On 3/6/2022 5:57 AM, Richard Damon wrote:
> On 3/5/22 11:03 PM, olcott wrote:
>> On 3/5/2022 9:57 PM, Richard Damon wrote:
>>> On 3/5/22 10:27 PM, olcott wrote:
>>>> On 3/5/2022 9:09 PM, Richard Damon wrote:
>>>>> On 3/5/22 9:39 PM, olcott wrote:
>>>>>> On 3/5/2022 7:56 PM, Richard Damon wrote:
>>>>>>> On 3/5/22 8:37 PM, olcott wrote:
>>>>>>>> On 3/5/2022 7:24 PM, Richard Damon wrote:
>>>>>>>>>
>>>>>>>>> On 3/5/22 8:03 PM, olcott wrote:
>>>>>>>>>> On 3/5/2022 6:46 PM, Richard Damon wrote:
>>>>>>>>>>>
>>>>>>>>>>> On 3/5/22 7:12 PM, olcott wrote:
>>>>>>>>>>>> On 3/5/2022 5:36 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 3/5/22 6:16 PM, olcott wrote:
>>>>>>>>>>>>>> On 3/5/2022 5:09 PM, Richard Damon wrote:
>>>>>>>>>>>>>>> On 3/5/22 5:51 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 3/5/2022 4:20 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> On 3/5/22 4:48 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 3/5/2022 3:39 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> On 3/5/22 4:04 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>> On 3/5/2022 1:30 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>>>> On 3/5/22 1:53 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>> On 3/5/2022 5:07 AM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> On 3/4/22 11:04 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 3/4/2022 10:03 PM, Python wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 3/4/2022 9:44 PM, Python wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> ...
>>>>>>>>>>>>>>>>>>>>>>>>>>>> True cannot possibly exist apart from
>>>>>>>>>>>>>>>>>>>>>>>>>>>> provable thus Gödel could not have possibly
>>>>>>>>>>>>>>>>>>>>>>>>>>>> found any expression of any language L that
>>>>>>>>>>>>>>>>>>>>>>>>>>>> is both true in L and unprovable in L.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> But still he did.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> You might as well have said that Gödel had a
>>>>>>>>>>>>>>>>>>>>>>>>>> dog that was three cats.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> Nope. I didn't say that. Stop dodging and lying.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> Gödel exhibited an expression can be true in L
>>>>>>>>>>>>>>>>>>>>>>>>> and unprovable
>>>>>>>>>>>>>>>>>>>>>>>>> in L.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> He did not. This is impossible.
>>>>>>>>>>>>>>>>>>>>>>>> It is inherently true at the deepest
>>>>>>>>>>>>>>>>>>>>>>>> philosophical level that analytic truth only
>>>>>>>>>>>>>>>>>>>>>>>> exists as connections between expressions of
>>>>>>>>>>>>>>>>>>>>>>>> language.
>>>>>>>>>>>>>>>>>>>>>>>> The proof of this is that no valid
>>>>>>>>>>>>>>>>>>>>>>>> counter-examples exist.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Denial of actual Truth is a good sign of insanity.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Please seek Professional Help.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Saying something is wrong just because you
>>>>>>>>>>>>>>>>>>>>>>> disagree with the conclusion is NOT the
>>>>>>>>>>>>>>>>>>>>>>> Analytical Logic way.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> To Refute a logical argument, you need to show
>>>>>>>>>>>>>>>>>>>>>>> what particular step breaks the rules of logic
>>>>>>>>>>>>>>>>>>>>>>> that are enforce in the field that the argument
>>>>>>>>>>>>>>>>>>>>>>> is presented in, or which assumed premise that
>>>>>>>>>>>>>>>>>>>>>>> went into the argument is not True in that field.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Of course to do this, you need to actually read
>>>>>>>>>>>>>>>>>>>>>>> and understand the argument, and not just depend
>>>>>>>>>>>>>>>>>>>>>>> on your own gut feelings.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> I am saying that the only way that we can know
>>>>>>>>>>>>>>>>>>>>>> that an analytic expression of language is true is
>>>>>>>>>>>>>>>>>>>>>> that it is either defined to be true or it is
>>>>>>>>>>>>>>>>>>>>>> derived by applying truth preserving operations to
>>>>>>>>>>>>>>>>>>>>>> expressions of language that are defined to be true.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> The above statements prove themselves to be true
>>>>>>>>>>>>>>>>>>>>>> categorically on the basis of the meaning of their
>>>>>>>>>>>>>>>>>>>>>> words.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> And you are confusing KNOWLEDGE with TRUTH.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> For something to be a KNOWN TRUTH, it needs some
>>>>>>>>>>>>>>>>>>>>> way to 'Prove' it, either an Analytic Proof, or a
>>>>>>>>>>>>>>>>>>>>> proper observation for Emperical/Synthetic Proof.
