On Friday, 2 September 2022 at 03:09:57 UTC+8, wij wrote:
> On Friday, 2 September 2022 at 00:35:30 UTC+8, Ben Bacarisse wrote:
> > wij <wyni...@gmail.com> writes:
> >
> > > On Thursday, 1 September 2022 at 19:27:51 UTC+8, Ben Bacarisse wrote:
> > >> wij <wyni...@gmail.com> writes:
> > >>
> > >> > On Thursday, 1 September 2022 at 03:05:16 UTC+8, Keith Thompson wrote:
> > >> <cut>
> > >> >> No, lim(x->c) f(x)=L does not imply f(c)=L.
> > >> >
> > >> > Exactly what I mean.
> > >> > This is my point: Limit cannot yield an equal conclusion.
> > >> It can if someone wants it to. The reals are closed under taking the
> > >> least upper bounds of sets of rationals. That's how they are defined..
> > >> It's the whole point of the set. There are other ways to express the
> > >> same definition -- Cauchy sequences converge in the reals, every Dedkind
> > >> cut is a real, and so on -- but it's the basic definition of the set we
> > >> use to do calculus.
> > >
> > > The real is not closed (lots of irrationals cannot be expressed with finite
> > > symbols). Maybe you are talking sub-set of ℝ.
> > No. I think you need to learn a bit more. "Closed", on it's own is
> > meaningless in this context. And I meant what I said. R is closed with
> > respect to taking least upper bounds of sets or rationals. It is
> > defined so that limits do what I say they do and not what you want them
> > to do.
> > > I don't use Dedkind-cut, Cauchy sequence, theories. 1. They cannot construct
> > > all real numbers as claimed. 2.These theories, as definition of real number,
> > > should presume no real numbers exist (they seem to do circular
> > > argument).
> > They are not circular. I can't type of a whole book on the reals for
> > you. Read one. The definitions are not circular.
> > >> (There's a mass of very interesting history here with the motivation
> > >> that numbers like pi and e should be actual numbers. They are not
> > >> algebraic, so the set we want needs to be closed under some operation
> > >> other than taking roots of polynomials.)
> > >
> > > Number is operation/algorithm/expression (my number view). So e can be a
> > > number, pi should be 'unreachable' by its definition. Like my definition of ∞,
> > > assigning a symbol for it should work. e can be defined (of course, different
> > > definition here). I don't use (high-level) set theory.
> > > √2 is essentially an expression, so a number. One can see it as a
> > > 'name'.
> > Your mission, should you choose to accept it, is to work out the details
> > and try to persuade someone that it's worth looking at. (And I read
> > below you claim to have done the first half of that.)
> > >> > 'lim' should always stick to its expression, 'lim' cannot be
> > >> > removed.
> > >> In the reals, that just gives us lots of ways to write numbers. The
> > >> rules of real arithmetic can not distinguish between
> > >>
> > >> lim_{n->oo} 1/n!
> > >>
> > >> and
> > >>
> > >> lim_{n->oo) (1+1/n)^n
> > >>
> > >> and so we say they are equal. Do you say they are equal? (Please
> > >> answer this question. You are not very diligent when it comes to
> > >> answering clarifying questions.)
> > >>
> > >> Similarly, the rules of real arithmetic don't let us distinguish between
> > >> lim_{n->oo} Sum_{k=1,n} 9/10^k and 1. That's why we write 0.999... = 1.
> > >>
> > >> No one objects to "leaving the lim there" except on grounds of
> > >> convenience. After all, people have told you repeatedly that
> > >> 0.999... is just another way to write a limit. We could ditch the
> > >> ... and always write the limit, but there would still be no way to tell
> > >> the difference between that long form number "lim_{n->oo} Sum_{k=1,n}
> > >> 9/10^k" and "1". People will just write "1" because it's shorter.
> > >
> > > I guess a typo in the reply above.
> > Yes. Mike spotted it as well. But you did not answer the question.
> > You want to keep the limit and never say that it's actually equal to
> > something (at least I think that's what you meant). So if I write
> >
> > lim+{n->oo} sum_{k=1,n} 1/k! and lim_{n->oo) (1+1/n)^n
> >
> > can you tell is they are the same? Are they equal? Can we not just
> > give them a name and write e for either of them instead? Please answer.
> I just say something previously missed. Your wording is like always very 'political'.
>
> ExprA: Σ(n->∞) 1/n!
> ExprB: lim(n->∞) (1+1/n)^n
>
> Yes, we can give them a name for sure. But, from the point of definition,
> ExprA and ExprB are different.
