Gödel's proof relies on self-contradictory expressions of language

[olcott]

Gödel's proof it relies upon a definition of incompleteness that
requires formal systems to be able to prove self-contradictory
expressions of language.

> Kurt Gödel's incompleteness theorem demonstrates that mathematics
> contains true statements that cannot be proved. His proof achieves
> this by constructing paradoxical mathematical statements. To see how
> the proof works, begin by considering the liar's paradox: "This
> statement is false." This statement is true if and only if it is
> false, and therefore it is neither true nor false.
>
> Now let's consider "This statement is unprovable." If it is provable,
> then we are proving a falsehood, which is extremely unpleasant and is
> generally assumed to be impossible. The only alternative left is that
> this statement is unprovable. Therefore, it is in fact both true and
> unprovable. Our system of reasoning is incomplete, because some truths
> are unprovable.
>
> https://www.scientificamerican.com/article/what-is-goumldels-proof/

"14 Every epistemological antinomy can likewise be used for a similar
undecidability proof." (Gödel 1931:40)

Does it make sense that formal systems are required to prove
epistemological antinomies (AKA self-contradictory expressions) or
should these expressions be rejected as non sequitur?

*The valid/sound deductive inference model seems to think that latter:*
∀F ∈ Formal_Systems ∀C ∈ WFF(F) ((F ⊢ C) ↔ True(F, C))
∀F ∈ Formal_Systems ∀C ∈ WFF(F) ((F ⊢ ¬C) ↔ False(F, C))
∀F ∈ Formal_Systems ∀C ∈ WFF(F) (((F ⊬ C) ∧ (F ⊬ ¬C)) ↔ NonSequitur(F, C))

*Non Sequitur*
https://en.wikipedia.org/wiki/Formal_fallacy)
In philosophy, a formal fallacy, deductive fallacy, logical fallacy or
non sequitur[1] (Latin for "it does not follow")

By simply disallowing symbolic logic to diverge from the valid/sound
deductive inference model Gödel Incompleteness and Tarski Undefinability
cease to exist.

*Gödel, Kurt 1931*
*On Formally Undecidable Propositions of Principia Mathematica*
*And Related Systems*

https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf)

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

[Richard Damon]

On 5/7/23 1:43 PM, olcott wrote:
> Gödel's proof it relies upon a definition of incompleteness that
> requires formal systems to be able to prove self-contradictory
> expressions of language.
>
> > Kurt Gödel's incompleteness theorem demonstrates that mathematics
> > contains true statements that cannot be proved. His proof achieves
> > this by constructing paradoxical mathematical statements. To see how
> > the proof works, begin by considering the liar's paradox: "This
> > statement is false." This statement is true if and only if it is
> > false, and therefore it is neither true nor false.
> >
> > Now let's consider "This statement is unprovable." If it is provable,
> > then we are proving a falsehood, which is extremely unpleasant and is
> > generally assumed to be impossible. The only alternative left is that
> > this statement is unprovable. Therefore, it is in fact both true and
> > unprovable. Our system of reasoning is incomplete, because some truths
> > are unprovable.
> >
> > https://www.scientificamerican.com/article/what-is-goumldels-proof/
>
> "14 Every epistemological antinomy can likewise be used for a similar
> undecidability proof." (Gödel 1931:40)
>
> Does it make sense that formal systems are required to prove
> epistemological antinomies (AKA self-contradictory expressions) or
> should these expressions be rejected as non sequitur?
>

Note, Consistant Formal Systems will reject actual epistemological
antinomies as non-Truth Bearing, and thus your premise is incorrect.

Formal Systems, to be consistent, only need to be able to prove every
True statement, and disprove every False statement, since BY DEFINITION,
an epistemological antinomy can neither be True or False, a Formal Logic
system doesn't need (and in fact CAN'T) prove or disprove an
epistemological antinomy, because such a statement won't be a Truth
Bearer, and thus neither True or False.

The thing you seem to be too stupid to understand is that Godel doesn't
use the Liar's paradox in its paradox form where it IS an
epistemological antinomy, but has transformed it from being about the
truth of the statement (and thus the antinomy) to a statement about the
provability of the statement, which breaks the paradox.

The

> *The valid/sound deductive inference model seems to think that latter:*
> ∀F ∈ Formal_Systems ∀C ∈ WFF(F) ((F ⊢ C)  ↔ True(F, C))
> ∀F ∈ Formal_Systems ∀C ∈ WFF(F) ((F ⊢ ¬C) ↔ False(F, C))
> ∀F ∈ Formal_Systems ∀C ∈ WFF(F) (((F ⊬ C) ∧ (F ⊬ ¬C)) ↔ NonSequitur(F, C))

Wrong, unless you mean COMPETE FORMAL SYSTEM,

Replace the "Prove" symbol, with the "Establishes" relationship, which
changes the requirement from a finite set of steps, to any (possibly
infinite) set of sets, and the statment holds for any formal system.

C is True in F, if there is a (possibly infinite) sequence of steps in F
from its Truth Makers

You are missing the fact that it is shown that it is possible for a
statement C to be TRUE, because there is a (possibly infinte) chain of
semantic connections from the Truth Makers of the system. through valid
logical inferances, to the statement C, but there might not be a valid
PROOF of the statement, which is a FINITE chain of semantic connections
from the Truth Makers of the system through valid logical inferences.

>
> *Non Sequitur*
> https://en.wikipedia.org/wiki/Formal_fallacy)
> In philosophy, a formal fallacy, deductive fallacy, logical fallacy or
> non sequitur[1] (Latin for "it does not follow")
>
> By simply disallowing symbolic logic to diverge from the valid/sound
> deductive inference model Gödel Incompleteness and Tarski Undefinability
> cease to exist.

Nope, because it DOESN'T, but only because you don't understand what a
sound or valid proof actually is, or the defintion of Truth.

>
> *Gödel, Kurt 1931*
> *On Formally Undecidable Propositions of Principia Mathematica*
> *And Related Systems*
>
> https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf)
>

Which just shows that you don't understand what you are reading.

Please point out to the step where he used an UNSOUND or INVALID logical
step. Not just where he says words that you disagree with, but performs
an actual logical step that is incorrect.

Your silence on this shows that you don't have a leg to stand on because
you are the one that doesn't hold to sound and valid logical inference
rules.

[olcott]

Try and find an example of this.

Tarski used the actual Liar Paradox to derive his comparable proof.
https://plato.stanford.edu/entries/goedel-incompleteness/#TarTheUndTru

> Please point out to the step where he used an UNSOUND or INVALID logical
> step. Not just where he says words that you disagree with, but performs
> an actual logical step that is incorrect.
>
> Your silence on this shows that

You have not yet understood the prerequisites.

> you don't have a leg to stand on because
> you are the one that doesn't hold to sound and valid logical inference
> rules.

[Richard Damon]

What DO you correctly understand. You haven't shown ANY actual knowledge
of the paper, but only the ability to pull pieces out of it totally out
of context.

>
> Tarski used the actual Liar Paradox to derive his comparable proof.
> https://plato.stanford.edu/entries/goedel-incompleteness/#TarTheUndTru

Nope, he DERIVES the Liar Paradox as a statement that must be true if a
"Definition of Truth" per his rules exists (that is, if there is a
deterministic method that always determines in a finite number of
operations if a given statement is true or false).

Note that he says what he used to get to that step, it isn't a "premise"
of his proof, it is a conclusion derived as part of a proof by
contradiction, which you don't seem to understand as even being possible.

That the assumption of a definition exists leads to an impossible
conclusion shows that such a definition can not exist.

>
>> Please point out to the step where he used an UNSOUND or INVALID
>> logical step. Not just where he says words that you disagree with, but
>> performs an actual logical step that is incorrect.
>>
>> Your silence on this shows that
>
> You have not yet understood the prerequisites.

What Prerequisites? That we accept your flawed logic?

YOU are the one making the claim, so YOU have the burden of proof, and
untill you provide it, the claim that you haven't established you claim
is correct, and it is also correct to call you claim that you HAVE
established it a LIE.

You are just showing that you don't understand how formal logic works.
If you want to claim the fundamental logic that it is based on is some
how flawed, you first need to actually demonstrate the actual error, and
then you need to show that your alternate "logic" can actualy perform in
the desired fields.

You have just failed to do any of that.

In fact, the mere fact that you are claiming to rely on an inappropriate
field of philosophy shows how off you are. Epistimoligy doesn't tell us
by what method things are ACTUALLY True or False, but talks about how
*WE* can *KNOW* if something is true of false.

Unless you are going to try to argue that if no one know the truth of
something, that thing doesn't actually HAVE a Truth value, the fact that
you are confusing Knowledge with Truth shows you are just too stupid to
understand what you are actually talking about.

[olcott]

On 5/7/2023 3:00 PM, Richard Damon wrote:

Try and find an example of this. (I pasted it in the wrong place)

[olcott]

You have Tarski correctly. Tarski rejects that truth can be correctly
formalized because he can't prove that a non-truth bearer is true.

That is like rejecting Geometry upon the failure to prove that a circle
is a square.

[Richard Damon]

So, can YOU prove that a non-truth bearer is true?

Because he shows that if truth can be formalized, then the liar's
paradox MUST be a True statement.

>
> That is like rejecting Geometry upon the failure to prove that a circle
> is a square.
>

Nope, just shows your stupidity and not understanding the argument.

It seems the concept of the proof by contradiction is beyond you. He
shows that there can not be a "Definition of Truth" (per his definition
of what that is) because if there was, then it would require that the
liar's paradox be true, becuase it can be proven using the existance of
that definition.

That would be like in Geometry trying to add a new axiom/assumption to
the system, but finding out if you do that there exists a circle that is
a square.

[Richard Damon]

Its definitional. In a formal system, a statement is only a member of
that system if it is either True or False. A statement is only part of a
formal system if it is established true or false in that system.

Since Truth is based on having a (possibly infinite) series of valid and
sound inferences from the Truth Makers of a system to the Statement (or
to its complement for falsehood), and a self-referential statement can
NEVER have all of its premises established, since it is one of them,
that can never happen.

[olcott]

Thus when we hypothesize that a formal system is powerful enough to have
its own provability predicate in F then when G asserts its own
unprovability in F we understand that this would require a sequence of
inference steps in F that prove that they themselves do not exist.

