real sense ∞ is NOT "unique",
>>>> there are MANY infinities that all have some common properties, but they
>>>> are not all the same.
>>>>>
>>>>> [note1] You can refute "-∞<0<∞", but you would need to establish a 'theory' to
>>>>> explain this assumed intermediate step. You need to explain lots more than simple
>>>>> denial, explain enough to replace what is in the textbooks. Note that,
>>>>> whatever you do is likely your own, not what the textbooks mean to say.
>>>>> [note2] What does "arbitrarily close to" mean? Note the word 'distance' in
>>>>> https://en.wikipedia.org/wiki/Limit_(mathematics)
>>>>> Where is the approaching value of x from? Is it not the density property
>>>>> guarantee you "For any two different numbers there exists a DIFFERENT
>>>>> number", so you can find another x in between? What is wrong if we use
>>>>> rational number x to approach like ancient Pythagorean did? Is the
>>>>> result still valid by the limit argument? Why in the final, 'approach'
>>>>> becomes 'equal', the x EQUALS to c, f(c)=L, the density property is
>>>>> abandoned?
>>>>> The limit theory has no concept of equality in its premise, it cannot
>>>>> logically deduce the equality conclusion, except limit defines equality.
>>>>> [note3] All decimals are real number, including infinitely long decimals. This is
>>>>> general recognition, not any standard. Finite number of symbols cannot
>>>>> define all of them. Actually, lots (technically almost all) of real numbers
>>>>> are each not definable. What the Dedekind-cut like theory claims is false
>>>>> (I don't need to look inside deep, circular arguments/definition should exist).
>>>
>>> Why you repeatedly skip the other nonetheless important questions? If I won't get answer,
>>> I will keep asking to remind you of your understanding is really like POO Halt.
>>> Those questions are clear and simple. You just showed you evade in various ways
>>> possible and keep talking about your definition/first principle/math (bias, really).
>> When the later question are based on the error done first, they can't
>> actually be answered.
>
> Understood now.
>
>> Fundamentally, your issue is that The Real Number System is definied one
>> way, but you want something different. You seem to want an Extended Real
>> Number System, but won't accept that some of the rules you like from the
>> Reals don't work anymore.
>
> You have revealed your understanding, I won't ask. But, please don't talk too much
> and so affirmatively what you don't really understand.

On Friday, 19 August 2022 at 17:55:29 UTC+2, richar...@gmail.com wrote:
> What don't I understand?
>
> The Real Number System HAS a Definition, and rules (like its Algebra)
>
> If you claim to be working in the Real Number system, you need to follow
> its rules. (This includes that infinity is NOT a Real Number).
>
> If you want to Extend the Real Number System, fine, Just say you are
> working in an Extended Real Number System. But when you do, remember
> that you have to figure out what Algebra works in that system. You can't
> just assume the Alrgebra of the Real Number System works.
>
> This is why clarity is important.
>
> I don't have a clear enough understanding of your extentions (not sure
> if you do either, but that is less important), but it IS an extension,
> and so needs to be treated as one.
>
> You might want to study some of the Extended Real Number Systems and the
> Trans-Finite number systems to see if anyone else has worked in the same
> direction you are thinking.

Jesus fucking christ. This moron will drive himself off a cliff just because the rules said so.

No! You absolutely DON'T have to follow any rules! The intensional properties of your system don't matter as long as your extensional properties are satisfied.

This is how interfaces work! Adhere to the contract externally - do whatever the hell you want internally. It's just irrelevant implementation detail!

On 8/19/22 12:44 PM, Skep Dick wrote:
> On Friday, 19 August 2022 at 17:55:29 UTC+2, richar...@gmail.com wrote:
>> What don't I understand?
>>
>> The Real Number System HAS a Definition, and rules (like its Algebra)
>>
>> If you claim to be working in the Real Number system, you need to follow
>> its rules. (This includes that infinity is NOT a Real Number).
>>
>> If you want to Extend the Real Number System, fine, Just say you are
>> working in an Extended Real Number System. But when you do, remember
>> that you have to figure out what Algebra works in that system. You can't
>> just assume the Alrgebra of the Real Number System works.
>>
>> This is why clarity is important.
>>
>> I don't have a clear enough understanding of your extentions (not sure
>> if you do either, but that is less important), but it IS an extension,
>> and so needs to be treated as one.
>>
>> You might want to study some of the Extended Real Number Systems and the
>> Trans-Finite number systems to see if anyone else has worked in the same
>> direction you are thinking.
>
> Jesus fucking christ. This moron will drive himself off a cliff just because the rules said so.
>
> No! You absolutely DON'T have to follow any rules! The intensional properties of your system don't matter as long as your extensional properties are satisfied.
>
> This is how interfaces work! Adhere to the contract externally - do whatever the hell you want internally. It's just irrelevant implementation detail!

But if your system is open, so others can work in it, you can't do that.

I will note, that the whole discission with you has just proved your a
Hypocrite, as you keep on trying to get people to accept YOUR
definitions, but you refuse to even try to adapt to their.

Without Rules, there is no logic. (At its core, logic is applying Rules
to knowledge to get more Knowledge).

