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

On 2024-11-02 11:43:02 +0000, olcott said:

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

On 2024-11-01 12:19:03 +0000, olcott said:
 

On 11/1/2024 5:42 AM, Mikko wrote:

On 2024-10-30 12:46:25 +0000, olcott said:
 

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.

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

 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.

From the proof of the exstence of Russell's set it is easy
to prove that 1 = 2. As long as no proof of 1 = 2 from a
self-reference in a formal system is shown there is no
reason to think that such system is inconsisten. And the
existence of insonstent formal systems does not mean that
the notion of a formal system is inconsistent.

When we handle this self-reference differently then this issue
is resolved.

No proof ot that, either.

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.
 

The foundation of all these theories is classical logic.
 

 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.

Classical logic does not substantially diverge from the model of
syllogism. It iextends it for situations that cannot be covered
with syllogistic logic. Presentational differences follow mainly
from the needs of the additional coverage.
No error has been shown in classical logic.
--
Mikko