> Since Truth is based on having a (possibly infinite) series of valid and
> sound inferences from the Truth Makers of a system to the Statement (or
> to its complement for falsehood), and a self-referential statement can
> NEVER have all of its premises established, since it is one of them,
> that can never happen.

Alternatively if an expression involves an infinite sequence of steps
then this sequence never resolves to true or false.

This sentence is not true.
It is not true about what?
It is not true about being not true.
It is not true about being not true about what?
It is not true about being not true about being not true...

[Richard Damon]

But it can.

For instance, if it requires the testing of all natural numbers, each
one taking a finite number of steps, that that DOES resolve in an
infinite number of steps, and thus IS established in the logic system.

If there is no positive number that when put into the divide by 2 or 3x
+ 1 steps that never reaches 1, then the Collatz conjecture is true,
even if it can only be verifies by testing EVERY number (that is an
infinite number of them). Things like the Collatz conjecture MUST be
Truth Beares, as either a number with the property exists or it doesn't,
so if no such number exists, we can say it is true that no such number
exists (or all numbers do the opposite) even if we can't formally prove
that result.

The fact that you tiny brain can't handle how infinity works doesn't
keep infinite steps, that actually arrive at the connection, from
establishing a statement as True.

>
> This sentence is not true.
> It is not true about what?
> It is not true about being not true.
> It is not true about being not true about what?
> It is not true about being not true about being not true...
>

First, you are not working from the Truth Makers, so isn't even a proper
sequence of steps (You have shown this previous error in understanding
the definition of a semantic connection).

Second, this shows why Non-Truth-Bearers don't ever connect, even in an
infinite number of steps, between the truth makers and the statement.

Thirdly, this statement NEVER gets to the end, not even after an
infinite number of steps, so does't connect in an infinite number of
steps. There is a difference between ACTUALLY connecting, but needing an
infinite number of steps, and never connecting.

So, your use of a straw man just show how little you understand what you
are talking about.

Just like two parrallel lines (in plane Geometry) NEVER meet, but always
have the same distance between them, even "at infinity" (which you can
never actually get to in ordinary plane Geometry, it is only a limit,
just like when we are talking about domains like the Reals.

That differs from things like 1/x and 0, which DO meet "at infinity".

[olcott]

Count to infinity is not computable.

> If there is no positive number that when put into the divide by 2 or 3x
> + 1 steps that never reaches 1, then the Collatz conjecture is true,
> even if it can only be verifies by testing EVERY number (that is an
> infinite number of them). Things like the Collatz conjecture MUST be
> Truth Beares, as either a number with the property exists or it doesn't,
> so if no such number exists, we can say it is true that no such number
> exists (or all numbers do the opposite) even if we can't formally prove
> that result.
>

Maybe for these kind of things there are truths that can never be verified.

> The fact that you tiny brain can't handle how infinity works doesn't
> keep infinite steps, that actually arrive at the connection, from
> establishing a statement as True.
>
>>
>> This sentence is not true.
>> It is not true about what?
>> It is not true about being not true.
>> It is not true about being not true about what?
>> It is not true about being not true about being not true...
>>
>
> First, you are not working from the Truth Makers, so isn't even a proper
> sequence of steps (You have shown this previous error in understanding
> the definition of a semantic connection).
>

Self-contradictory expressions are not truth bearers you can either
comprehend that or fail to comprehend that.

> Second, this shows why Non-Truth-Bearers don't ever connect, even in an
> infinite number of steps, between the truth makers and the statement.
>

OK so we agree on this.

> Thirdly, this statement NEVER gets to the end, not even after an
> infinite number of steps, so does't connect in an infinite number of
> steps. There is a difference between ACTUALLY connecting, but needing an
> infinite number of steps, and never connecting.
>

OK so we agree on this too.

You still do not acknowledge that you understand that when G asserts its
own unprovability in F proving G requires a sequence of inference steps

in F that prove that they themselves do not exist.

*x := y means x is defined to be another name for y*
https://en.wikipedia.org/wiki/List_of_logic_symbols
This seem to be the only way that we get actual self-reference
all of the textbooks merely approximate self-reference with ↔

∃G ∈ F (G := (F ⊬ G))
There exists a G in F that proves its own unprovabilty in F
Maybe you have a more accurate way to translate the symbols.

> So, your use of a straw man just show how little you understand what you
> are talking about.
>
> Just like two parrallel lines (in plane Geometry) NEVER meet, but always
> have the same distance between them, even "at infinity" (which you can
> never actually get to in ordinary plane Geometry, it is only a limit,
> just like when we are talking about domains like the Reals.
>
> That differs from things like 1/x and 0, which DO meet "at infinity".

[Richard Damon]

Didn't say it was, but not all things that are True are Computable.

>
>> If there is no positive number that when put into the divide by 2 or
>> 3x + 1 steps that never reaches 1, then the Collatz conjecture is
>> true, even if it can only be verifies by testing EVERY number (that is
>> an infinite number of them). Things like the Collatz conjecture MUST
>> be Truth Beares, as either a number with the property exists or it
>> doesn't, so if no such number exists, we can say it is true that no
>> such number exists (or all numbers do the opposite) even if we can't
>> formally prove that result.
>>
>
> Maybe for these kind of things there are truths that can never be verified.

