On 1/23/23 6:39 PM, olcott wrote:
> On 1/23/2023 4:54 PM, Richard Damon wrote:
>> On 1/23/23 5:23 PM, olcott wrote:
>>> On 1/23/2023 10:51 AM, Richard Damon wrote:
>>>> On 1/23/23 10:18 AM, olcott wrote:
>>>>> On 1/20/2023 4:54 PM, Richard Damon wrote:
>>>>>> On 1/20/23 5:16 PM, olcott wrote:
>>>>>>> On 1/20/2023 4:09 PM, Richard Damon wrote:
>>>>>>>> On 1/20/23 5:02 PM, olcott wrote:
>>>>>>>>> On 1/20/2023 2:46 PM, Richard Damon wrote:
>>>>>>>>>> On 1/20/23 2:31 PM, olcott wrote:
>>>>>>>>>>> On 1/19/2023 8:34 PM, Richard Damon wrote:
>>>>>>>>>>>> On 1/19/23 2:12 PM, olcott wrote:
>>>>>>>>>>>>> On 1/17/2023 5:44 PM, Richard Damon wrote:
>>>>>>>>>>>>>> On 1/17/23 11:39 AM, olcott wrote:
>>>>>>>>>>>>>>> On 1/16/2023 7:51 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>> No, because I am showing that G is TRUE, not PROVABLE.
>>>>>>>>>>>>>>>> Truth can use infinte sets oc connections, proofs can't.
>>>>>>>>>>>>>>>> Only YOU have perposed that we think about infinite proofs.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Formal systems cannot ever use infinite connections from
>>>>>>>>>>>>>>> their
>>>>>>>>>>>>>>> expressions of language to their truth maker axioms thus
>>>>>>>>>>>>>>> eliminating
>>>>>>>>>>>>>>> these from consideration as any measure of true "in the
>>>>>>>>>>>>>>> system".
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Source? or is this just another of your made up "Facts"
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> You can't even remember that you said this?
>>>>>>>>>>>>
>>>>>>>>>>>> No, I said they can't have infinite PROOFS, not infinite
>>>>>>>>>>>> connections to Truth.
>>>>>>>>>>>>>
>>>>>>>>>>>>>> WHERE in the definition of a "Formal System" does it say
>>>>>>>>>>>>>> that the connecti0on must be finite.
>>>>>>>>>>>>>
>>>>>>>>>>>>> You said that formal system cannot have infinite proofs.
>>>>>>>>>>>>> Did you change your mind?
>>>>>>>>>>>>
>>>>>>>>>>>> Right ***PROOF*** not ***TRUTH***
>>>>>>>>>>>>
>>>>>>>>>>>> Truth can be based on an infinite chain of connections,
>>>>>>>>>>>> proofs can not.
>>>>>>>>>>>
>>>>>>>>>>> Truth *in a formal system* cannot be based on infinite
>>>>>>>>>>> connections because formal systems are not allowed to have
>>>>>>>>>>> infinite connections.
>>>>>>>>>>
>>>>>>>>>> Says Who ***FOR TRUTH***
>>>>>>>>>>
>>>>>>>>>> You reference does not provide that data, so I guess you are
>>>>>>>>>> just making it up, and thus showing you to be a LIAR.
>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> Haskell Curry establishes that truth in a theory (AKA formal
>>>>>>>>>>> system) is anchored in the elementary theorems (AKA axioms)
>>>>>>>>>>> of this formal system.
>>>>>>>>>>
>>>>>>>>>> Right, ANCHORED TO, not limited to. Statments other than the
>>>>>>>>>> elementary theorems are True, and they are true if they have a
>>>>>>>>>> connection (not limited to finite) to these Truths.
>>>>>>>>>>
>>>>>>>>>> Where does he say True statements must have a FINITE
>>>>>>>>>> connection to the elementary theorems.
>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> A theory (over (f) is defined as a conceptual class of these
>>>>>>>>>>> elementary
>>>>>>>>>>> statements. Let::t be such a theory. Then the elementary
>>>>>>>>>>> statements
>>>>>>>>>>> which belong to ::t we shall call the elementary theorems
>>>>>>>>>>> of::t; we also
>>>>>>>>>>> say that these elementary statements are true for::t. Thus,
>>>>>>>>>>> given ::t,
>>>>>>>>>>> an elementary theorem is an elementary statement which is
>>>>>>>>>>> true. A theory
>>>>>>>>>>> is thus a way of picking out from the statements of (f a certain
>>>>>>>>>>> subclass of true statements.
>>>>>>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>>>>>>>>
>>>>>>>>>>> Perhaps you believe that you are enormously much brighter
>>>>>>>>>>> than Haskell Curry ?
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> You don't understand what he is saying,
>>>>>>>>>>
>>>>>>>>>> He is saying these statements are True in F, as a given.
>>>>>>>>>
>>>>>>>>> Wrongo !!!
>>>>>>>>>
>>>>>>>>>     The terminology which has just been used implies that the
>>>>>>>>>     elementary statements are not such that their truth and
>>>>>>>>>     falsity are known to us without reference to::t.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>> Right, they aren't just true in the Statement class, but are
>>>>>>>> only considerdd true because we are in the Theory F.
>>>>>>>>
>>>>>>>
>>>>>>> F is not the theory T is the theory.
>>>>>>>
>>>>>>>
>>>>>>
>>>>>>
>>>>>> Red Herring.
>>>>>>
>>>>>> F is the Theory in Godels descussion.
>>>>>>
>>>>>
>>>>> https://en.wikipedia.org/wiki/Metamathematics
>>>>> LP := ~True(LP) is untrue yet that does not make it true.
>>>>>
>>>>> When we examine this at the meta level we escape the
>>>>> self-contradiction
>>>>> and can say that it is true that LP is untrue.
>>>>
>>>>
>>>> Excpet that untrue is not ~True() in classical logic, which makes
>>>> statements either True or False, or makes them Not a Truth Bearer,
>>>> which makes them not in the domain of the True predicate.
>>>>
>>>> You need to move to tri-value logic to do this, at which point you
>>>> loose the relationship that ~True(x) -> False(x)
>>>>
>>>
>>> True / false and not a truth bearer.
>>
>> That is your TRI-value logic.
>>
>
> It is true by logical necessity.
> Every expression of language must necessarily be
> true, false, neither true nor false.

