From Newsgroup: comp.theory

Procedural description is better than average formal axiomatic system:
1. The abstract terms/symbols in average formal axiomatic system are often not
used properly because they are so called 'abstract math. concept', not much
one can deduce anything objectively from it, and so often are practically
worse than in plain natural language.
2. Average (current) formal system cannot describe procedural fact.
Peano Theorem could be described in C language as:
NaturnalNumber N[]={0}; // assuming N is a dynamic array
NaturnalNumber next_natural_number(NaturnalNumber n) {
return mak_next(N,m); // make a next natural number of m from N
}
bool is_natural_num(n) {
if(is_format_correct(n)==false) { return false; }
if(contain(N,n)) return true;
N m= get_max(N);
for(;;) {
m=mak_next(N,m);
insert(N,m); // insert m to N
if(n==m) return true;
}
return false; // unreachable
}
So, in the procedural view above, natural number is essentially just a specification of the format of the symbol called number, and we can use next_natural_number(..). And, 'symbol' in procedural language implicitly suggests that it is finitely long (so, we cannot call is_natural_num(..)). While with the natural number defined by Peano Axiom, formally proving ∞∉ℕ is
difficult, if not impossible... all the way down to infinity and other math.
In math., 'natural number' is an abstract concept. In reasoning, errors can easily arise when abstract arguments in natural language involve two sets of natural numbers (intentionally or unintentionally). For example, if a "other natural number" set N is defined as "even numbers" {0, 2, 4, 8,...}, then the numbers 2 and 8 in N are not even numbers as defined by N. Reasoning of comparison can lead to confusion about the meaning of "even numbers." These errors are less likely to occur if expressed in a program (or with improved notation, such as N<0,+1>). The "natural numbers" commonly used in math. induction are often from a different natural numbers system.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On Mon, 2025-09-15 at 08:23 +0800, wij wrote:

Procedural description is better than average formal axiomatic system:

 1. The abstract terms/symbols in average formal axiomatic system are often not
    used properly because they are so called 'abstract math. concept', not much
    one can deduce anything objectively from it, and so often are practically
    worse than in plain natural language.

 2. Average (current) formal system cannot describe procedural fact.

Peano Theorem could be described in C language as:

  NaturnalNumber N[]={0};   // assuming N is a dynamic array

  NaturnalNumber next_natural_number(NaturnalNumber n) {
   return mak_next(N,m);    // make a next natural number of m from N
  }

  bool is_natural_num(n) {
   if(is_format_correct(n)==false) { return false; }
   if(contain(N,n)) return true;
   N m= get_max(N);
   for(;;) {
     m=mak_next(N,m);   
     insert(N,m);        // insert m to N
     if(n==m) return true;
   }
   return false; // unreachable
  }

So, in the procedural view above, natural number is essentially just a specification of the format of the symbol called number, and we can use next_natural_number(..). And, 'symbol' in procedural language implicitly suggests that it is finitely long (so, we cannot call is_natural_num(..)). While with the natural number defined by Peano Axiom, formally proving ∞∉ℕ is
difficult, if not impossible... all the way down to infinity and other math.

In math., 'natural number' is an abstract concept. In reasoning, errors can easily arise when abstract arguments in natural language involve two sets of natural numbers (intentionally or unintentionally). For example, if a "other  
natural number" set N is defined as "even numbers" {0, 2, 4, 8,...}, then the
numbers 2 and 8 in N are not even numbers as defined by N. Reasoning of comparison can lead to confusion about the meaning of "even numbers." These errors are less likely to occur if expressed in a program (or with improved notation, such as N<0,+1>). The "natural numbers" commonly used in math. induction are often from a different natural numbers system.

From about, we can easily define real number as such:
EN::= Same as the natural numbers defined above, but the number (symbol) can
be infinitely long.
ℝ::= {x| x=p/q or -p/q, p,q∈EN, q≠0}
Note that real number is but an extension of rational number that includes infinitely long symbols (no way to explicitly appear).
There are lots of different theories (either inconsistent or not closed) about what real number is. A simple,intuitive one is handy to have and we actually  already knew it naturally. Why bother The Emperor's New Clothes?
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 9/14/2025 5:23 PM, wij wrote:

Procedural description is better than average formal axiomatic system:

[...]

Zero aside for a moment, but a tree might be a fairly decent start? A
two ary tree:


0
/ \
/ \
/ \
1 2
/ \ / \
/ \ / \
3 4 5 6
..........................

on and on... Three ary would be:


0
/|\
/ | \
/ | \
1 2 3
........................

on and on...


The tree levels and their numbers already have their inherent properties
in them? the root node is zero.





--- Synchronet 3.21a-Linux NewsLink 1.2