I am keeping this post in both sci.logic and comp.theory
because it focuses on a similar idea to the Curry/Howard
correspondence between formal systems and computation.
Computation and all of the mathematical and logical operations
of mathematical logic can be construed as finite string
transformation rules applied to finite strings.
The semantics associated with finite string tokens can
be directly accessible to expression in the formal language.
It is basically an enriched type hierarchy called a knowledge
ontology.
A computation can be construed as the tape input to a
Turing machine and its tape output. All of the cases
where the output was construed as a set of final machine
states can be written to the tape.
I am not sure but I think that this may broaden the scope
of a computable function, or not.
The operations of formal systems can thus be directly
performed by a TM. to make things more interesting the
tape alphabet is UTM-32 of a TM equivalent RASP machine.
On 10/27/2024 6:38 AM, Richard Damon wrote:
On 10/26/24 9:22 PM, olcott wrote:
On 10/26/2024 8:04 PM, Richard Damon wrote:
On 10/26/24 5:57 PM, olcott wrote:
On 10/26/2024 10:48 AM, Richard Damon wrote:
On 10/26/24 8:59 AM, olcott wrote:
On 10/26/2024 2:52 AM, Mikko wrote:
On 2024-10-25 14:37:19 +0000, olcott said:
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On 10/25/2024 3:14 AM, Mikko wrote:
On 2024-10-24 16:07:03 +0000, olcott said:
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On 10/24/2024 9:06 AM, Mikko wrote:
On 2024-10-22 15:04:37 +0000, olcott said:
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On 10/22/2024 2:39 AM, Mikko wrote:
On 2024-10-22 02:04:14 +0000, olcott said:
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On 10/16/2024 11:37 AM, Mikko wrote:
On 2024-10-16 14:27:09 +0000, olcott said:
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The whole notion of undecidability is anchored in ignoring the fact that
some expressions of language are simply not truth bearers.
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A formal theory is undecidable if there is no Turing machine that
determines whether a formula of that theory is a theorem of that
theory or not. Whether an expression is a truth bearer is not
relevant. Either there is a valid proof of that formula or there
is not. No third possibility.
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After being continually interrupted by emergencies
interrupting other emergencies...
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If the answer to the question: Is X a formula of theory Y
cannot be determined to be yes or no then the question
itself is somehow incorrect.
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There are several possibilities.
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A theory may be intentionally incomplete. For example, group theory
leaves several important question unanswered. There are infinitely
may different groups and group axioms must be true in every group.
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Another possibility is that a theory is poorly constructed: the
author just failed to include an important postulate.
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Then there is the possibility that the purpose of the theory is
incompatible with decidability, for example arithmetic.
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An incorrect question is an expression of language that
is not a truth bearer translated into question form.
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When "X a formula of theory Y" is neither true nor false
then "X a formula of theory Y" is not a truth bearer.
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Whether AB = BA is not answered by group theory but is alwasy
true or false about specific A and B and universally true in
some groups but not all.
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See my most recent reply to Richard it sums up
my position most succinctly.
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We already know that your position is uninteresting.
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Don't want to bother to look at it (AKA uninteresting) is not at
all the same thing as the corrected foundation to computability
does not eliminate undecidability.
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No, but we already know that you don't offer anything interesting
about foundations to computability or undecidabilty.
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In the same way that ZFC eliminated RP True_Olcott(L,x)
eliminates undecidability. Not bothering to pay attention
is less than no rebuttal what-so-ever.
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No, not in the same way.
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Pathological self reference causes an issue in both cases.
This issue is resolved by disallowing it in both cases.
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Nope, because is set theory, the "self-reference"
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does exist and is problematic in its several other instances.
Abolishing it in each case DOES ELIMINATE THE FREAKING PROBLEM.
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Yes, IN SET THEORY, the "self-reference" can be banned, by the nature of the contstruction.
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That seems to be the best way.
It works for sets, but not for Computations, due to the way things are defined.
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In Computation Theory it can not, without making the system less than Turing Complete, as the structure of the Computations fundamentally allow for it,
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Sure.
So, you ADMIT that your computation system you are trying to advocate is less than Turing Complete?
I never said that.
That means that the Halting Problem isn't a problem.
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and in a way that is potentially undetectable.
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I really don't think so it only seems that way.
Of course it is.
The method of assigning meaning to the symbols can be done is a meta- system that the system doesn't know about, and thus its meaning is unknowable to the logic system.
When the only way that you learn is to memorize things from books
you make huge mistakes. It is the typical convention to assign
meaning in a way that the systems is unaware of. This is not the
only possible way. It is a ridiculously stupid way that causes
all kinds of undetectable semantic errors.
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You don't seem to understand that fact, but the fundamental nature of being able to encode your processing in the same sort of strings you process makes this a possibility.
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Not at all. Tarski made this mistake of saying this and
everyone believed him.
It does not make these things undetectable, it merely
allows failing to detect.
No, it makes things undetectable, unless you allow the system to just reject ALL statements, even if they are not actually "self-referential" to be considered "bad".
When we encode natural langugae as formal language
"This sentence is not true"
becomes:
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
A detected error.
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Dues to the nature of its relationship to Mathematics and Logic, it turns out that and logic with certain minimal requirements can get into a similar situation.
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I think that I can see deeper than the Curry/Howard Isomorphism.
Computations and formal systems are in their most basic foundational essence finite string transformation rules.
You don't undertstand what you see.
Part of the problem is that while Compuation Theory and Formal Logic System do have large parts that are just finite string transformation rules, they have other parts that are not.
You won't be able to show this. Try to define any computation
that cannot be expressed as a tape input and a tape output.
A TM takes its tape as input and has a set of final states and
or a tape output. The final states could be written to the tape.
You memorize from textbooks and I see deeper than textbooks say.
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Your only way to remove it from these fields is to remove that source of "power" in the systems, and the cost of that is just too high for most people, thus you plan just fails.
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Detection then rejection.
But since detection is impossible, you can not get to rejection.
Detection is "impossible" only because of foundational misconceptions.
Once you allow the creation of the statement, you can't reject it later and still have the claim of handling "All".
Sure you can. As long as the error is detected before final
output all is well.
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Of course, you understanding is too crude to see this issue, so it just goes over your head, and your claims just reveal your ignorance of the fields.
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Sorry, that is just the facts, that you seem to be too stupid to understand.
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In other words you can correctly explain every single detail
conclusively proving how finite string transformation rules
are totally unrelated to either computation and formal systems.
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That isn't what I said, and just proves your stupidity.
You mind is just too small to handle these discussions.
You can't even form sound rebuttals. The main rebuttal that
you have is essentially anchored in ad hominem. Your rebuttals
never have anything in the ballpark of sound reasoning.
*The form of your best rebuttals*
I memorized X from a book and you are not doing it that way
therefore you are stupid and ignorant.
The philosophy of computation begins with existing ideas and
sees what happens when these ideas are reformulated.
-- Copyright 2024 Olcott "Talent hits a target no one else can hit; Geniushits a target no one else can see." Arthur Schopenhauer
The philosophy of computation reformulates existing ideas on a new basis ---
Re: The philosophy of computation reformulates existing ideas on a new basis ---