Right, which means can't be proven.

That shows your definition, where all truth must be provable, is INCORRECT.

>
>> The fact that you tiny brain can't handle how infinity works doesn't
>> keep infinite steps, that actually arrive at the connection, from
>> establishing a statement as True.
>>
>>>
>>> This sentence is not true.
>>> It is not true about what?
>>> It is not true about being not true.
>>> It is not true about being not true about what?
>>> It is not true about being not true about being not true...
>>>
>>
>> First, you are not working from the Truth Makers, so isn't even a
>> proper sequence of steps (You have shown this previous error in
>> understanding the definition of a semantic connection).
>>
>
> Self-contradictory expressions are not truth bearers you can either
> comprehend that or fail to comprehend that.

Right, But saying you aren't provable isn't being "Self-Contradictory".

You just are showing you don't understand that actual meaning of that word.

>
>> Second, this shows why Non-Truth-Bearers don't ever connect, even in
>> an infinite number of steps, between the truth makers and the statement.
>>
>
> OK so we agree on this.
>
>> Thirdly, this statement NEVER gets to the end, not even after an
>> infinite number of steps, so does't connect in an infinite number of
>> steps. There is a difference between ACTUALLY connecting, but needing
>> an infinite number of steps, and never connecting.
>>
>
> OK so we agree on this too.
>
> You still do not acknowledge that you understand that when G asserts its
> own unprovability in F proving G requires a sequence of inference steps
> in F that prove that they themselves do not exist.

You are still stuck on your lie that you can assert as true something
which can not be proven in that system.

You just agreed that some true statements can not be "verified"

>
> *x := y means x is defined to be another name for y*
> https://en.wikipedia.org/wiki/List_of_logic_symbols
> This seem to be the only way that we get actual self-reference
> all of the textbooks merely approximate self-reference with ↔

So, you REALLY don't understand logic. "↔" doesn't mean anything like
self-reference, it means always has the same truth value, even if just
as a coencident, or because the two sides derive from a common source
(but aren't the same statement).

>
> ∃G ∈ F (G := (F ⊬ G))
> There exists a G in F that proves its own unprovabilty in F
> Maybe you have a more accurate way to translate the symbols.

No, the statements says that there exists a G in F that STATES that G in
not provable in F.

Where is the "Proves" operator in that statement?

G isn't "Proving" that it is unprovable, it is just asserting it.

Note also, this isn't Godel's G, so mostly irrelevant. In Godel's proof,
G (in F) is just a statement about there not existing a natural number
that satisfies a particular relationship. It is only in Meta-F that we
get any of the meaning about "proving" statements,

[olcott]

None-the-less

> as a coencident, or because the two sides derive from a common source
> (but aren't the same statement).
>
>>
>> ∃G ∈ F (G := (F ⊬ G))
>> There exists a G in F that proves its own unprovabilty in F
>> Maybe you have a more accurate way to translate the symbols.
>
> No, the statements says that there exists a G in F that STATES that G in
> not provable in F.

That sounds about right.

>
> Where is the "Proves" operator in that statement?
>
> G isn't "Proving" that it is unprovable, it is just asserting it.
>

States is a better word that proves or asserts.

> Note also, this isn't Godel's G, so mostly irrelevant. In Godel's proof,
> G (in F) is just a statement about there not existing a natural number
> that satisfies a particular relationship. It is only in Meta-F that we
> get any of the meaning about "proving" statements,
>

It matches this
...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ...(Gödel 1931:39-41)

>>
>>> So, your use of a straw man just show how little you understand what
>>> you are talking about.
>>>
>>> Just like two parrallel lines (in plane Geometry) NEVER meet, but
>>> always have the same distance between them, even "at infinity" (which
>>> you can never actually get to in ordinary plane Geometry, it is only
>>> a limit, just like when we are talking about domains like the Reals.
>>>
>>> That differs from things like 1/x and 0, which DO meet "at infinity".
>>
>

[Richard Damon]

IF there isn't an actual self-reference, you can't claim it to be
self-contradictory.

>
>> as a coencident, or because the two sides derive from a common source
>> (but aren't the same statement).
>>
>>>
>>> ∃G ∈ F (G := (F ⊬ G))
>>> There exists a G in F that proves its own unprovabilty in F
>>> Maybe you have a more accurate way to translate the symbols.
>>
>> No, the statements says that there exists a G in F that STATES that G
>> in not provable in F.
>
> That sounds about right.

So you argee?

>
>>
>> Where is the "Proves" operator in that statement?
>>
>> G isn't "Proving" that it is unprovable, it is just asserting it.
>>
>
> States is a better word that proves or asserts.
>
>> Note also, this isn't Godel's G, so mostly irrelevant. In Godel's
>> proof, G (in F) is just a statement about there not existing a natural
>> number that satisfies a particular relationship. It is only in Meta-F
>> that we get any of the meaning about "proving" statements,
>>
>
> It matches this
> ...We are therefore confronted with a proposition which asserts its own
> unprovability. 15 ...(Gödel 1931:39-41)

Right, we have a statement, IN META-F, that shows that G in F implies,
through logic in meta-F, that G is not provable in F.

