Re: The philosophy of computation reformulates existing ideas on a new basis ---

On 11/2/24 7:43 AM, olcott wrote:

On 11/2/2024 4:09 AM, Mikko wrote:

On 2024-11-01 12:19:03 +0000, olcott said:
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On 11/1/2024 5:42 AM, Mikko wrote:

On 2024-10-30 12:46:25 +0000, olcott said:
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ZFC only resolved Russell's Paradox because it tossed out
the incoherent foundation of https://en.wikipedia.org/wiki/ Naive_set_theory

>
Actually Zermelo did it. The F and C are simply minor improvements on
other aspects of the theory.

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Thus establishing the precedent that replacing the foundational
basis of a problem is a valid way to resolve that problem.

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No, that does not follow. In particular, Russell's paradox is not a
problem, just an element of the proof that the naive set theory is
inconsistent. The problem then is to construct a consistent set
theory. Zermelo proposed one set theory and ZF and ZFC are two other
proposals.
>

 My view is that the same kind of self-reference issue that
showed naive set theory was inconsistent also shows that the
current notion of a formal system is inconsistent. When we
handle this self-reference differently then this issue is
resolved.

And if you think that, you need to show the contradiction that results.
You need to use the rules, EXACTLY AS DEFINED, and show that you can prove two contradictory results.

 When a formal system is ONLY a sequence of truth preserving
operations applied to a consistent set of expressions that
have been stipulated to be true then expressions that would
otherwise show incompleteness are rejected because they have
no path to true or false.

And what step that wasn't such a thing was used in the incompleteness proof?
Formal Logic *IS* just the results of sequences of what it defines to be Truth Preserving operations, on the defined set of statements that have been stipulated to be true, or proven true by the sequence of operations on those that have been.
The problem is that Formal Logic allows the sequence of operations that make something to be true infinite, but only allows finite sequences of operations to be use to make a proof, or to make something known.

 

The foundation of all these theories is classical logic.
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 The key error of classical logic is that it diverged from the
model of the syllogism where there is always a path to true or
false or the syllogism is ill-formed.
 

The problem with syllogism, is that it greatly restricts the operations that one can do.
It can't quite perform all the logic of even first-order logic.
If you want to limit yourself to syllogism, go ahead, but just realize that you just boxed yourself out of problems that need higher powered logic to work.
IF I remember right, you just lost the fullness of arithmetic, but maybe kept things of the order of primary school math.
Of course, it seems that is all thet logic you understand, so the loss isn't meaningful to you.