(1) x ∉ Provable if and only if p // assumption
(2) x ∈ True if and only if p // assumption
(3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
(4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
(5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)
(6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)
(7) x ∈ True
(8) x ∉ Provable
(9) x̄ ∉ Provable

https://liarparadox.org/Tarski_275_276.pdf

Do you notice that the axiom on line (5) simply contradicts thus refutes
the assumption on line (3) ?

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/28/23 10:40 PM, olcott wrote:
>
> (1) x ∉ Provable if and only if p // assumption
> (2) x ∈ True if and only if p // assumption
> (3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)
> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)
> (7) x ∈ True
> (8) x ∉ Provable
> (9) x̄ ∉ Provable
>
> https://liarparadox.org/Tarski_275_276.pdf
>
> Do you notice that the axiom on line (5) simply contradicts thus refutes
> the assumption on line (3) ?
>

So, you don't understand what he is saying. (1) and (2) are not
"Assumptions".

You also are neglection to note that he changed "Theories" in the
middle, and moved to a meta-theory of higher order, which changes the
definition of provable.

You don't seem to understand that provability can be a function of the
"Theory" you are in.

Stopped at first error.

On 3/28/2023 9:40 PM, olcott wrote:
>
> (1) x ∉ Provable if and only if p // assumption
> (2) x ∈ True if and only if p // assumption
> (3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)

The above contrives a statement that is true and not provable.

> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)

This axiom proves that there are no such true and unprovable statements:
(3) ~Provable(x) ↔ True(x)
(5) Provable(x) → True(x)

> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)
> (7) x ∈ True
> (8) x ∉ Provable
> (9) x̄ ∉ Provable
>
> https://liarparadox.org/Tarski_275_276.pdf
>
> Do you notice that the axiom on line (5) simply contradicts thus refutes
> the assumption on line (3) ?
>

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/29/23 10:06 AM, olcott wrote:
> On 3/28/2023 9:40 PM, olcott wrote:
>>
>> (1) x ∉ Provable if and only if p // assumption
>> (2) x ∈ True if and only if p // assumption
>> (3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
>
> The above contrives a statement that is true and not provable.

No, the above STATES a statement that is true and not provable based on
PROVEN statements.

Note, (1) and (2) are PROVEN statements from earlier in the paper.

You just don't seem to be able to understand that paper

>
>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)
>
> This axiom proves that there are no such true and unprovable statements:
> (3) ~Provable(x) ↔ True(x)

And why is that axiom correct?

That is YOUR problem, you have ASSUMED that not provable means not true,
and that ASSUMPTION is just incorrect.

That false assumption makes your whole logic system inconsistent.

You have just ADMITTED that you whole logic system is based on a FALSE
ASSUMPTION.

YOU HAVE FAILED.

All you have done is proven that you don't understand the basics of logic.

> (5) Provable(x) → True(x)
>
>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)
>> (7) x ∈ True
>> (8) x ∉ Provable
>> (9) x̄ ∉ Provable
>>
>> https://liarparadox.org/Tarski_275_276.pdf
>>
>> Do you notice that the axiom on line (5) simply contradicts thus
>> refutes the assumption on line (3) ?
>>
>

On 3/29/23 10:06 AM, olcott wrote:
>
> This axiom proves that there are no such true and unprovable statements:
> (3) ~Provable(x) ↔ True(x)
> (5) Provable(x) → True(x)

And your mere stating this proves you don't know what you are talking
about, as that is NOT a axiom in the field Tarski is talking about.

PERIOD. DEFINITION.

Axioms are the ACCEPTED "self evident" statements of a field, which this
is not.

Thus proving your ignorance, and stupidity.