Nothing "Self-Contradictory" about that, since it is perfectly valid for
G to be true in F, but not provable in F, but only in some more advance
meta-system (which shows that it is true in F).

[olcott]

*Stanford Encyclopedia of Philosophy Self-Reference*

Diagonal lemma.
Let S be a theory extending first-order arithmetic. For every formula
ϕ(x) there is a sentence ψ such that S ⊢ ψ ↔ ϕ⟨ψ⟩.

Here the notation S ⊢ α means that α is provable in the theory S, and
ϕ⟨ψ⟩ is short for ϕ(⟨ψ⟩). Assume a formula ϕ(x) is given that is intended
to express some property of sentences – truth, for instance. Then the
diagonal lemma gives the existence of a sentence ψ satisfying the
biimplication ψ ↔ ϕ⟨ψ⟩. The sentence ϕ⟨ψ⟩ can be thought of as
expressing that the sentence ψ has the property expressed by ϕ(x). The
biimplication thus expresses that ψ is equivalent to the sentence
expressing that ψ has property ϕ. One can therefore think of ψ as a
sentence expressing of itself that it has property ϕ. In the case of
truth, it would be a sentence expressing of itself that it is true. The
sentence ψ is of course not self-referential in a strict sense, but
mathematically it behaves like one.
https://plato.stanford.edu/entries/self-reference/

No one in math ever directly expresses actual self-reference, they only
approximate it not because actual-self-reference cannot be expressed
merely because it is unconventional to express actual self-reference.

*Here is actual self-reference*

x := y means x is defined to be another name for y

https://en.wikipedia.org/wiki/List_of_logic_symbols

∃G ∈ WFF(F) (G := ¬(F ⊢ G))
There exists G a WFF of F that states it is unprovable in F.

...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ... (Gödel 1931:39-41)

Gödel intended his actual G to be isomorphic to the above self-
referential expression.

>>
>>> as a coencident, or because the two sides derive from a common source
>>> (but aren't the same statement).
>>>
>>>>
>>>> ∃G ∈ F (G := (F ⊬ G))
>>>> There exists a G in F that proves its own unprovabilty in F
>>>> Maybe you have a more accurate way to translate the symbols.
>>>
>>> No, the statements says that there exists a G in F that STATES that G
>>> in not provable in F.
>>
>> That sounds about right.
>
> So you argee?
>

I was struggling to find the best wording and yours is better.

>>
>>>
>>> Where is the "Proves" operator in that statement?
>>>
>>> G isn't "Proving" that it is unprovable, it is just asserting it.
>>>
>>
>> States is a better word that proves or asserts.
>>
>>> Note also, this isn't Godel's G, so mostly irrelevant. In Godel's
>>> proof, G (in F) is just a statement about there not existing a
>>> natural number that satisfies a particular relationship. It is only
>>> in Meta-F that we get any of the meaning about "proving" statements,
>>>
>>
>> It matches this
>> ...We are therefore confronted with a proposition which asserts its
>> own unprovability. 15 ...(Gödel 1931:39-41)
>
> Right, we have a statement, IN META-F, that shows that G in F implies,
> through logic in meta-F, that G is not provable in F.
>

...We are therefore confronted with a proposition which asserts its own
unprovability. 15 ... (Gödel 1931:39-41)

Gödel intended his actual G to be isomorphic to the above self-
referential expression.

When we have a powerful F that can directly express the unprovability of
G in F (isomorphic to Gödel's G in PM) then this G is self-
contradictory.

> Nothing "Self-Contradictory" about that, since it is perfectly valid for
> G to be true in F, but not provable in F, but only in some more advance
> meta-system (which shows that it is true in F).

G cannot be satisfied in F therefore G is not true in F.

In mathematical logic, a formula [of a formal system] is satisfiable
[in this formal system] if it is true under some assignment of values
to its variables [in this formal system].
https://en.wikipedia.org/wiki/Satisfiability

>>
>>
>>>>
>>>>> So, your use of a straw man just show how little you understand
>>>>> what you are talking about.
>>>>>
>>>>> Just like two parrallel lines (in plane Geometry) NEVER meet, but
>>>>> always have the same distance between them, even "at infinity"
>>>>> (which you can never actually get to in ordinary plane Geometry, it
>>>>> is only a limit, just like when we are talking about domains like
>>>>> the Reals.
>>>>>
>>>>> That differs from things like 1/x and 0, which DO meet "at infinity".
>>>>
>>>
>>
>

[Richard Damon]

Right, we discover IN THE META that from the statement G we can prove
that a CONSEQUENCE of G (only demonstratable in the Meta) is that G has
the same truth value as its unprovability.

>
> Gödel intended his actual G to be isomorphic to the above self-
> referential expression.

Nope, you are over-simplifying things. The level of indirection is
important, as your direct statement first needs to be shown to be a
Truth Bearer. The mathematical statement is PROVABLE to be a Truth
Bearer by the properties of Mathematics, and THAT proof can be done in
F, which lets us say that G is TRUE in F.

The need to go into the meta to get that meaning is what makes it work.
As you have shown, if you can establish using just the system that a
statement asserts its own unprovabilty, then we can't prove that
statement to be a Truth Bearer or we have the self-contradiction, since
if it is false, we can prove it to be true, but because of that it must
be true, and that logic becomes a proof in the system that it is true,
so it must be false. Therefore, we can't prove it to be a Truth Bearer.

