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

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.'

Only because it seems to create a trivially small system.

 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.

So?

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

That isn't what I said. I said that you system, to be decidable, couldn't include the mathematics of the Natural Numbers.
Show me how you generate those, and do not also allow for the Godel proof of incompleteness or undeciability.

 

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The language of such a formal system is an extended form of the Montague Grammar of natural language semantics. I came up with this mostly in the last two years. I have been working on it for 22 years.
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The Montague Grammar Rudolf Carnap Meaning postulates are organized in a knowledge ontology inheritance hierarchy. https:// en.wikipedia.org/ wiki/ Ontology_(information_science)

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And the problem is that either your claim is wrong, or your logic system is just shown to be too small to be useful for many of the things we want to be able to do because it can't support the mathematics of Natural Numbers.
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 It can say anything that can be said. It is the complete set
of all general knowledge that can be expressed in language.

So, it include the statement "This statement is not true"?

 

You don't seem to understand that all the properties you don't like about Logic Systems are all conditioned on the ability for the system to have a certain level of power in their ability to do logic.

 Semantic logical entailment is rich enough to say anything
that can be said.

Really, so how do you derive mathematics from it?

 

"Tpy" systems that have been limited below that level will not experiance the problems, but also are too weak to do the problems we typically want to do with logic.
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This ultimate shows your fundamental misunderstanding of what you are talking about, especially your inability to handle abstractions, and things that can create "infinities".