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

On 5/6/25 1:46 PM, olcott wrote:

On 5/6/2025 6:20 AM, Richard Damon wrote:

On 5/6/25 12:27 AM, olcott wrote:

On 5/5/2025 10:31 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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The mathematics of natural numbers (as I have already explained)
begins with basic facts about natural numbers and only applies
truth preserving operations to these basic facts.
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When we begin with truth and only apply truth preserving
operations then WE NECESSARILY MUST END UP WITH TRUTH.
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When we ALWAYS end up with TRUTH then we NEVER end up with UNDECIDABILITY.
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Its not that hard, iff you pay enough attention.
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But we do, because decidability requires finite steps to get the answer, but Trurh can come from an infinite number of steps.
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 True(x) accepts any x that can be derived by applying truth
preserving operations to the set of Basic Facts that comprise
the entire body of general knowledge that can be expressed
in language and rejects everything else.

But it can't.

 True(x) is really Known_to_be_True(x).

Which means you are admitting to lying.
Things can be true that are not known to be true.

True(~x) is really Known_to_be_False(x).
 It cannot have any undecidability its is membership
algorithm for the set of all general knowledge that
can be expressed using language.
 

Your problem is you are sticking yourself into a non-existant logic system, but are too stupid to understand that is what you have done.
Your definition is self-contradictory.
You can not enumerate all the basic facts of knowledge when you restrict what logic ooperation can be done as you claim.
The problem is that you end up with an infinite number of basic facts, because you lost the operations that let us finitely define the Natural Nubmers.