>>>>>>>>>>>>>>>>>>>>> But it is True before it was proven, if it was true.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Proofs can not change the Truth value of statements
>>>>>>>>>>>>>>>>>>>>> (unless they are statements about something having
>>>>>>>>>>>>>>>>>>>>> been proven), they just move them from an unproven
>>>>>>>>>>>>>>>>>>>>> (and likely unknown, but maybe presumed) statement
>>>>>>>>>>>>>>>>>>>>> to being Knowledge and Proven True.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> It is OBVIOUS that something can be True but not
>>>>>>>>>>>>>>>>>>>>> Known.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Goldbach's conjecture is one of the oldest and
>>>>>>>>>>>>>>>>>>>> best-known unsolved problems in number theory and
>>>>>>>>>>>>>>>>>>>> all of mathematics. It states that every even whole
>>>>>>>>>>>>>>>>>>>> number greater than 2 is the sum of two prime numbers.
>>>>>>>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Goldbach%27s_conjecture
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> The truth that we begin with is the proof that the
>>>>>>>>>>>>>>>>>>>> Goldbach criteria is met with every element of the
>>>>>>>>>>>>>>>>>>>> sequence even whole numbers greater than 2 that has
>>>>>>>>>>>>>>>>>>>> been tested.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Testing an element N of this sequence begins with
>>>>>>>>>>>>>>>>>>>> the element N as its premise then testing every pair
>>>>>>>>>>>>>>>>>>>> of primes that are numerically less that this
>>>>>>>>>>>>>>>>>>>> element to see if the sum of any pair = N.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> ∀N ∈ (even whole numbers greater than 2)
>>>>>>>>>>>>>>>>>>>> ∃!x ∈ Prime_Numbers < N
>>>>>>>>>>>>>>>>>>>> ∃!y ∈ Prime_Numbers < N
>>>>>>>>>>>>>>>>>>>> Sum(x,y) == n
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> ∀N ∈ (even whole numbers greater than 2)
>>>>>>>>>>>>>>>>>>>> Every case where Goldbach's criteria is met is a
>>>>>>>>>>>>>>>>>>>> formal proof from N to the satisfaction of the above
>>>>>>>>>>>>>>>>>>>> expression, thus proving its truth.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> So, how do you PROVE that this holds for ALL even
>>>>>>>>>>>>>>>>>>> numbers? Remember, THAT is the conjecture, not that
>>>>>>>>>>>>>>>>>>> it works for 'lots' of values.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> The fact that we haven't found an exception, doesn't
>>>>>>>>>>>>>>>>>>> mean that an exception doesn't exist, just that we
>>>>>>>>>>>>>>>>>>> haven't found it yet.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Until you show you can test EVERY even number, you
>>>>>>>>>>>>>>>>>>> can't use that as your proof, and so your method has
>>>>>>>>>>>>>>>>>>> failed to actually prove it.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Therefore not all Truth needs to be Provable.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> If any element N of (even whole numbers greater than
>>>>>>>>>>>>>>>>>>>> 2) cannot be proven to satisfy the Goldbach criteria
>>>>>>>>>>>>>>>>>>>> then Goldbach's conjecture is false.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> For Goldbach true cannot possibly exist apart from
>>>>>>>>>>>>>>>>>>>> provable is proven.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> WRONG
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Goldbach conjecture MUST be either True or it is FALSE.
