On 4/22/23 5:34 PM, olcott wrote:
> On 4/22/2023 4:26 PM, Richard Damon wrote:
>> On 4/22/23 5:10 PM, olcott wrote:
>>> 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.
>>>
>>>
>>
>> Who says G can not be satisified in F?
>>
>
> To derive G in F requires a set of inference steps in F that proves that
> these same inference steps do not exist in F.

So, you don't know what satisified means a guess, since it has nothing
to do with "deriving". Deriving tend to refer to PROOFS, not Truth.

Note too, Truth can come from in infinite set of steps in the system, so
might not be actually knowable with just the system.

You clearly don't understand how the Meta-System can let us know more
about the system then we can learn just from the system itself.

You are just proving your ignorance.

>
>> In fact, Godel's G has no model variables in it that need to be
>> satisfied. G is just UNCONDITIONALLY TRUE in all the models of F.
>>
>> I don't think you actually understand what you are talking about.
>>
>> That just shows how ignorant you are.
>