But, if the link to its assertion of unprovability requires going
outside F, then we break that cycle, as we didn't prove it "IN F", so it
can still be true.

>
> When we have a powerful F that can directly express the unprovability of
> G in F (isomorphic to Gödel's G in PM) then this G is self-
> contradictory.

Except that no such F can exist, since the relationship that G is
looking at is based on the full set of Truth Makers of F, and the ONLY
way to convert that relationship to the concept that G is expressing its
own unprovability is with the specific set of assigments of the Truth
Makers in F to specific numbers. That assignment can NOT be "in F",
without breaking F and making it have a non-finite set of Truth Makers.

>
>> Nothing "Self-Contradictory" about that, since it is perfectly valid
>> for G to be true in F, but not provable in F, but only in some more
>> advance meta-system (which shows that it is true in F).
>
> G cannot be satisfied in F therefore G is not true in F.

Why do you say that G cannnot be statisfied in F?

DO you even really know what that means? (Not just your (never) learned
by rote statement you make below).

>
> In mathematical logic, a formula [of a formal system] is satisfiable
> [in this formal system] if it is true under some assignment of values
> to its variables [in this formal system].
> https://en.wikipedia.org/wiki/Satisfiability

And since G is ALWAYS true in F, it meets that definition.

Remember, G is the statement that there exists no natural number that
satisfies a specific relationship.

It has been shown that no such number CAN exist, therefore G is TRUE.

ALWAYS.

Thus, G is statisified.

[olcott]

Not at all. I boiled them down to their barest essence. Gödel's G was
intended to be and is isomorphic to a self-contradictory expression.

This is dead obvious in Tarski's comparable proof where he flat out
states that he is anchoring his proof in the actual Liar Paradox.

[Richard Damon]

So, you are just PROVING that you don't understand how logic actually
works and are falling for your own Straw man Error.

The error has been pointed out to you many times, but it seems you are
too Stupid AND Ignorant to be able to understand why you are wrong.

It is clear that you don't even understand the meaning of "Isomorphic",
as you are misusing it.

Also, as has been pointed out, you don't understand what Tarski is
saying, again likely because you have brainwashed yourself so much with
your rote learning of your own errors, that you refuse to see what is
actaully there.

All you are doing is cementing you place as a laughing stock of the
logical community. Your ideas are DEAD, as you will also be soon.

[olcott]

On 5/10/2023 6:24 AM, Richard Damon wrote:
> On 5/9/23 10:46 PM, olcott wrote:
>> On 5/9/2023 8:30 PM, Richard Damon wrote:
>>> On 5/9/23 8:18 PM, olcott wrote:
>>
>>>> Gödel intended his actual G to be isomorphic to the above self-
>>>> referential expression.
>>>
>>> Nope, you are over-simplifying things.
>>
>> Not at all. I boiled them down to their barest essence. Gödel's G was
>> intended to be and is isomorphic to a self-contradictory expression.
>>
>> This is dead obvious in Tarski's comparable proof where he flat out
>> states that he is anchoring his proof in the actual Liar Paradox.
>>
>>
>
> So, you are just PROVING that you don't understand how logic actually
> works and are falling for your own Straw man Error.
>

No I am proving to have a deeper understanding of these things than most
others have.

When we understand that he sums up his own proof as

...We are therefore confronted with a proposition which asserts its
own unprovability. 15 ... (Gödel 1931:39-41)

Then we can see that he intended his G to be isomorphic to a G that
...which asserts its own unprovability. 15 ... (Gödel 1931:39-41)

and he intended this be self contradictory
...14 Every epistemological antinomy can likewise be used for a
similar undecidability proof...(Gödel 1931:39-41)

Here is how G asserts its own unprovability in F is self-contradictory:
Proving G requires a sequence of inference steps in F that prove that

they themselves do not exist.

That you continue to fail to understand this is not my mistake it is
your mistake.

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems

https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf


Since Tarski directly stated that he is anchoring his comparable proof
in the actual Liar Paradox I have provided sufficient support for my
position.

Ever since 1936 the world has been convinced that the notion of Truth
is not formally definable entirely on the basis that Tarski could not
prove that the non-truth bearer of the Liar Paradox is true.

[Richard Damon]

On 5/10/23 10:27 AM, olcott wrote:
> On 5/10/2023 6:24 AM, Richard Damon wrote:
>> On 5/9/23 10:46 PM, olcott wrote:
>>> On 5/9/2023 8:30 PM, Richard Damon wrote:
>>>> On 5/9/23 8:18 PM, olcott wrote:
>>>
>>>>> Gödel intended his actual G to be isomorphic to the above self-
>>>>> referential expression.
>>>>
>>>> Nope, you are over-simplifying things.
>>>
>>> Not at all. I boiled them down to their barest essence. Gödel's G was
>>> intended to be and is isomorphic to a self-contradictory expression.
>>>
>>> This is dead obvious in Tarski's comparable proof where he flat out
>>> states that he is anchoring his proof in the actual Liar Paradox.
>>>
>>>
>>
>> So, you are just PROVING that you don't understand how logic actually
>> works and are falling for your own Straw man Error.
>>
>
> No I am proving to have a deeper understanding of these things than most
> others have.

Nope, just that you are so dumb you don't know what you don't understand.

>
> When we understand that he sums up his own proof as
>    ...We are therefore confronted with a proposition which asserts its
>    own unprovability. 15 ... (Gödel 1931:39-41)

No, that isn't a "summary" of his proof, but a STEP in the proof.

From G in F, we can prove in Meta-F, that G

>
> Then we can see that he intended his G to be isomorphic to a G that
> ...which asserts its own unprovability. 15 ... (Gödel 1931:39-41)

Nope, you don't seem to understand what a chain of logic is.

>
> and he intended this be self contradictory
> ...14 Every epistemological antinomy can likewise be used for a
> similar undecidability proof...(Gödel 1931:39-41)

No, it just shows that you have no idea what he is talking about, or the
meaning of the words are that you are using.

YOU are the one guilt of trying to put words in other peoples mouthes,
to then try to disprove those altered words.

In other words, your whole arguement is bassed on asserting a Strawman
Falacy.

>
> Here is how G asserts its own unprovability in F is self-contradictory:
> Proving G requires a sequence of inference steps in F that prove that
> they themselves do not exist.

Except that the ACTUAL statement of G isn't in any way
"Self-Contradictiory", so your "Isomorphism" / "Equivalence" is just
your pathologica lie.

>
> That you continue to fail to understand this is not my mistake it is
> your mistake.

Nope, You are the one making the mistake.

It is a demonstarted principle, that if EVERYONE disagrees with you, you
are likely wrong. Even the greatest who came up with new ideas, were
able to get at least a FEW of the smartest to understand what they were
talking about.

You have only gotten agreement from a couple at the bottom, and people
you have "tricked" by the misuse of words, and who don't actually agree
with your ideas.

>
> Gödel, Kurt 1931.
> On Formally Undecidable Propositions of Principia Mathematica And
> Related Systems
>
> https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf
>
> Since Tarski directly stated that he is anchoring his comparable proof
> in the actual Liar Paradox I have provided sufficient support for my
> position.
>

Nope, In fact, he is using the non-truth bearing of the Liars Paradox
for his proof,

You are again showing your stupiity.

> Ever since 1936 the world has been convinced that the notion of Truth
> is not formally definable entirely on the basis that Tarski could not
> prove that the non-truth bearer of the Liar Paradox is true.
>

Nope, and you are just proving that YOU have no idea what Truth actually
is, or what Logic actually is.

You are just proving your utter ignorance and stupidity with your
pathological lying about these things.

Your eternal destiny is to be known (for as long as you are remembered)
as the ignorant pathologica liar who totally misunderstands how logic works.

YOU FAIL.

[olcott]

I say that Tarski is using the Liar Paradox as the basis of his proof
and you say no I am wrong the truth is that Tarski is using the Liar
Paradox as the basis of his proof?

[olcott]

On 5/10/2023 6:29 PM, Richard Damon wrote:
> On 5/10/23 10:27 AM, olcott wrote:
>> On 5/10/2023 6:24 AM, Richard Damon wrote:
>>> On 5/9/23 10:46 PM, olcott wrote:
>>>> On 5/9/2023 8:30 PM, Richard Damon wrote:
>>>>> On 5/9/23 8:18 PM, olcott wrote:
>>>>
>>>>>> Gödel intended his actual G to be isomorphic to the above self-
>>>>>> referential expression.
>>>>>
>>>>> Nope, you are over-simplifying things.
>>>>
>>>> Not at all. I boiled them down to their barest essence. Gödel's G was
>>>> intended to be and is isomorphic to a self-contradictory expression.
>>>>
>>>> This is dead obvious in Tarski's comparable proof where he flat out
>>>> states that he is anchoring his proof in the actual Liar Paradox.
>>>>
>>>>
>>>
>>> So, you are just PROVING that you don't understand how logic actually
>>> works and are falling for your own Straw man Error.
>>>
>>
>> No I am proving to have a deeper understanding of these things than most
>> others have.
>
> Nope, just that you are so dumb you don't know what you don't understand.
>

I say that incorrectly. I have a deeper understanding OF THE ESSENCE OF
HIS PROOF. It is commonly understood that Gödel's actual proof is
isomorphic to {a proposition which asserts its own unprovability}. It is
also commonly understood that this is self-contradictory.

What is not commonly understood is that formal systems that cannot prove
self-contradictory expressions are not in any way deficient.

[Richard Damon]

YOU have been saying that because Tarski, erroneosly, finds that logic
can't prove the liar's paradox, his proof must be wrong, i.e there is
no definition of Truth.


I say that his proof shows that if a Definition of Truth (meaning a
determinate procedure to determine if any statement is true or false)
existed, then it would be possible to prove that the liar's paradox is a
true statement, which is NOT correct, thus there can not exist a
definition of truth.


You words are implying that we call systems incomplete or truth
undefinable because they can't resolve the Liar's Paradox.

The actual fact is that truth is undefinable, because such a definition
of truth creates the INCORRECT determination of a resolution of the
Liar's Paradox.

You are just showing you lack of understanding of what people are
saying, and a refusal to listen, (because you are afraid to learn you
are wrong) which leads to your stupidity.