>>>>>>>>>>>>>>>>>> Unless there exists a proof that each (even whole
>>>>>>>>>>>>>>>>>> number greater than 2) meets the Goldbach criteria it
>>>>>>>>>>>>>>>>>> is not true. If even one of these proofs does not
>>>>>>>>>>>>>>>>>> exist then it is not true.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> True cannot possibly exist apart from provable.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> WRONG.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> IT is an EMPRICAL FACT, that the Goldbach conjecture
>>>>>>>>>>>>>>>>> must be either true of false.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> It is an analytical fact not an empirical fact.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> If there exists a single (even whole number greater than
>>>>>>>>>>>>>>>> 2) that cannot possibly be proved to meet the Goldbach
>>>>>>>>>>>>>>>> criteria then the Goldbach conjecture is not true.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Otherwise every (even whole number greater than 2) can
>>>>>>>>>>>>>>>> be proved to meet the Goldbach criteria, thus True
>>>>>>>>>>>>>>>> depends on Provable.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Right, if one exception exists, the conjecture is not
>>>>>>>>>>>>>>> true, but false.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> We do not have, and may never have, a proof that no such
>>>>>>>>>>>>>>> exception exists.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Thus, by your definition, YOU can not treat the 'Truth'
>>>>>>>>>>>>>>> of the conjecture as a Truth Bearer, because you don't
>>>>>>>>>>>>>>> KNOW if it is actually provable, let alone if it is
>>>>>>>>>>>>>>> provably true.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Unless every element of the set of (even whole number
>>>>>>>>>>>>>> greater than 2) provably meets the Goldbach criteria, the
>>>>>>>>>>>>>> Goldbach conjecture is not true, thus proving that its
>>>>>>>>>>>>>> truth depends on its provability.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> WRONG.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Repeating your statement doesn't make it True.
>>>>>>>>>>>>>
>>>>>>>>>>>>> I will now DEMAND that you PROVE your statement or be able
>>>>>>>>>>>>> to just claim you are are liar and a Hypocrite.
>>>>>>>>>>>>
>>>>>>>>>>>>  >>>> If there exists a single (even whole number greater
>>>>>>>>>>>> than 2) that
>>>>>>>>>>>>  >>>> cannot possibly be proved to meet the Goldbach
>>>>>>>>>>>> criteria then the
>>>>>>>>>>>>  >>>> Goldbach conjecture is not true.
>>>>>>>>>>>>
>>>>>>>>>>>> If >= 1 element is not provable then false entails
>>>>>>>>>>>> < 1 elements are not provable entails
>>>>>>>>>>>> all elements are provable.
>>>>>>>>>>>
>>>>>>>>>>> You aren't making sense, because you don't understand the
>>>>>>>>>>> difference between True and Provable.
>>>>>>>>>>>
>>>>>>>>>>> Yes, If we can show >= 1 numbers that fail the requirements
>>>>>>>>>>> we have proven that the conjecture is False.
>>>>>>>>>>
>>>>>>>>>> If a proof exists (even if we never know it) that >= 1 numbers
>>>>>>>>>> that fail the requirements then it is false (even if we never
>>>>>>>>>> know it).
>>>>>>>>>>
>>>>>>>>>> Only if no proof exists (even if we never know it) that fail
>>>>>>>>>> the requirements then it is true.
>>>>>>>>>>
>>>>>>>>>> This is simply other other way of saying that unless every
>>>>>>>>>> element of the set can be proven to meet the requirements
>>>>>>>>>> (even if we never know it) then the conjecture is False.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> You just don't understand about infinitie sets do you. You are
>>>>>>>>> not guarenteed to be able to make a finite proof that all of an
>>>>>>>>> infinite set meets the requirements.