Nope. You can also use a two level division like you actually talk about.

Statments are either Truth Bearers or they are Not

Truth Bearers are either True or they are False.

>
>
>>>
>>>> Note, most of mathematics is based on the two-value logic system.
>>>>
>>>>
>>>
>>> Thus forcing it to classify "not a truth bearer" incorrectly.
>>> If all you have is a hammer the unscrewing a screw becomes quite
>>> destructive.
>>
>> Nope, a "statement" can be well formed, and thus MUST be a "Truth
>> Bearer" or it isn't and is NOT a "Truth Bearer"
>>
>> By ignoring that mathematically defined statement ARE "Truth Bearers",
>> you logic system is just broken.
>>
>>>
>>>>>
>>>>> https://plato.stanford.edu/entries/tarski-truth/#195DefOff
>>>>>
>>>>> It looks like model theory is required to determine the truth of
>>>>> some mathematical expressions, this had it origins in Tarski's
>>>>> definition of truth.
>>>>>
>>>>> ∃n ∈ ℕ (N > 3)       // does not seem to need model theory
>>>>> ∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory
>>>>>
>>>>
>>>> ∃ is a symbol out of model theory, so hard to not need model theory.
>>>>
>>>> Quoting from your reference:
>>>>
>>>> Model theory by contrast works with three levels of symbol. There
>>>> are the logical constants ( = , ¬ , & for example), the variables
>>>> (as before), and between these a middle group of symbols which have
>>>> no fixed meaning but get a meaning through being applied to a
>>>> particular structure. The symbols of this middle group include the
>>>> nonlogical constants of the language, such as relation symbols,
>>>> function symbols and constant individual symbols. They also include
>>>> the quantifier symbols  ∀ and ∃, since we need to refer to the
>>>> structure to see what set they range over.
>>>
>>> I just showed how to explicitly specify what they range over: ∃n ∈ ℕ
>>
>> Which means you are using "Model Theory"
>>
>> Maybe you don't understand those words.
>
> Model theory is used to define things that are not otherwise defined.
> When they are otherwise defined there is no need for model theory.

So, you don't understand what Model Theory is (or mixing different
definitions)
>
>>>
>>>>> G is true in F iff it cannot be shown that G is true in F
>>>>>
>>>>
>>>> Nope, you don't understand what G is. The Definition of G in F does
>>>> NOT refer in any way determinable in F to the statement G.
>>>>
>>>
>>> ∃n ∈ ℕ (n > 3) // Is this true or false?
>>> How do you know?
>>
>> Simple, 4 exists (S(S(S(S(0)))), 4 > 3, 4 ∈ ℕ, thus the statement is
>> True. Like many (but not all) True statements, it can be proven.
>>
>>>
>>> Generically how does ascertain that that any logic expression is true or
>>> false?
>>
>> Note, "Ascertain" means you are talking about KNOWLEDGE, not Truth.
>>
>> Truth doesn't need to be ascertained to be true, it just is.
>>
>> It needs to be ascertained to be KNOWN.
>>
>> It is a TRUE statement that either all even numbers greater than 2 are
>> the sum of 2 primes or there exists at least one that is not. We don't
>> know which one of them is true right now, but we do know that one of
>> them is.
>>
>> This seems to be one of your core problems, confusing what can be
>> known to be true with what IS true.
>>
>>>
>>> Most generically an analytical expression of formal or natural
>>> language is only true if it has a semantic connection to its truth
>>> maker axioms.
>>
>> Right, but that connection might not be known, or might even be infinite.
>>
>> It is only KNOWLEDGE or PROOF that requires a finite connection.
>>>
>>> The "truth maker axioms" of natural language are the definition of the
>>> meaning of its words.
>>
>>
>> No, the accepted Truth Maker Axioms of the Theory (not what their
>> words mean in Natural Language) determine what is true.
>
> of natural language such as English
> of natural language such as English
> of natural language such as English
> of natural language such as English

Which has been proven unsuitable for logic.

Given the statement:

If this sentence is true, Peter Olcott is a moron.

This is a valid logical statement of natural language form.

By the meaning of the words, it is TRUE, because if the sentence IS
true, then by the DEFINITION of True, it must be actually True.

Thus, since it HAS been proven true, its implication must be correct,
and thus YOU ARE A MORON.

The "flaw" in the statement is that Natural Language isn't suitable to
fully express logic.

>
>>
>> Your reliance on "Natural Language" is what has actually been proven
>> to lead to problems.
>>
>
> The entire body of all analytical knowledge can only be expressed using
> language. Hardly any of this is currently expressed using formal
> language. All knowledge is necessarily true by definition.

Right, TECHNICAL/FORMAL language, not NATURAL language.

You are incorrect that hardly any of this is expressed using formal
language, and that is a major part of your problem. Words that are words
in "Natuaral Language" are frequently refined to a formal definition for
particular usage. If you don't understand that formal definition, or
even more important WHICH formal definition is needed for a given
statement, you won't understand it.

And All Knowledge being necessarily true is NOT a universal definition,
in fact, one of the problems of the study of knowledge is how to avoid
the introduction into "Knowledge" of things that we THINK are True but
are actually incorrect. We WANT everything that we (think we) know to be
actually true, but factually, since there IS a human element in the
aquisition of knowledge, there is a possibility of error and of thinking
we know something that isn't true.

>
>>>
>>> The truth maker axioms for the above expression is the definition of
>>> the ordered set of natural numbers:
>>>
>>> https://www.britannica.com/science/Peano-axioms
>>>
>>
>> You understand that Godel showed that under the Peano-axioms, he
>> proved that their exists truths that can not be proven.
>
> We can make the Gödel number of "I just ate some chicken" using the
> adjacent ASCII values. This too cannot be proven in the Peano-axioms.

Nope. You don't understand what Godel does. Not understanding something
does not make it not true, you are just serving your Herring with Red Sauce.

Yes, you can set up a system where you use Godel's math to create a
number that represents the statement "I just ate some chicken", but that
statement has nothing to do with Godel's proof.

The fact you can throw out Red Herring that means nothing doesn't
discount the proof, it just proves you don't understand what you are
talking about.

>
>> It becomes a consequence of the induction axiom that allows him to be
>> able to define the primative recursive relationship that shows that
>> you can not prove within the theory that no nmber exists that matches
>> that theory, and also create an extention to that theory (that is used
>> to create that relationship) that allows us to actually prove that
>> statment must be true, and also that no proof of this can exist in the
>> base theory.
>>
>
> Its a mere gimmick.
> He acknowledged that the Liar Paradox forms an equivalent proof.

No, it isn't. If you think it is, then SHOW that it is. But to do that,
you need to understand what he did and where he "just used a gimmick"

All your statments show is that you just don't understand what he is
saying and are such a pathological liar that you will make up excuses to
cover that.

>
>> The induction property that proves it only comes in the extension (the
>> meta-theory) and is not in the base theory,
>
> "This sentence is not true" is self-evidently untrue yet that does not
> make the sentence true within the scope of self-contradiction.

Right, but he doesn't say it does.

He used the FORM of the statement to build a completely different
statement about provability that IS a Truth Bearer, and isn't
self-contradictory.

>
>> so the base theory can't make the proof, but can evaluate for every
>> term, thus making the INFINITE chain that makes it true in the Theory.
>>
>
> It is not an infinite chain, it is simply that the sentence is true
> outside of the scope of self-contradiction and impossible to evaluate
> within the scope of self-contradiction.

Nope, it is an infinite chain.

Your problem is you don't even understand the actual statment you are
talking about and are talking about something which isn't what it
actually is.

>
>> Peano ARITHMATIC changed that induction axiom to a first order logic
>> definition, weaking what the theory can do, but allows it to appear to
>> be complete, but NOT express ALL the properties of the Natural Numbers.
>>
>
> Natural numbers themselves never had the property of provability.

????

There are MANY proofs of properties of natural numbers.

>
> *The five Peano axioms are*
> (1) Zero is a natural number.
> (2) Every natural number has a successor in the natural numbers.
> (3) Zero is not the successor of any natural number.
> (4) If the successor of two natural numbers is the same, then the two
>     original numbers are the same.
> (5) If a set contains zero and the successor of every number is in the
>     set, then the set contains the natural numbers.
>
> https://www.britannica.com/science/Peano-axioms

Right, and from those you can prove a lot of properties of the Natural
Numbers.

I think you are just showing you don't even understand what provability
means.

Godel shows that from those 5 axioms and the basic principles of logic
(and a lot of theorems proven from them) he can prove that there are
some statments which are True, that can not be proven.

>
>> I beleive that it becomes the (or at least one of the) largest logic
>> system that retains "Completeness" while sitll being "Consistent".
>> (But it can't prove itself to be consistent)
>>
>> This is of course, over you head, so you wil  either deny it or just
>> ignore the refuation and go off on some other tack.
>