Re: Mathematical incompleteness has always been a misconception --- Ultimate Foundation of Truth

On 3/3/2025 10:21 PM, olcott wrote:

On 3/3/2025 9:01 PM, dbush wrote:

On 3/3/2025 9:49 PM, olcott wrote:

On 3/3/2025 7:06 PM, dbush wrote:

On 3/3/2025 7:57 PM, olcott wrote:

On 3/3/2025 9:08 AM, Mikko wrote:

On 2025-03-01 19:58:21 +0000, olcott said:
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On 3/1/2025 2:45 AM, Mikko wrote:

On 2025-02-28 22:04:31 +0000, olcott said:
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On 2/28/2025 4:04 AM, Mikko wrote:

On 2025-02-26 01:33:48 +0000, olcott said:
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On 2/25/2025 5:58 PM, Richard Damon wrote:

On 2/25/25 1:40 PM, olcott wrote:

On 2/25/2025 12:15 PM, joes wrote:

Am Mon, 24 Feb 2025 20:02:49 -0600 schrieb olcott:

On 2/24/2025 6:12 PM, Richard Damon wrote:

On 2/24/25 6:11 PM, olcott wrote:

On 2/24/2025 6:27 AM, Richard Damon wrote:

On 2/23/25 11:39 PM, olcott wrote:

On 2/23/2025 8:50 PM, Richard Damon wrote:

On 2/23/25 1:08 PM, olcott wrote:

On 2/22/2025 9:56 PM, Richard Damon wrote:

On 2/22/25 1:42 PM, olcott wrote:

On 2/22/2025 3:25 AM, Mikko wrote:

On 2025-02-22 04:44:35 +0000, olcott said:

On 2/21/2025 7:05 PM, Richard Damon wrote:

On 2/21/25 6:19 PM, olcott wrote:

On 2/20/2025 2:54 AM, Mikko wrote:

On 2025-02-18 03:59:08 +0000, olcott said:

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Sure I do.
A Systems is semantically sound if every statement that can be proven
is actually true by the systems semantics,

That is very good.
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in other words, the system doesn't allow the proving of a false
statement.

That is not too bad yet ignores that some expressions might not have
any truth value.

Which has nothing to do with "soundness".

When any system assumes that every expression is true or false and is
capable of encoding expressions that are neither IT IS STUPIDLY WRONG.

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In honour of Gödel this is usually called "incomplete".
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Where "incomplete" has always been an idiom for stupid wrong.
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No, only in your faulty logic.
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Incomplete means that there are some truths that can't be proven in the system.
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That comes from stupidly failing to require {true in the system}
to require {proven in the system}. Fix this one stupid mistake
and all of incompleteness goes away.

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No, that merely means that "true in the system" is incomplete in some
systems (e.g., natural numbers). There are sentences that are true in
practical applications of the system but not in the system itself.
That is not a defect as it does not prevent useful practical aplications.

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The bottom line here is that expressions that do not have
a truth-maker are always untrue. Logic screws this up by
overriding the common meaning of terms with incompatible
meanings. Provable(common) means has a truth-maker.

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Logic doesn't care about truths and truth makers except in the (usually
uninteresting) special cases where truth makers are found in the logic
itself.

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Incompleteness(math) and Undecidability(logic) are
artifacts of defining the term provable(math)
in a way that is inconsistent with provable(common)
{shown to be definitely true by whatever means}.

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No such inconsistency is shown.
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If True(X) means has a truth-maker and Provable(X)
means shown to have a truth-maker then the only
difference between Provable(X) and True(X) are unknown
truths. Some of these may be unknowable truths requiring
an infinite set of finite proofs.
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And those systems containing those unknowable truths are by definition incomplete.

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In other words unless you are all knowing something is wrong with you.
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There's nothing wrong with a system that can't show all truths are in fact true in a finite amount of time.  Such system can be and are in fact useful.
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It seems like you simply don't like the term that was selected to describe such a system.  You can give it another name if it makes you happy.  That doesn't change what the name refers to.  It just means that incomplete(math) and some_other_term_that_po_likes mean the exact same thing.

 We could create a language that only includes the word "the"
and such an artificial contrivance could be said to be incomplete.
We can see that this use of "incomplete" is no more than an
artificial contrivance.
 Alternatively we could create a formal system that contains the
sum total of all human general knowledge that can be expressed
using language. Such a system is only incomplete in that it
does not contain unknown things.
 

By definition, a system that contains a true statement that has *only* an infinite connection to its truthmaker is termed an incomplete system.
Robinson arithmetic is incomplete because the true statement "no number is equal to its successor" which can be built from the language of the system has *only* an infinite connection to the axioms of the system.