On 3/28/2023 9:40 PM, olcott wrote:
>
> (1) x ∉ Provable if and only if p // assumption
> (2) x ∈ True if and only if p // assumption
> (3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)

"we can derive the following theorems from the definition of truth (cf.
Ths. 1 and 5 in § 3):" (Tarski)

> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)
> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)

> (7) x ∈ True
> (8) x ∉ Provable
> (9) x̄ ∉ Provable
>
> https://liarparadox.org/Tarski_275_276.pdf
>
> Do you notice that the axiom on line (5) simply contradicts thus refutes
> the assumption on line (3) ?
>

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/29/23 8:55 PM, olcott wrote:
> On 3/28/2023 9:40 PM, olcott wrote:
>>
>> (1) x ∉ Provable if and only if p // assumption
>> (2) x ∈ True if and only if p // assumption
>> (3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
>
> "we can derive the following theorems from the definition of truth (cf.
> Ths. 1 and 5 in § 3):" (Tarski)

Did you miss that he switched to a Meta-Theory with more power?

>
>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)
>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)
>
>> (7) x ∈ True
>> (8) x ∉ Provable
>> (9) x̄ ∉ Provable
>>
>> https://liarparadox.org/Tarski_275_276.pdf
>>
>> Do you notice that the axiom on line (5) simply contradicts thus
>> refutes the assumption on line (3) ?
>>
>

You sem to be missing what x and p are. That there exist SOME SPECIFIC
sentence x that has this property, statements (1), (2), (3) do not hold
for all statements, but for a specific statement that must exist.

Also, (3) is NOT in "contraditions" to (5), as (5) says that if x is
provable, then it must be true, (and if x is false it can not be proven).

The Truth values of x being True but x not being provable is a CONSITENT
set of truth value that do not lead to any contradictions.

Since x is not provable, then statment (5) has a false premise, and thus
says nothing about its conclusion, so no contradiction.

Only by adding that provable(x) is and only if True(x) do you get a
contradiciton.

All you re doing is proving that you don't understand what you are
talking about, and that you logic is contraictory.

(And that you have the emotional stability of a three-year-old)

On 3/29/2023 7:55 PM, olcott wrote:
> On 3/28/2023 9:40 PM, olcott wrote:
>>
>> (1) x ∉ Provable if and only if p // assumption
>> (2) x ∈ True if and only if p // assumption
>> (3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
>
> "we can derive the following theorems from the definition of truth (cf.
> Ths. 1 and 5 in § 3):" (Tarski)
>
>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)
>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)
>
>> (7) x ∈ True
>> (8) x ∉ Provable
>> (9) x̄ ∉ Provable
>>
>> https://liarparadox.org/Tarski_275_276.pdf
>>
>> Do you notice that the axiom on line (5) simply contradicts thus
>> refutes the assumption on line (3) ?
>>
>

The whole theory metatheory thing is a scam that avoids confronting the
reality that self-contradictory expressions of language are simply not
truth bearers.

This sentence is not true: "This sentence is not true" is true in the
meta-theory.

Trying to resolve self-contradictory sentences with para-consistent
logic or any other means is like continuing to trying to "bake" an
actual angel food cake using only house bricks for ingredients.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/29/23 9:39 PM, olcott wrote:
> On 3/29/2023 7:55 PM, olcott wrote:
>> On 3/28/2023 9:40 PM, olcott wrote:
>>>
>>> (1) x ∉ Provable if and only if p // assumption
>>> (2) x ∈ True if and only if p // assumption
>>> (3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
>>
>> "we can derive the following theorems from the definition of truth
>> (cf. Ths. 1 and 5 in § 3):" (Tarski)
>>
>>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails
>>> true(x)
>>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails
>>> true(x̄)
>>
>>> (7) x ∈ True
>>> (8) x ∉ Provable
>>> (9) x̄ ∉ Provable
>>>
>>> https://liarparadox.org/Tarski_275_276.pdf
>>>
>>> Do you notice that the axiom on line (5) simply contradicts thus
>>> refutes the assumption on line (3) ?
>>>
>>
>
> The whole theory metatheory thing is a scam that avoids confronting the
> reality that self-contradictory expressions of language are simply not
> truth bearers.

So, you don't understand the concept of different "Theories" can exist
based on different sets of axioms.

The fact that you can't actually point out an ACTUAL contradiction, only
your perceived contradictions based on your FALSE assumption that True
implies Proveabl, shows that you have nothing to go on.