For example, you were asking for a "Well-Defined Definition", but that
ment you were IMPOSING A RULE about how to define things.

The problem with you method is that you have no idea if what you are
doing makes any sense, as if you don't know and follow the constraints
in the methods you are using, you have no promise they will give
reasonable results.

It seems you have already jumped of the cliff.

Note, The Rules don't tell you what you HAVE to do, they tell you what
you need to do to get the results promised by the system.

You are ALWAYS free to alter the system to look at something new, but
HONESTY requires that you make it clear how you are devating, and that
you take into account you deviations (a good reason to make them clear).

I am not saying you HAVE to use the Real Numbers, but if you claim to be
doing so, you need to follow the Rules of them or you are just a LIAR.

You can use what every methods you want, as long as you are honest about
them and what you are doing, and give others the right to do the same.

On Thursday, August 18, 2022 at 11:02:48 PM UTC-7, Skep Dick wrote:
> On Friday, 19 August 2022 at 06:17:23 UTC+2, dklei...@gmail.com wrote:
> > Still meaningless. There is no "last" digit.
> There is a last digit to ANYONE who can look past completed infinities!
>
> That is you (according to you).
>
You don't like this but I say whether infinite objects are real or not is a
matter for philosophers - not you or me.

When I contemplate an infinite I see no last member. There are always
more members. There are just as many more members as there are
original members.

On Friday, August 19, 2022 at 4:46:10 AM UTC-7, Skep Dick wrote:
>
> Show me the number 0! Not the symbol representing it.
>
Most mathematicians would, given this context, take 0 to mean
the empty set. Then the successor of any number N would be N
plus the set whose only member is N.

On 19/08/2022 14:57, Ben Bacarisse wrote:
>>> [...] I'm not sure you gave enough
>>> information for anyone who does /not/ know the rules to work it out.
>> I didn't. But you introduced Hackenbush!
> Only because that's the interpretation I needed to square what you
> wrote. If they are a not Hackenbush numbers, then why do you say that
> R(L) > 0? The problem, for the reader, is why "Walker" numbers have
> RL(LR) = 1/3 but R(L) > 0.

One interpretation of strings of R's and L's is as Hackenbush
numbers; there are others. But 0 is represented by the empty string,
so [in the scenario I posed] no other string is 0. In particular, it
has the sign given by its first letter; in "R(L)", you start by going
right and can never get back to the origin.

> Given the name you used ("balanced binary"), the most natural
> interpretation is of a binary representation who infinite strings are
> the limits of infinite digit sums which works for RL(LR) but not,
> apparently, for R(L).

I don't think "balanced binary" implies anything about limits,
an invention of real analysis. We're just playing games with strings.
You can interpret that either as having fun, or as a genuine game [eg
a simplified version of Hackenbush].

> And if they /are/ Hackenbush numbers, then saying that RL(LR) = 1/3 is
> really throwing a spanner in the works (for the learner), because (but
> here I am out of my confort zone) RL(LR) might be "identified" with 1/3
> because there is no closer Hackenbush number to 1/3.

I don't expect learners to glean "everything" about surreals
from a page or so of text; but they've been pointed at Conway's and
Knuth's books. From that PoV, everything you need to know about
"RL(LR)" can be gleaned from the game

{ 0, RLL, RLLRL, RLLRLRL, ... | R, RL, RLLR, RLLRLR, ... }.

Perhaps worth noting that there are two possible interpretations of
[eg] "RL(LR)"; the one that mathematicians latch onto, that "(...)"
means an infinite sequence of what's in the parentheses, and the one
I prefer, that it's a sequence of a length chosen by the player who
moves into that sequence. In my version, you therefore never have
to deal with [actual] infinities or with limits.

> That's analogous
> to a Dedekind cut, but it isn't one because we are not constructing the
> reals. Writing RL(LR) = 1/3 in this context is, in my option, not
> helpful.

For real x, it /is/ a Dedekind cut, in which we throw into the
left set all finite binary rationals < x and into the right set all
those > x. At least, it starts that way, but those not close to x are
dominated, and there is a theorem that dominated options may be omitted.
After that, we're into a philosophical jungle, which I don't propose to
enter here.
>> [I'm not proposing to write a textbook here; anyone with a
>> mathematical background and who is not already familiar with the
>> surreals can look it up. I just wanted to give some pointers.]
> Then I think you need to give a keyword to help people look it up. Did
> you try searching for "balanced binary"?

Ah. No, sorry. I assumed, obviously incorrectly, that it was
a sufficiently-well understood concept, at least in CS. It's discussed
in Knuth's "The Art of Computer Programming" somewhere [but I don't
have that on my shelves so can't give an actual reference]. As penance,
I point you and others at:

https://www.youtube.com/watch?v=mPn2AdMH7UQ

which is Knuth talking about the surreals, with follow-up links to more
of Knuth and also to a talk by Conway.

[FTAOD, I don't propose to continue this discussion much longer,
else it will turn from a small number of modest articles into a huge
thesis amounting to a detailed exposition of Conway's work.]

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