>
> e^k= ∑(n->∞) k/n! should be written as "e^k≒ ∑(n->∞) k/n!" to be more precise,
> because this equation as I know come from a long deduction and
> it has irrational 'remainder' omitted. ExprB is simple: (1+1/∞)^∞ (my notation)
> So, simply put, ExprA and ExprB are not equal (no true equation can link them).
>
> As said, as infinity is defined, the rest should be hopefully deterministic.
> > > From the idea of information theory
> > > A= lim(x->∞) 1-1/n = 0.999...
> > > B= lim(x->∞) 1-1/2^n = 0.999...
> > > C= lim(x->∞) 1-1/10^n= 0.999...
> > > D= lim(x->∞) 1-2/10^n= 0.999...
> > >
> > > Before one determine A=B=C=D=0.999..., All such numbers(expressions) are
> > > distinguishable, SO WE CAN DISCUSS them.
> > Provided we have a meaning for the symbols. You have not given one
> > here. Standard textbooks give a meaning from which we conclude that
> > A=B=C=D=1 so you would be well-advised to use different symbols..
> >
> > I can't stress this enough. Unless you are a crank trying to say that
> > the world is wrong about the reals, you a defining something new, so you
> > should use new notation.
> > > (We don't invent rules to make them
> > > equal for no GOOD and SOUND reason).
> > > Yes, all these could come down to the definition of 'equal'. I use the definition:
> > > A=B ::= The occurrence of A can be replaced by B, vice versa.
> > That's reasonable (except for a few details that are not really
> > important here). 0.999... can be used in place of 1 since they denote
> > the same real number.
> >
> > In Wij-numbers 0.999___ =/= 1. Note the new notation so no one think
> > you are talking about the reals and limit of partial sums.
> > > To make my idea more clear, starting from infinity should be simpler:
> > > 123...
> > > 566...
> > > ....34
> > None of these have a conventional meaning and you have not explained
> > your number system so I can't comment. What happens if I add them,
> > divide them, subtract one from another etc.?
> > > These are all infinities (infinitely many, probably more than the first look).
> > > They are infinity (number) because they each can be valid expression and
> > > TM can be designed to represent them.... I found the simple and good way to
> > > handle infinity is making it unique. Therefore,
> > >
> > > '∞' ::=
> > > 1. ∀n∈ℕ, n<∞
> > > 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
> > >
> > > From this definition, others should be clear and hopefully deterministic
> > > (to save long discussion of 0.999... issues discussed).
> > That's a start. But I still don't know the rules for this new number
> > system. I can lookup other standard systems with infinite numbers, but
> > you won't say if you mean any of those. I expect not, since you want
> > this to be your very own invention. What's more, they are usually
> > extension to the reals, so 0.999... = 1 in them as well.
> >
> > Maybe you want to extend the reals so can have 0.999... and 0.999___ as
> > well?
> > >> You need to say, explicitly, that you are not talking about the real
> > >> numbers. And you need to say what numbers you /are/ talking about. And
> > >> yo need to lay out the rules or arithmetic that enable someone to use
> > >> your set whilst as the same time distinguishing between lim_{n->oo}
> > >> Sum_{k=1,n} 9/10^k" and "1". It's a lot of work. Are you up for it?
> > >>
> > >> --
> > >> Ben.
> > >
> > > I already made my idea clear in
> > > https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
> > Great. I'll assume you have done all the work. Is there any reason I
> > should read it?
> > > and mentioned many times. It was originally written for me to follow, so
> > > something may not be clear to readers. But, I think my idea should be clear
> > > enough so far. Adding these explanation to that file ruins its
> > > purpose.
> > But my posts have been about why you are wrong about the reals. I've
> > not said you are wrong about your own numbers. I think it's /likely/
> > that you are wrong about them, but I'd have to have a reason to read
> > about them first. And I don't see a reason. Why are they interesting?
> > > Real numbers are not all constructable.
> > True.
> > > I don't think real numbers can be better
> > > addressed without infinity (think about the vast number of irrationals not
> > > addressable). Some real number 'explode', some 'implode'.
> > > What I am talking about is really noting but from the definition of infinity.
> > > All should be followed, not some new theory, maybe just new
> > > recognition.
> > >
> > > As you can see, I cannot name what I thought differently. Because we
> > > are dealing the same thing (basically, the measurement of the real
> > > world, the distance between two points in the real space). What is in
> > > text-book is inconsistent, useful when the authority says you must say
> > > the same to gain ??? or avoid punishment (If I were taking an exam. I
> > > would answer the same as you would. Why not? somebody pay for it and
> > > I want the prize), or useful when nothing involving infinity.
> > Is there someone who can read over your text before posting? I find
> > your writing very hard to follow. I don't know what these remarks mean.
> >
> > --
> > Ben.

Correction: e^k ≒ ∑(n=0,∞) k^n/n!