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Science and religion are in full accord but science and faith are in complete discord.

tech / sci.math / Re: There exists a G such that G is logically equivalent to its own unprovability in F

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
>
>

Subject	Author
There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	olcott
Re: There exists a G such that G is logically equivalent to its own	Richard Damon
Re: There exists a G such that G is logically equivalent to its own	Python