Program that solve infinity problems

[wyn...@gmail.com]

I had just written a short article, which is also a programming plan
about how to handle infinity. The basic idea (except to make known) is that
NUMBER IS OPERATION.

class ANum { // number is a tree node
int m_optype;
int m_arg1;
int m_arg2;
};

This article already defy what most of us are familiar with, which made me
a bit uneasy.
What I know I did not know well is 1. Is the plan mathematically correct?
2. what else the math (number system) is out there I missed 3. typo and
wording errors. 4. what kind of question can we ask, particularly typed
by keyboard, and to what extend the terminal output we can accept.
We do not want the whole tree dumped. 5. skill of search tree pruning
required. 6. what is the result can we expect, from the article?.

Understanding the essence of the article may take time, especially those
believe 0.999...=1 or lim(n->∞){1/n}=0, or not yet cared about.

---------- The article is short but I think is still a bit too long to put here
So I just put snipped paragraphs. The full article is file Infinity.txt
can be found in the main directory:
https://sourceforge.net/projects/cscall/files/libwy-0.59.3.tgz/download

...[Snipped]
+------------+
| Defining ∞ |
+------------+
From the initial understanding of [Prop1], there are many infinities. Some trial
analysis exhibits that there might exists a base infinity. On the other hand,
facing an equation like e=(1+1/∞)^∞ (or similar equations), we must ask:
Is that 'finite' value e unique? I believe we would like it unique. Therefore, ∞
must be unique as well. So, adding ∞ to the algebraic system is doable.

Definition of ∞:
1. ∀n∈ℕ, n<∞
2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞

The benefit of handling ∞ this way is 1.safe guaranteed 2.self-contained, not
much is needed to explain. What is left should probably be interpretation
issues.

---------------------------------------------------
Test1:
Let f(x)=∑(n=0,x){n} = x*(x+1)/2 // sum of natural numbers
g(x)=∑(n=0,x){2n}= x*(x+1) // sum of even numbers
h(x)=∑(n=0,x){2*n+1} = (x+1)^2 // sum of odd numbers

g(x)+h(x)=∑(n=0,x){(2*n)+(2*n+1)}=
=∑(n=0,x){4*n+1} =x*(x+1)+(x+1)^2=(x+1)*(2*x+1) // sum of odd and even
// numbers
// Using ∞
f(∞)=∞*(∞+1)/2 // sum of natural numbers
g(∞)=∞*(∞+1) // sum of even numbers
h(∞)=(∞+1)^2 // sum of odd numbers, h(∞)-g(∞)=∞+1
g(∞)+h(∞)= // sum of odd and even
=∑(n=0,∞){(2*∞)+(2*∞+1)}= ∞*(∞+1)+(∞+1)^2=(∞+1)*(2*∞+1)
=∑(n=0,∞){4*∞+1} =(∞+1)*(2*∞+1) // 1+5+9+...+(4*∞+1)

∴ The sum of natural numbers is not equal to the sum of odd numbers plus the
sum of even numbers (there are ambiguity issues)

...[Snipped]
+------------------------+
| Modifying Peano axioms |
+------------------------+
The sub-goal of the text so far is teaching computers a way to help us handle
math problems. For now, the main issues are ∞ and infinite progressions.
Therefore, Peano axioms need be modified:

1. 0∈ℕ, INT_MAX∈ℕ // INT_MAX is the maximal number in ℕ. This number is
// arbitrarily large but can be reached by counting
// and can be viewed as a pseudo-infinity. Axiom 3a has
// explained this point. Programs with INT_MAX set to
// greater than S^63(0)=63 should work fine for many
// infinity problems.
// [Note] INT_MAX can in theory be a finite value that
// can never be reached in reality, e.g. Axiom 3a
// can generate (allow) the successor of any given
// 'maximum', is still not INT_MAX.
∞∈𝔸

2. ∀x∈ℕ∪𝔸, x=P(S(x)) // S(x) represents the succesor function of x, P(x) the
// predecessor function of x. P and S are inverse function
// of each other. Uniqueness is implied by the definition
// of function.
3a. ∀x∈ℕ
if(x==INT_MAX) S(x)=∞-INT_MAX // Range jump. ∞-INT_MAX∈𝔸
else S(x)∈ℕ
3b. ∀x∈𝔸
if(x==∞-INT_MAX) P(x)=INT_MAX // Range jump. ∞-INT_MAX is the smallest
// (non-fractional) number in 𝔸.
else P(x)∈𝔸 // S(∞)∈𝔸 holds as well, just not yet used

4. Others are roughly the same as the average definition of Peano axioms.
No need to be completely formal for now.

[wyn...@gmail.com]

wyn...@gmail.com於 2019年11月12日星期二 UTC+8上午11時41分25秒寫道：

What does these to do with C++? I personally believe, the original (spirit?)
C++ is very mathematical, like 'class' (realization of the idea of set and
concept) and operator overloads, all felt like a computer version of abstract
algebra, besides the author of C++ is a mathematician. There is a subtle link
between C++ and mathematics. Who would know the outcome and effects to C++?

I decided to finish the headaching math stuff to start thinking about
program implements. But before that, I need opinions of the definition of
real number to finish it, not the officials, anything you feel right,
comfortable is welcome. And, could someone recommend some computer algebra
programs for me to try out things?