Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable

On 5/5/25 10:26 PM, olcott wrote:

On 5/5/2025 8:11 PM, Richard Damon wrote:

On 5/5/25 11:31 AM, olcott wrote:

On 5/5/2025 6:04 AM, Richard Damon wrote:

On 5/4/25 10:23 PM, olcott wrote:

When we define formal systems as a finite list of basic facts and allow semantic logical entailment as the only rule of inference we have systems that can express any truth that can be expressed in language.
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Also with such systems Undecidability is impossible. The only incompleteness are things that are unknown or unknowable.

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Can such a system include the mathematics of the natural numbers?
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If so, your claim is false, as that is enough to create that undeciability.
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It seems to me that the inferences steps that could
otherwise create undecidability cannot exist in the
system that I propose.'

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Only because it seems to create a trivially small system.
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 When I told you that the system comprises the entire
set of all general knowledge that can be expressed in
language many many times, you must have a mental defect
to to think that this system is very small.

No, the problem is that you are imagining doing something you can not do. The problem is your logic is self-inconsistant, as you then limit it by only what it provable by semantic logical entailment (per your understanding of it) but much of knowledge is about systems that are not so restricted, so that knowledge can not be finitely represented in such a system.
How does your system have mathematics? Do you list the ENTIRE addition table, all Aleph0 x Aleph0 entries of it?

 

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For example: "This sentence is not true" cannot be
derived by applying semantic logical entailment to
basic facts. It is rejected as semantically unsound
on this basis.

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So?
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Try to show any complete concrete example using
a system of basic facts and applying semantic logical
entailment where undecidability can be derived.

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That isn't what I said. I said that you system, to be decidable, couldn't include the mathematics of the Natural Numbers.
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 It does includes the mathematics of natural numbers
expressed as basic facts and truth preserving
operations applied to these basic facts.

Then it isn't finite, as the operations of Set Theory that produce the Natural Numbers are not expressable by your logic system, thus you set of basic facts needs to be infinite in size.

 When you start with truth and only apply truth
preserving operations you necessarily only end
up with truth. This means that you NEVER end
up with any undecidability.

Sure you do. What non-truth perserving operation did Godel or Turing use?

 The Liar Paradox: "this sentence is not true"
is rejected as untrue because it cannot be derived
by applying only truth preserving operations to
basic facts.
 

So?
Computation Theory is based on Truth Perserving operations, and there is no Turing Machine that can compute the Halting Property for all possible programs (given via a representation) does not exist, and thus the problem is uncomputable, and it is possible for any Formal System that supports the properties of the Natural Numbers to produce a Primiative Recursive Relationship G, such that the statement that "there does not exist a natural number g, such that g satisfies G", which will be true, but can not be proven in that system.
Your problem is you are just too ignorant of what you talk about, and assume (stupidly) that anything you don't understand can't be true.
Sorry, you are just proving your stupidity and ignornace, and made yourself into a pathological liar by not caring about the truth.