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

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.
 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.
 

But we do, because decidability requires finite steps to get the answer, but Trurh can come from an infinite number of steps.
Since you admit that mathematics is based on Truth Perserving operations, how do you prove the statement that no natural number g satisfies the progperty G derived from the system you are in by Godel's proof?
Godel shows it must be true by how G is constructed in a meta-system that knows the full finite list of facts about the system, and also shows that it can not be proven in that system.
Your problem is you just don't understand what you are talking about, and just assume things that are not correct.