Re: Peano Axioms anchored in First Grade Arithmetic on ASCII Digit String pairs

On 10/26/2024 3:02 AM, Mikko wrote:

On 2024-10-25 13:31:16 +0000, olcott said:
 

On 10/25/2024 3:01 AM, Mikko wrote:

On 2024-10-24 14:28:35 +0000, olcott said:
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On 10/24/2024 8:51 AM, Mikko wrote:

On 2024-10-23 13:15:00 +0000, olcott said:
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On 10/23/2024 2:28 AM, Mikko wrote:

On 2024-10-22 14:02:01 +0000, olcott said:
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On 10/22/2024 2:13 AM, Mikko wrote:

On 2024-10-21 13:52:28 +0000, olcott said:
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On 10/21/2024 3:41 AM, Mikko wrote:

On 2024-10-20 15:32:45 +0000, olcott said:
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The actual barest essence for formal systems and computations
is finite string transformation rules applied to finite strings.

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Before you can start from that you need a formal theory that
can be interpreted as a theory of finite strings.

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Not at all. The only theory needed are the operations
that can be performed on finite strings:
concatenation, substring, relational operator ...

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You may try with an informal foundation but you need to make sure
that it is sufficicently well defined and that is easier with a
formal theory.
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The minimal complete theory that I can think of computes
the sum of pairs of ASCII digit strings.

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That is easily extended to Peano arithmetic.
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As a bottom layer you need some sort of logic. There must be unambifuous
rules about syntax and inference.
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I already wrote this in C a long time ago.
It simply computes the sum the same way
that a first grader would compute the sum.
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I have no idea how the first grade arithmetic
algorithm could be extended to PA.

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Basically you define that the successor of X is X + 1. The only
primitive function of Peano arithmetic is the successor. Addition
and multiplication are recursively defined from the successor
function. Equality is often included in the underlying logic but
can be defined recursively from the successor function and the
order relation is defined similarly.
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Anyway, the details are not important, only that it can be done.
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First grade arithmetic can define a successor function
by merely applying first grade arithmetic to the pair
of ASCII digits strings of [0-1]+ and "1".
https://en.wikipedia.org/wiki/Peano_axioms
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The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
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When we boil this down to its first-grade arithmetic foundation
this would seem to mean that there are some cases where the
sum of a pair of ASCII digit strings cannot be computed.

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No, it does not. Incompleteness theorem does not apply to artihmetic
that only has addition but not multiplication.
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The incompleteness theorem is about theories that have quantifiers.
A specific arithmetic expression (i.e, with no variables of any kind)
always has a well defined value.
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So lets goes the next step and add multiplication to the algorithm:
(just like first grade arithmetic we perform multiplication
on arbitrary length ASCII digit strings just like someone would
do with pencil and paper).
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Incompleteness cannot be defined. until we add variables and
quantification: There exists an X such that X * 11 = 132.
Every detail of every step until we get G is unprovable in F.

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Incompleteness is easier to define if you also add the power operator
to the arithmetic. Otherwise the expressions of provability and
incompleteness are more complicated. They become much simpler if
instead of arithmetic the fundamental theory is a theory of finite
strings. As you already observed, arithmetic is easy to do with
finite strings. The opposite is possible but much more complicated.

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The power operator can be built from repeated operations of
the multiply operator.

 It is possible but to say that x is the z'th power of y is overly
complicated with a first order formula using just addition and
multiplication.
 

Just imagine c functions that have enough memory to compute
sums and products of ASCII strings of digits using the same
method that people do.
char* sum(char* x, char *y);
char* product(char* x, char *y);
char* difference(char* x, char *y); // never returns < 0
char* power_of(char* base, char * power);
I am not going to bother with the rest of the steps.
We simply multiply  base times itself power times.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer