Re: Gödel's actual proof and deriving all of the digits of the actual Gödel numbers

On 10/31/24 8:15 AM, olcott wrote:

On 10/31/2024 4:45 AM, Mikko wrote:

On 2024-10-30 12:13:43 +0000, olcott said:
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On 10/30/2024 4:57 AM, Mikko wrote:

On 2024-10-29 13:25:34 +0000, olcott said:
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On 10/29/2024 2:38 AM, Mikko wrote:

On 2024-10-28 14:04:24 +0000, olcott said:
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On 10/28/2024 3:35 AM, Mikko wrote:

On 2024-10-27 14:29:22 +0000, olcott said:
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On 10/27/2024 4:02 AM, Mikko wrote:

On 2024-10-26 13:57:58 +0000, olcott said:
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On 10/25/2024 11:07 PM, Richard Damon wrote:

On 10/25/24 7:06 PM, olcott wrote:

On 10/25/2024 5:17 PM, Richard Damon wrote:

On 10/25/24 5:52 PM, olcott wrote:

On 10/25/2024 10:52 AM, Richard Damon wrote:

On 10/25/24 9:31 AM, olcott wrote:

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. Will a terabyte be enough to store
the Gödel numbers?
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Likely depends on how big of a system you are making F.
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I am proposing actually doing Gödel's actual proof and
deriving all of the digits of the actual Gödel numbers.
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Then try it and see.
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You do understand that the first step is to fully enumerate all the axioms of the system, and any proofs used to generate the needed properties of the mathematics that he uses.
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Gödel seems to propose that his numbers are
actual integers, are you saying otherwise?
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Not at all, just that they may be very large numbers.

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Are they less than one GB each? I want to see the c
code that computes them. I want to know how many bytes
of ASCII digits strings they are.

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The memory needs are easier to estimate if you use a different
numbering system:
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1. Encode all formulas with the 94 visible ASCII characters.
2. Encode the 94 ASCII characters with two decimal digits.
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Just encode them as actual ASCII and you have a 94-ary number
system in half the space.
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In addition to the 94 ASCII characters you may use 6 other characters.
To encode a proof you need one character (e.g. semicolon or one of
the 6 non-ASCII characters) for separator. Some uses of this encodeing
are much simpler if the code 00 is used as a separator and a filler
that is not a part of a formula. That way you can use formulas that are
shorter than the space for them. For example, proofs are easier to handle
if every sentence of the proof is padded to the same length. Leading
zeros should be meaningless anyway.
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At the end of the page http://iki.fi/mikko.levanto/lauseke.html
I have an arithmetic expression that evaluates to a 65600 digits
number. With one leading zero the number can be split in to 21867
groups of three digits. Each group encodes one character of the
expression.
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Gödel numbers of proofs are larger, possibly much arger, than Gödel
numbers of formulas.
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Lets at least see the exact sequence of steps as applied
to ASCII digits. He says he is basing this on arithmetic
lets see this actual arithmetic even is applied to variables.
What are the 100% completely specified steps with zero details
left out where elements of the set of arithmetic operations
applied to ASCII digits can possibly say things totally outside
of the scope of arithmetic operations?

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Gödel did not use ASCII digits. The rules of his numbering can
found in textbooks of logic.

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In other words this is too difficult for you.

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"In other words" is too difficult for you. You should not use those
words.
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https://www.liarparadox.org/G%C3%B6del_Sentence(1931).pdf

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That page is not relevant to our immediate context. Note that it
uses symbols that are already defined earlier in the opus.
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I think that the assumption that it is anchored in
arithmetic is incorrect until I see the details of
it anchored in actual arithmetic.

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Depends on what you mean by "it" and "anchored".
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Exactly what additional basic operations are require besides this
to actual algorithmically perform every step of his whole proof?
char* sum(x, char* y)
char* product(x, char* y)
char* exponent(x, char* y)

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In those operations x should have a type. More specifically, the same
type as y and the function.
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 Yet arithmetic does not have types and the proof
is supposed to be about numbers.

But C programs need types.
IF you want "typeless", then you can't have the "char* stuff.
Note, the "Godel Numbers" are NOT about "finite string representation", but how to encode a finite string into a single meaningful value that can be processed.
is proof is based on establishing that mathematical operations can be the equivalent of finite string operations, and thus some of the known properties of Natural Numbers can be used.
I presume you agree that it should be possible to write a program that can check if a given proof is valid?
All that Primative Recurse Relationship that the Godel statement is about is such a theorem prover, and the theorem it is checking is that no input will be accepted by this program as a valid proof. He is showing that such a meta-relationship can be constructed in the language.

 

In addition to these operations you need comparisons:
bool equal(char* x, char* y)
bool greater(char* x, char* y)
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Formulas and in particular the undecidable formulas contain universal
and existential quantifiers. THere is no way to iimplement those in C.
But Gödel numbers can be computed and proofs checked without them.
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