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

On 5/6/25 10:43 AM, olcott wrote:

On 5/6/2025 3:30 AM, Alan Mackenzie wrote:

[ Followup-To: set ]
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In comp.theory olcott <polcott333@gmail.com> wrote:

On 5/5/2025 3:12 PM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 5/5/2025 2:34 PM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 5/5/2025 1:52 PM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 5/5/2025 1:19 PM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 5/5/2025 11:05 AM, Alan Mackenzie wrote:

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[ .... ]

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Follow the details of the proof of Gödel's Incompleteness
Theorem, and apply them to your "system".  That will give you
your counter example.

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My system does not do "provable" instead it does "provably true".

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I don't know anything about your "system" and I don't care.  If
it's a formal system with anything above minimal capabilities,
Gödel's Theorem applies to it, and the "system" will be incomplete
(in Gödel's sense).

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I reformulate the entire notion of "formal system"
so that undecidability ceases to be possible.

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Liar.  That is impossible.

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[ Irrelevant nonsense snipped. ]

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When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.
Is that too difficult for you?

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Not at all.  One of the truths you inescapably end up with is Gödel's
Theorem.  Either that, or the system is self-contradictory or too weak
to do anything at all.

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Gödel's theorem cannot possibly be recreated when
True(x) is defined to apply truth preserving
operations to basic facts.

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On the contrary, whether or not True(x) can be so defined, Gödel's
theorem cannot be avoided.

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[ .... ]

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That would appear to be well beyond your level of understanding.  You
ought to show some respect towards those who do understand these things.

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I have spent 22 years focusing on pathological self-reference.
My understanding really is deeper.

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It might be a little deeper than it was, but that's not saying very much.
The concept of proof by contradiction, for example, is way beyond you.
Even the very idea of a mathematical proof, its status, its significance
is beyond you.  You don't even understand what it is you're lacking.

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Those 22 years have been suboptimally spent.

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As I said, you ought to show a bit of respect to those who understand
these mathematical things.

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So you don't understand that when True(x) is
defined to only apply truth preserving operations
to basic facts that are stipulated to be true
that every input including random gibberish
and self-contradiction IS DECIDABLE AS TRUE OR ~TRUE.

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That's like being challenged by a young child to understand some detail
of his newest fantasy.  Except you're not a child, and ought to have an
adult's sense of proportion and reality, and a sense of your own
limitations.  You're lacking these.
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 In other words you cannot possibly point out any actual mistake
because what I said is proven completely true entirely on the
basis of the meaning of its words.

No, the problem is you don't understand the meaning of the words you use, which HAVE a defined meaning in the field, which differ from how you are using it, and thus you are just spouting nonsense.

 When starting with truth and ONLY truth preserving operations
are applied ONLY truth (not undecidability) is derived.

Wrong, undecidability does come. I think you don't understand what that means. It doesn't mean a statement that can be neigther true or false, but a statement we can not decide on its truth value, but it has one.

 If I was wrong then you could provide a counter-example.
Because I am correct no valid counter-example exists.

Godel' G.
That there does not exist a natural number g that satifies the particular primative recursive relationship that was create in the meta-math.
This statement can be proven in the meta-math, but not in the base system, but is still true there, established by an INFINITE series of truth preserving operations.
Something you have shown yourself to stupid to understand,
Go ahead, try to show how that statement is NOT undecidable in the base system.

 

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

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