>
> This sentence is not true: "This sentence is not true" is true in the
> meta-theory.

Which isn't a sentence that has been used, so just shows you are going
ba k to your strawman.

>
> Trying to resolve self-contradictory sentences with para-consistent
> logic or any other means is like continuing to trying to "bake" an
> actual angel food cake using only house bricks for ingredients.
>

What "self-contradictory" statements?

Note, a statement refering to its provability is different than a
statement refering to its truthfullness.

All you have done is show that you "think" you understand things that
you don't and you make FALSE statements about them due to your lack of
understanding.

Your mind seems stuck in the mud of systems that are too simple to
handle the proofs you want to talk about.

You are just PROVING your ignorance and stupdity.

On 3/29/2023 8:39 PM, olcott wrote:
> On 3/29/2023 7:55 PM, olcott wrote:
>> On 3/28/2023 9:40 PM, olcott wrote:
>>>
>>> (1) x ∉ Provable if and only if p // assumption
>>> (2) x ∈ True if and only if p // assumption
>>> (3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
>>
>> "we can derive the following theorems from the definition of truth
>> (cf. Ths. 1 and 5 in § 3):" (Tarski)
>>
>>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails
>>> true(x)
>>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails
>>> true(x̄)
>>
>>> (7) x ∈ True
>>> (8) x ∉ Provable
>>> (9) x̄ ∉ Provable
>>>
>>> https://liarparadox.org/Tarski_275_276.pdf
>>>
>>> Do you notice that the axiom on line (5) simply contradicts thus
>>> refutes the assumption on line (3) ?
>>>
>>
>
> The whole theory metatheory thing is a scam that avoids confronting the
> reality that self-contradictory expressions of language are simply not
> truth bearers.
>
> This sentence is not true: "This sentence is not true" is true in the
> meta-theory.
>
> Trying to resolve self-contradictory sentences with para-consistent
> logic or any other means is like continuing to trying to "bake" an
> actual angel food cake using only house bricks for ingredients.
>

we can derive the following theorems from the definition of truth
(5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)

proves that this is false

We shall show that the sentence x is actually undecidable
and at the same time true.
(3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)

Provable(x) → True(x) // theorem
refutes
~Provable(x) ↔ True(x) // assumption

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/28/2023 9:40 PM, olcott wrote:
>
> (1) x ∉ Provable if and only if p // assumption
> (2) x ∈ True if and only if p // assumption

We shall show that the sentence x is actually undecidable and at the
same time true.

> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)

we can derive the following theorems from the definition of truth

> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)
> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)
> (7) x ∈ True
> (8) x ∉ Provable
> (9) x̄ ∉ Provable
>
> https://liarparadox.org/Tarski_275_276.pdf
>
> Do you notice that the axiom on line (5) simply contradicts thus refutes
> the assumption on line (3) ?
>

(5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
*proves that this is false*
(3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)

Provable(x) → True(x) // theorem
*refutes*
~Provable(x) ↔ True(x) // assumption

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/29/23 10:53 PM, olcott wrote:
> On 3/28/2023 9:40 PM, olcott wrote:
>>
>> (1) x ∉ Provable if and only if p // assumption
>> (2) x ∈ True if and only if p // assumption
>
> We shall show that the sentence x is actually undecidable and at the
> same time true.
>

Which he does, and you haven't shown how that isn't shown.

>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>
> we can derive the following theorems from the definition of truth

Which is fine.

>
>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)
>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)
>> (7) x ∈ True
>> (8) x ∉ Provable
>> (9) x̄ ∉ Provable
>>
>> https://liarparadox.org/Tarski_275_276.pdf
>>
>> Do you notice that the axiom on line (5) simply contradicts thus
>> refutes the assumption on line (3) ?
>>
>
> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
> *proves that this is false*
> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>

Nope, since x isn't Provable, (5) shows nothing. You just don't
understand how logic works.

If Sam is a cat, then Sam is a Mammel tells us something if Sam is a
cat, but not if Sam is not cat. Sam might be a dog, which is also a
mammal, thus a false preposition implying a true preposition is a valid
instance of an if/then clause. (The False premise doesn't force the
conclusion, but it doesn't force the conclusion false).

So, if x WAS provable, then we would know that x must be true, but from
(3) we know that this case can't be correct, as, by the structure of
that SPECIFIC x, if x was Provable, it couldn't be true. Thus we can
show that x must be unprovable and also true.

This is like Godels actual statement. From the implications (shown in
Meta-F) if a number existed that satisfies the defined primative
recursive relationship, then we have proof of the existance of a proof
for G (and from Meta-F could actually decode the number to get the
proof), which by the rule (5) says G must be true.

But from the statement itself, if such a number existed, then G must be
false, since G asserts that no such number exists.

This means the only possiblity that exists is that no such number
exsits, which actually DOES make G true, but no proof of this fact can
exist in F (as if one did exist, we could use meta-F to convert that
proof into a number that satisfies the relationship, makeing G false).

> Provable(x) → True(x)  // theorem
FALSE → TRUE // A Valid instance of if-then / implication

> *refutes*
> ~Provable(x) ↔ True(x) // assumption
TRUE ↔ TRUE // A Valid instance of if and only if.
>

Since a valid combination of values for the terms exists, there is no
contradiction that refutes the statements.

You are just showing you don't understand basic logic.

This has been evident for years.

On 3/29/23 10:47 PM, olcott wrote:
> On 3/29/2023 8:39 PM, olcott wrote:
>> On 3/29/2023 7:55 PM, olcott wrote:
>>> On 3/28/2023 9:40 PM, olcott wrote:
>>>>
>>>> (1) x ∉ Provable if and only if p // assumption
>>>> (2) x ∈ True if and only if p // assumption
>>>> (3) x ∉ Provable if and only if x ∈ True. // combine (1) and (2)
>>>
>>> "we can derive the following theorems from the definition of truth
>>> (cf. Ths. 1 and 5 in § 3):" (Tarski)
>>>
>>>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>>>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails
>>>> true(x)
>>>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails
>>>> true(x̄)
>>>
>>>> (7) x ∈ True
>>>> (8) x ∉ Provable
>>>> (9) x̄ ∉ Provable
>>>>
>>>> https://liarparadox.org/Tarski_275_276.pdf
>>>>
>>>> Do you notice that the axiom on line (5) simply contradicts thus
>>>> refutes the assumption on line (3) ?
>>>>
>>>
>>
>> The whole theory metatheory thing is a scam that avoids confronting the
>> reality that self-contradictory expressions of language are simply not
>> truth bearers.
>>
>> This sentence is not true: "This sentence is not true" is true in the
>> meta-theory.
>>
>> Trying to resolve self-contradictory sentences with para-consistent
>> logic or any other means is like continuing to trying to "bake" an
>> actual angel food cake using only house bricks for ingredients.
>>
>
> we can derive the following theorems from the definition of truth
> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>
> proves that this is false

Nope, just that it isn't provable.

>
> We shall show that the sentence x is actually undecidable
> and at the same time true.
> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>
> Provable(x) → True(x)  // theorem
>   refutes
> ~Provable(x) ↔ True(x) // assumption
>

Nope, it has an VALID answer of x is not provable and x is true.

You are just to stupid to understand that combination, because you have
gaslit yourself into believing that such a combination is imposible,
when it is actually quite possible.

I think you just don;t understand what → actually means.

On 3/29/2023 9:53 PM, olcott wrote:
> On 3/28/2023 9:40 PM, olcott wrote:
>>
>> (1) x ∉ Provable if and only if p // assumption
>> (2) x ∈ True if and only if p // assumption
>
> We shall show that the sentence x is actually undecidable and at the
> same time true.
>
>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>
> we can derive the following theorems from the definition of truth
>
>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails true(x)
>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails true(x̄)
>> (7) x ∈ True
>> (8) x ∉ Provable
>> (9) x̄ ∉ Provable
>>
>> https://liarparadox.org/Tarski_275_276.pdf
>>
>> Do you notice that the axiom on line (5) simply contradicts thus
>> refutes the assumption on line (3) ?
>>
>
> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
> *proves that this is false*
> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>
> Provable(x) → True(x)  // theorem
> *refutes*
> ~Provable(x) ↔ True(x) // assumption

When an axiom contradicts an assumption the assumption is tossed out as
erroneous.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/30/23 10:49 AM, olcott wrote:
> On 3/29/2023 9:53 PM, olcott wrote:
>> On 3/28/2023 9:40 PM, olcott wrote:
>>>
>>> (1) x ∉ Provable if and only if p // assumption
>>> (2) x ∈ True if and only if p // assumption
>>
>> We shall show that the sentence x is actually undecidable and at the
>> same time true.
>>
>>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>>
>> we can derive the following theorems from the definition of truth
>>
>>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails
>>> true(x)
>>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails
>>> true(x̄)
>>> (7) x ∈ True
>>> (8) x ∉ Provable
>>> (9) x̄ ∉ Provable
>>>
>>> https://liarparadox.org/Tarski_275_276.pdf
>>>
>>> Do you notice that the axiom on line (5) simply contradicts thus
>>> refutes the assumption on line (3) ?
>>>
>>
>> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>> *proves that this is false*
>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>>
>> Provable(x) → True(x)  // theorem
>> *refutes*
>> ~Provable(x) ↔ True(x) // assumption
>
> When an axiom contradicts an assumption the assumption is tossed out as
> erroneous.
>

But it doesn't contradict it, and that's YOUR problem.

Provable(x) -> True(x) means that *IF* x is in fact, provable, it must
be true.

If x isn't, in fact, provable, it means nothing.

That x is not provable if, and only if, it it true, means that, yes, if
x was provable, then it wouldn't be true, and thus we have a
contradiction, but if x isn't provable, it is true, and that doesn't
contradict the first statement.

Thus, since we have a valid result, of x being true, but not provable,
which is a valid case for both statement, we have no contradiction.

You obviously don't understand what a contradiction means.

That is like the pair of equartions,

x = 2 * y, and x = y + 1

Those both say x is expressed differently, but together we can "solve"
the system and say that x = 2, and y = 1.

In the same way

Provable(x) -> True(x)
~Provable(x) <-> True(x)

solves to X is True, and x is not provable.

You just don't like that answer.

Also, not, the "~Povable(x) <-> True(x)" is not an "assumption', but
that there exists an x that satisifies this is a PROVEN statement. You
just apparently can't understand that logic.

All you have been doing is proving that you reasoning is defective, and
that you are ignorant of the basics of logic, thus your "correct
reasoning" system is likely worthless.

On 3/30/2023 9:49 AM, olcott wrote:
> On 3/29/2023 9:53 PM, olcott wrote:
>> On 3/28/2023 9:40 PM, olcott wrote:
>>>
>>> (1) x ∉ Provable if and only if p // assumption
>>> (2) x ∈ True if and only if p // assumption
>>
>> We shall show that the sentence x is actually undecidable and at the
>> same time true.
>>
>>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>>
>> we can derive the following theorems from the definition of truth
>>
>>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails
>>> true(x)
>>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails
>>> true(x̄)
>>> (7) x ∈ True
>>> (8) x ∉ Provable
>>> (9) x̄ ∉ Provable
>>>
>>> https://liarparadox.org/Tarski_275_276.pdf
>>>
>>> Do you notice that the axiom on line (5) simply contradicts thus
>>> refutes the assumption on line (3) ?
>>>
>>
>> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>> *proves that this is false*
>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>>
>> Provable(x) → True(x)  // theorem
>> *refutes*
>> ~Provable(x) ↔ True(x) // assumption
>
> When an axiom contradicts an assumption the assumption is tossed out as
> erroneous.
>

When True(x) is a consequence of Provable(x) then
True(x) cannot be a consequence of ~Provable(x).

True() and Provable() must be evaluated semantically and not merely as
propositional logic.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/30/23 7:08 PM, olcott wrote:
> On 3/30/2023 9:49 AM, olcott wrote:
>> On 3/29/2023 9:53 PM, olcott wrote:
>>> On 3/28/2023 9:40 PM, olcott wrote:
>>>>
>>>> (1) x ∉ Provable if and only if p // assumption
>>>> (2) x ∈ True if and only if p // assumption
>>>
>>> We shall show that the sentence x is actually undecidable and at the
>>> same time true.
>>>
>>>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>>>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>>>
>>> we can derive the following theorems from the definition of truth
>>>
>>>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails
>>>> true(x)
>>>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails
>>>> true(x̄)
>>>> (7) x ∈ True
>>>> (8) x ∉ Provable
>>>> (9) x̄ ∉ Provable
>>>>
>>>> https://liarparadox.org/Tarski_275_276.pdf
>>>>
>>>> Do you notice that the axiom on line (5) simply contradicts thus
>>>> refutes the assumption on line (3) ?
>>>>
>>>
>>> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>>> *proves that this is false*
>>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>>>
>>> Provable(x) → True(x)  // theorem
>>> *refutes*
>>> ~Provable(x) ↔ True(x) // assumption
>>
>> When an axiom contradicts an assumption the assumption is tossed out
>> as erroneous.
>>
>
> When True(x) is a consequence of Provable(x) then
> True(x) cannot be a consequence of ~Provable(x).

WRONG.

If x IS True, then it doesn't matter what it is a consequence of.

You don't understand how logic works.

>
> True() and Provable() must be evaluated semantically and not merely as
> propositional logic.
>

So, you don't understand how logic works.

On 3/30/23 7:16 PM, Richard Damon wrote:
> On 3/30/23 7:08 PM, olcott wrote:
>> On 3/30/2023 9:49 AM, olcott wrote:
>>> On 3/29/2023 9:53 PM, olcott wrote:
>>>> On 3/28/2023 9:40 PM, olcott wrote:
>>>>>
>>>>> (1) x ∉ Provable if and only if p // assumption
>>>>> (2) x ∈ True if and only if p // assumption
>>>>
>>>> We shall show that the sentence x is actually undecidable and at the
>>>> same time true.
>>>>
>>>>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>>>>> (4) either x ∉ True or x̄ ∉ True; // axiom: ~true(x) ∨ ~true(~x)
>>>>
>>>> we can derive the following theorems from the definition of truth
>>>>
>>>>> (5) if x ∈ Provable, then x ∈ True; // axiom: provable(x) entails
>>>>> true(x)
>>>>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: provable(x̄) entails
>>>>> true(x̄)
>>>>> (7) x ∈ True
>>>>> (8) x ∉ Provable
>>>>> (9) x̄ ∉ Provable
>>>>>
>>>>> https://liarparadox.org/Tarski_275_276.pdf
>>>>>
>>>>> Do you notice that the axiom on line (5) simply contradicts thus
>>>>> refutes the assumption on line (3) ?
>>>>>
>>>>
>>>> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>>>> *proves that this is false*
>>>> (3) x ∉ Provable if and only if x ∈ True. // ~Provable(x) ↔ True(x)
>>>>
>>>> Provable(x) → True(x)  // theorem
>>>> *refutes*
>>>> ~Provable(x) ↔ True(x) // assumption
>>>
>>> When an axiom contradicts an assumption the assumption is tossed out
>>> as erroneous.
>>>
>>
>> When True(x) is a consequence of Provable(x) then
>> True(x) cannot be a consequence of ~Provable(x).
>
> WRONG.
>
> If x IS True, then it doesn't matter what it is a consequence of.
>
> You don't understand how logic works.

One thing to remember, for (3) "x" isn't "any ol statement", but a very
particualr one, and as such, Provable(x) and True(x) are constants, not
variables.

>
>>
>> True() and Provable() must be evaluated semantically and not merely as
>> propositional logic.
>>
>
> So, you don't understand how logic works.
>
>

Which since x is a defined senetece, the "evaluation" is trivial, as the
statement IS True, and not Provable.