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.

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.