>>>>>>>>>
>>>>>>>>
>>>>>>>> If there is a single element of the set of (even whole numbers
>>>>>>>> greater than 2) that cannot be proven to meet the requirements
>>>>>>>> then the Goldbach conjecture fails
>>>>>>>>
>>>>>>
>>>>>> where A is (even whole numbers greater than 2)
>>>>>> and P is Provably meets Goldbach criteria
>>>>>> ¬Goldbach ⇔ ∃x ∈ A, ¬P(x).
>>>>>>
>>>>>>>> is merely another way of saying that
>>>>>>>>
>>>>>>>> unless every element of the set of (even whole numbers greater
>>>>>>>> than 2) can be proven to meet the requirements then the Goldbach
>>>>>>>> conjecture fails.
>>>>>>>>
>>>>>>>
>>>>>>> You still don't get it, do you.
>>>>>>
>>>>>> You are the one that fails to understand how existential
>>>>>> quantification is translated into universal quantification.
>>>>>>
>>>>>> https://sites.math.washington.edu/~aloveles/Math300Winter2011/m300Quantifiers.pdf
>>>>>>
>>>>>>
>>>>>> (1) ¬[∀x ∈ A, P(x)] ⇔ ∃x ∈ A, ¬P(x). // from the link
>>>>>>
>>>>>> where A is (even whole numbers greater than 2)
>>>>>> and P is Provably meets Goldbach criteria
>>>>>>
>>>>>> negate both sides of (1)
>>>>>> ∀x ∈ A, P(x) ⇔ ¬∃x ∈ A, ¬P(x).
>>>>>>
>>>>>> Goldbach ⇔ every element of (even whole numbers greater than 2)
>>>>>> Provably meets Goldbach criteria.
>>>>>>
>>>>>> is a correct transformation of: ¬Goldbach ⇔ ∃x ∈ A, ¬P(x).
>>>>>>
>>>>>
>>>>> And where do you get PROVABLE form any of that.
>>>>>
>>>>> You still have to PROVE an attribute for every member of the
>>>>> infinite set of Evens.
>>>>>
>>>>
>>>> Like I said you simply do not understand existential and universal
>>>> quantification well enough to translate between them.
>>>>
>>>> A is (even whole numbers greater than 2)
>>>> P is Provably meets Goldbach criteria
>>>>
>>>> On 3/5/2022 5:09 PM, Richard Damon wrote:
>>>>  > On 3/5/22 5:51 PM, olcott wrote:
>>>>  >> If there exists a single (even whole number greater than 2) that
>>>>  >> cannot possibly be proved to meet the Goldbach criteria then the
>>>>  >> Goldbach conjecture is not true.
>>>>  >
>>>>  > Right, if one exception exists, the conjecture is not true, but
>>>> false.
>>>>
>>>> My above paragraph translated into math:
>>>> ¬Goldbach ⇔ ∃x ∈ A, ¬P(x).
>>>>
>>>> People that do understand the math of it will be able to transform
>>>> that into this: Goldbach ⇔ ∀x ∈ A, P(x)
>>>>
>>>
>>> Which means that Goldbachs can be proven False if there exists a
>>> number that is even > 2 that doesn't meet the requirement.
>>>
>>> FINE.
>>>
>>> That means that IF Goldbachs Conjecture is False, that fact will be
>>> provable.
>>>
>>> How do you get to the fact that if it is True, it must be provable,
>>
>> It is simply the translation of the existential quantification:
>> ¬Goldbach ⇔ ∃x ∈ A, ¬P(x)
>>    into universal quantification:
>> Goldbach ⇔ ∀x ∈ A, P(x)
>>
>
> Which doesn't say what you think it says.
>
where A is (even whole numbers greater than 2)
and P is Provably meet the Goldbach criteria

You agreed with this:
¬Goldbach ⇔ ∃x ∈ A, ¬P(x)

Math correctly transforms it into this
Goldbach ⇔ ∀x ∈ A, P(x)

⇔ if and only if;
https://en.wikipedia.org/wiki/List_of_logic_symbols

Goldbach if and only if
all elements of the set of
(even whole numbers greater than 2)
Provably meet the Goldbach criteria

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer