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

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.

 Can such a system include the mathematics of the natural numbers?
 If so, your claim is false, as that is enough to create that undeciability.
 

It seems to me that the inferences steps that could
otherwise create undecidability cannot exist in the
system that I propose.
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.
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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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)

 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.
 

It can say anything that can be said. It is the complete set
of all general knowledge that can be expressed in language.

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.

"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.
 This ultimate shows your fundamental misunderstanding of what you are talking about, especially your inability to handle abstractions, and things that can create "infinities".

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer