Re: Mathematical incompleteness has always been a misconception --- Tarski

On 2025-02-21 23:19:10 +0000, olcott said:

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

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

On 2/12/2025 4:21 AM, Mikko wrote:

On 2025-02-11 14:07:11 +0000, olcott said:
 

On 2/11/2025 3:50 AM, Mikko wrote:

On 2025-02-10 11:48:16 +0000, olcott said:
 

On 2/10/2025 2:55 AM, Mikko wrote:

On 2025-02-09 13:10:37 +0000, Richard Damon said:
 

On 2/9/25 5:33 AM, Mikko wrote:

Of course, completness can be achieved if language is sufficiently
restricted so that sufficiently many arithemtic truths become inexpressible.
 It is far from clear that a theory of that kind can express all arithmetic
truths that Peano arithmetic can and avoid its incompletness.

 WHich, it seems, are the only type of logic system that Peter can understand.
 He can only think in primitive logic systems that can't reach the complexity needed for the proofs he talks about, but can't see the problem, as he just doesn't understand the needed concepts.

 That would be OK if he wouldn't try to solve problems that cannot even
exist in those systems.

 There are no problems than cannot be solved in a system
that can also reject semantically incorrect expressions.

 The topic of the discussion is completeness. Is there a complete system
that can solve all solvable problems?

 When the essence of the change is to simply reject expressions
that specify semantic nonsense there is no reduction in the
expressive power of such a system.

 The essence of the change is not sufficient to determine that.

 In the same way that 3 > 2 is stipulated the essence of the
change is that semantically incorrect expressions are rejected.
Disagreeing with this is the same as disagreeing that 3 > 2.

 That 3 > 2 need not be (and therefore usually isn't) stripualted.

 The defintion of the set of natural numbers stipulates this.
 

It follows from the traditional meanings of "3", "2", and ">".
Therefore the above statement is meaningless.
 

The
result depends on all of the change. But as long as we don't even
know whether that kind of change is possible at all the details are
impossible to determine.

 LP := ~True(LP) has never been more than nonsense.

 More specifically, your nonnsense. The symbol ":=" usually means definition
but requires that the symbol on the left side (in this case "LP") is not
used on the right side (and also that it is not used in the definition of
any of the symbols on the right side).
 Usually languages of formal logic are constructed so that symbol that is
defined with an expression that starts with a negation operator cannot
be used as an argument to a function or a predicate.
 

Tarski (although otherwise quite brilliant) had a blind spot.

 Tarski did not use your nonsense.
 

 Tarski anchored his whole proof in the Liar Paradox.

More specifically, to the idea that the Liar Paradox does not have a
truth value. Do you reject that idea?
--
Mikko