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

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.
 

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.
When we begin with truth and only apply truth preserving
operations then WE NECESSARILY MUST END UP WITH TRUTH.
When we ALWAYS end up with TRUTH then we NEVER end up with UNDECIDABILITY.
Its not that hard, iff you pay enough attention.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer