On 4/22/2023 4:04 PM, Richard Damon wrote:
> On 4/22/23 4:47 PM, olcott wrote:
>> On 4/22/2023 3:43 PM, Richard Damon wrote:
>>> On 4/22/23 4:39 PM, olcott wrote:
>>>> On 4/22/2023 3:14 PM, Richard Damon wrote:
>>>>> On 4/22/23 4:10 PM, olcott wrote:
>>>>>> On 4/22/2023 3:06 PM, Richard Damon wrote:
>>>>>>> On 4/22/23 4:02 PM, olcott wrote:
>>>>>>>> On 4/22/2023 3:00 PM, Richard Damon wrote:
>>>>>>>>> On 4/22/23 3:54 PM, olcott wrote:
>>>>>>>>>> On 4/22/2023 2:44 PM, Richard Damon wrote:
>>>>>>>>>>> On 4/22/23 3:34 PM, olcott wrote:
>>>>>>>>>>>> On 4/22/2023 2:15 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 4/22/23 3:11 PM, olcott wrote:
>>>>>>>>>>>>>> On 4/22/2023 1:01 PM, Richard Damon wrote:
>>>>>>>>>>>>>>> On 4/22/23 1:13 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 4/22/2023 11:56 AM, Richard Damon wrote:
>>>>>>>>>>>>>>>>> On 4/22/23 12:45 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 4/22/2023 11:36 AM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>> On 4/22/23 12:27 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>> On 4/22/2023 11:12 AM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>>>> On 4/22/23 11:39 AM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>> On 4/22/2023 9:57 AM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 4/22/23 10:48 AM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 4/22/2023 9:38 AM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 4/22/23 10:28 AM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 4/22/2023 6:17 AM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 4/21/23 11:40 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 4/21/2023 9:45 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 4/21/23 9:41 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 4/21/2023 7:49 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 4/21/23 8:33 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ∃G ∈ F (G ↔ (G ⊬ F))
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> There exists a G such that G is
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> logically equivalent to its own
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> unprovability in F
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> *If we assume that there is such a G in
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> F that means that*
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G is true means there is no sequence of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> inference steps that satisfies G in F.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G is false means there is a sequence of
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> inference steps that satisfies G in F.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> *Thus the above G simply does not exist
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in F*
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> So?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I finally learned enough model theory to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> correctly link provability to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> truth in the conventional model theory way.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Doesn't seem so, you don't seem to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> understand the difference. You seem to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> confuse Truth with Knowledge.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I finally approximated {G asserts its own
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> unprovability in F}
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> using conventional math symbols in their
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> conventional way.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Except that isn't what G is, you only think
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that because you can't actually understand
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> even the outline of Godel's proof, so you
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> take pieces out of context.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> G never asserts its own unprovability.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> The statement that we now have a statement
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that asserts its own unprovablity, as a
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> simplification describing a statment
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> DERIVED from G, and that derivation happens
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in Meta-F, and is about what can be proven
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in F.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Since Godel's G isn't of that form, but
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only can be used to derive a statment IN
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> META-F that says that G is not provable
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> in F, your argument says nothing about
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Godel's G.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> F ⊢ GF ↔ ¬ProvF (┌GF┐).
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> I have finally created a G that is
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> equivalent to
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Panu Raatikainen's SEP article.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> So?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Did you read that article?
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Also, you don't understand what those
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> terms mean, because G being true doesn't
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> mean there is no sequence of inference
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> steps that satisfies G in F, but there is
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> no FINITE sequence of inference steps
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> that satisfies G in F.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ∃G ∈ F (G ↔ (G ⊬ F))
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Because we can see that every finite or
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> infinite sequence in F that
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> satisfies the RHS of ↔ contradicts the LHS
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> a powerful F can infer that G
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> is utterly unsatisfiable even for infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> sequences in this more
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> powerful F.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Nope. Show the PROOF.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> You don't know HOW to do a proof, you can
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> only do arguement.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> ∃G ∈ F (G ↔ (G ⊬ F))
>>>>>>>>>>>>>>>>>>>>>>>>>>>> There exists a G in F such that G is
>>>>>>>>>>>>>>>>>>>>>>>>>>>> logically equivalent to its own
>>>>>>>>>>>>>>>>>>>>>>>>>>>> unprovability in F
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> A proof is any sequence of steps that shows
>>>>>>>>>>>>>>>>>>>>>>>>>>>> that its conclusion is a
>>>>>>>>>>>>>>>>>>>>>>>>>>>> necessary consequence of its premises.\
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> Boy are you wrong.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> A proof is a FINITE sequence of steps that
>>>>>>>>>>>>>>>>>>>>>>>>>>> shows that a given statement is a necessary
>>>>>>>>>>>>>>>>>>>>>>>>>>> consequence of the defined system.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> "Proof" doesn't have a  "Premise", it has a
>>>>>>>>>>>>>>>>>>>>>>>>>>> system.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> The statement may have conditions in it
>>>>>>>>>>>>>>>>>>>>>>>>>>> restricting when
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> ∃G ∈ F (G ↔ (G ⊬ F))
>>>>>>>>>>>>>>>>>>>>>>>>>>>> There exists a G in F such that G is
>>>>>>>>>>>>>>>>>>>>>>>>>>>> logically equivalent to its own
>>>>>>>>>>>>>>>>>>>>>>>>>>>> unprovability in F
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> If G is true then there is no sequence of
>>>>>>>>>>>>>>>>>>>>>>>>>>>> inference steps that satisfies G in F making
>>>>>>>>>>>>>>>>>>>>>>>>>>>> G untrue.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> no FINITE sequence, making G UNPROVABLE, and
>>>>>>>>>>>>>>>>>>>>>>>>>>> there IS an INFINITE sequence making it TRUE.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> This is possible.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> If G is false then there is a sequence of
>>>>>>>>>>>>>>>>>>>>>>>>>>>> inference steps that satisfies G in F making
>>>>>>>>>>>>>>>>>>>>>>>>>>>> G true.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> If G is false, then there is a finite
>>>>>>>>>>>>>>>>>>>>>>>>>>> sequence proving G, which forces G to be
>>>>>>>>>>>>>>>>>>>>>>>>>>> true, thus this is a contradiction.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> Because the RHS of ↔ contradicts the LHS
>>>>>>>>>>>>>>>>>>>>>>>>>>>> there is no such G in F.
>>>>>>>>>>>>>>>>>>>>>>>>>>>> Thus the above G simply does not exist in F.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> Nope, because we can have an infinite
>>>>>>>>>>>>>>>>>>>>>>>>>>> sequence that isn't finite, G can be True but
>>>>>>>>>>>>>>>>>>>>>>>>>>> not Provable.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> If G is false and ↔ is true this makes the RHS
>>>>>>>>>>>>>>>>>>>>>>>>>> false which negates the RHS making it say (G ⊢
>>>>>>>>>>>>>>>>>>>>>>>>>> F) which makes G true in F.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> Right, G can't be false, but it can be True.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> Thus ↔ cannot be satisfied thus no such G exists
>>>>>>>>>>>>>>>>>>>>>>>> in F.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Why do you say that?
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> I don't think you know what you terms mean.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> There exists a G in F such that G is true if and
>>>>>>>>>>>>>>>>>>>>>>> only if G is Unprovable.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> Logical equality
>>>>>>>>>>>>>>>>>>>>>> p q p ↔ q
>>>>>>>>>>>>>>>>>>>>>> T T   T // G is true if and only if G is Unprovable.
>>>>>>>>>>>>>>>>>>>>>> T F   F //
>>>>>>>>>>>>>>>>>>>>>> F T   F //
>>>>>>>>>>>>>>>>>>>>>> F F   T // G is false if and only if G is Provable.
>>>>>>>>>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Truth_table#Logical_equality
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> Row(1) There exists a G in F such that G is true
>>>>>>>>>>>>>>>>>>>>>> if and only if G is
>>>>>>>>>>>>>>>>>>>>>> unprovable in F making G unsatisfied thus untrue
>>>>>>>>>>>>>>>>>>>>>> in F.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> Row(4) There exists a G in F such that G is false
>>>>>>>>>>>>>>>>>>>>>> if and only if G is
>>>>>>>>>>>>>>>>>>>>>> provable in F making G satisfied thus true in F.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> If either Row(1) or Row(4) are unsatisfied then ↔
>>>>>>>>>>>>>>>>>>>>>> is false.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> But if neither row values can ACTUALLY EXIST, then
>>>>>>>>>>>>>>>>>>>>> the equality is true.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> If either Row(1) or Row(4) cannot have the same
>>>>>>>>>>>>>>>>>>>> value for p and q
>>>>>>>>>>>>>>>>>>>> (for whatever reason) then ↔ is unsatisfied and no
>>>>>>>>>>>>>>>>>>>> such G exists in F.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> So, you don't understand how truth tables work.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> You don't need to have all the rows with true being
>>>>>>>>>>>>>>>>>>> possible, you need all the rows that are possible to
>>>>>>>>>>>>>>>>>>> be True.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> To the best of my knowledge
>>>>>>>>>>>>>>>>>> ↔ is also known as logical equivalence meaning that
>>>>>>>>>>>>>>>>>> the LHS and the RHS
>>>>>>>>>>>>>>>>>> must always have the same truth value or ↔ is not true.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Right, and for that statement, the actual G found in F,
>>>>>>>>>>>>>>>>> the ONLY values that happen is G is ALWAYS true, an
>>>>>>>>>>>>>>>>> Unprovable is always true.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Thus the equivalence is always true.
>>>>>>>>>>>>>>>> I don't think that is the way that it works.
>>>>>>>>>>>>>>>> We must assume that the RHS is true and see how that
>>>>>>>>>>>>>>>> effects the LHS
>>>>>>>>>>>>>>>> We must assume that the RHS is false and see how that
>>>>>>>>>>>>>>>> effects the LHS
>>>>>>>>>>>>>>>> ((True(RHS) → True(LHS)) ∧ (False(RHS) → False(LHS))) ≡
>>>>>>>>>>>>>>>> (RHS ↔ LHS)
>>>>>>>>>>>>>>>> False(RHS) → True(LHS) refutes (RHS ↔ LHS)
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Nope, that isn't how it works.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Can you show me something that says that is how it works?
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> p ↔ q would seem to mean ((p → q) ∧ (q → p))
>>>>>>>>>>>>>> Here is a much clearer and conventional way of showing that
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Logical implication derives logical equivalence
>>>>>>>>>>>>>> p---q---(p ⇒ q)---(q ⇒ p)---(q ↔ p)
>>>>>>>>>>>>>> T---T------T----------T---------T
>>>>>>>>>>>>>> T---F------F----------T---------F
>>>>>>>>>>>>>> F---T------T----------F---------F
>>>>>>>>>>>>>> F---F------T----------T---------T
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> So, why does the fact that the last line is never used in
>>>>>>>>>>>>> this case cause a problem.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> ∃G ∈ F (G ↔ (G ⊬ F))
>>>>>>>>>>>>
>>>>>>>>>>>> I am just saying that according to the conventional rules of
>>>>>>>>>>>> logic the
>>>>>>>>>>>> above expression is simply false. There is no G that is
>>>>>>>>>>>> logically
>>>>>>>>>>>> equivalent to its own unprovability in F.
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> But Godel's G satisfies that.
>>>>>>>>>>>
>>>>>>>>>>> Remember, G is the statement that there does not exist a
>>>>>>>>>>> number g such that g statisifes a particular Primative
>>>>>>>>>>> Recursive Relationship (built in Meta-F, but using only
>>>>>>>>>>> operations defined in F).
>>>>>>>>>>>
>>>>>>>>>> There is no such G in F says the same thing, yet does not
>>>>>>>>>> falsely place
>>>>>>>>>> the blame on F.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Yes, but can you PROVE your statement? If not, you are just
>>>>>>>>> making unsubstantiated false claims, just like DT.
>>>>>>>>>
>>>>>>>>
>>>>>>>> I just proved it. The only gap in the proof was your lack of
>>>>>>>> understanding (an honest mistake not a lie) about how ↔ works.
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> Nope, how did you prove that no such G exists? You claims that
>>>>>>> row 4 can't be satisfied? it doesn't need to ever be used.
>>>>>>
>>>>>> Try and prove that with a source, in the mean time I will tentatively
>>>>>> assume that you are wrong. I proved that I am correct with the above
>>>>>> truth table yet this assumes: p ↔ q means ((p → q) ∧ (q → p))
>>>>>>
>>>>>
>>>>>
>>>>> WRONG, YOU are making the claim, so YOU need to prove it.
>>>>>
>>>> I may have been mistaken when I thought that more than one row of the
>>>> truth table needed to be satisfied. Furthermore in retrospect this
>>>> looks
>>>> like a dumb mistake that I did not notice as a dumb mistake until I
>>>> looked at the truth table for ∧. So we are back to row one.
>>>>
>>>> ∃G ∈ F (G ↔ (F ⊬ G))
>>>> If the RHS is satisfied then this means that there are no inference
>>>> steps in F that derive G, thus G cannot be shown to be true in F.
>>>>
>>>>
>>>
>>> Nope, there is no FINITE series of infernece steps in F that derive G.
>>>
>>
>> This G cannot be shown to be true in F.
>
> It can't be PROVEN in F, but it can be PROVEN to be true in F with a
> proof in Meta-F
>
> You just don't seem to understand how these Meta-systems work.
>

If G cannot be satisfied in F then G is not true in F.

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