[Richard Damon]

But that isn't what his proof is about,

You just have a deeper MISunderstanding of what he is saying because you
don't understand what he is saying at all, but are just trying to
understand the altered strawman arguement that you think you can understand,

YOU FAIL.

None of thes proofs are about a system being deficient for not being
able to resolve a self-contradictory statement or a non-truth-bearer.
The fact you think they are just shows that you are misunderstanding the
proofs.

Godel shows a statement, THAT IS TRUE, (and thus CAN'T be
self-contradictory) that can not be proven in that system. This meets
the DEFINTION of "Incompleteness" in Logic.

Tarski shows that there are some statements, that have a truth value,
that we can not know that truth value, because the mere existance of a
"Definition" (deterministic method) to test them with leads to the
contradiction that the Liar's Paradox must be True..

The problem isn't that he expects that a system should be able to
resolve the Liar's Paradox, but that a "Definition of Truth" leads to a
claimed resolution, namely that the Liar's Paradox IS True (which means
it also must be False). He shows that a "Definition of Truth" turns the
Liar's Paradox from a non-truth-bearer into a Truth Bearer that is True
(and thus also False).

Your failure to understand this just shows your stupidity.

[olcott]

On 5/11/2023 6:37 AM, Richard Damon wrote:
> On 5/10/23 11:01 PM, olcott wrote:
>> On 5/10/2023 6:29 PM, Richard Damon wrote:
>
>>>>
>>>> Gödel, Kurt 1931.
>>>> On Formally Undecidable Propositions of Principia Mathematica And
>>>> Related Systems
>>>>
>>>> https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf
>>>>
>>>> Since Tarski directly stated that he is anchoring his comparable proof
>>>> in the actual Liar Paradox I have provided sufficient support for my
>>>> position.
>>>>
>>>
>>> Nope, In fact, he is using the non-truth bearing of the Liars Paradox
>>> for his proof,
>>>
>>
>> I say that Tarski is using the Liar Paradox as the basis of his proof
>> and you say no I am wrong the truth is that Tarski is using the Liar
>> Paradox as the basis of his proof?
>>
>
> YOU have been saying that because Tarski, erroneosly, finds that logic
> can't prove the liar's paradox, his proof must be wrong, i.e  there is
> no definition of Truth.
>
>
> I say that his proof shows that if a Definition of Truth (meaning a
> determinate procedure to determine if any statement is true or false)
> existed, then it would be possible to prove that the liar's paradox is a
> true statement,

Where did you get that nutty idea?

[olcott]

He does this in the same way that this Liar Paradox is true:
This sentence is not true: "This sentence is not true"

It is not that the actual Liar Paradox is true it is only when the Liar
Paradox is applied to itself that it becomes true.

G states that it is unprovable in F and is unprovable in F because it is
self-contradictory in F.

When we test the same statement in metamathematics then it becomes true
because it escapes the self-contradiction the same way that the above
Liar Paradox escaped the self-contradiction.

The prerequisite to attaining this much deeper understanding of the
essence of Gödel's proof is
(1) Understanding that Gödel's G was intended to be and is isomorphic to
{a proposition which asserts its own unprovability [in PM]}

(2) Thus making Gödel's G isomorphic to a self-contradictory expression.

> Tarski shows that there are some statements, that have a truth value,
> that we can not know that truth value, because the mere existance of a
> "Definition" (deterministic method) to test them with leads to the
> contradiction that the Liar's Paradox must be True..
>

When you understand the essence of his proof you will understand that
Tarski's metatheory is merely applying the Liar Paradox to itself
outside of the scope of self-contradiction.

"This sentence is not true" is not a truth bearer.
This sentence is not true: "This sentence is not true" is true because
the inner sentence is not a truth bearer.

> The problem isn't that he expects that a system should be able to
> resolve the Liar's Paradox, but that a "Definition of Truth" leads to a
> claimed resolution, namely that the Liar's Paradox IS True (which means
> it also must be False). He shows that a "Definition of Truth" turns the
> Liar's Paradox from a non-truth-bearer into a Truth Bearer that is True
> (and thus also False).
>
> Your failure to understand this just shows your stupidity.
>

In this case it was Tarski's failure to understand.
Only when we boil things down to their barest essence (as I have had to
do as a software engineer for almost four decades) do we see these
things in their true light.

[Richard Damon]

FROM HIS PROOF!

He first does a lot of work to establish a number of properties.

He then makes a trial assumption that there could be a "Definition of
Truth" per his meaning.

He then goes through a few logical steps, and comes out with a proof
that the Liar's Paradox is True, given the assumption of a Definition of
Truth.

Since he knows this is impossible, he concludes that there can not be a
"Defintion of Truth".


You can't see that in his proof?

of course, your "reducing things to there essentials" which requires you
changing the meaning of things problably obliterates that part of the logic.

I can't help that you are too ignorant and stupid to understand what he
is saying.

[Richard Damon]

Nope.

The statement is that "there exists no natural number that satisfies a
particular primative recursive relationship". No natural number
satisifies that relationship, so the statment is TRUE.

Are you claiming that a statement that actually matches what is might
not be true? or that a statment which you claim to have a different
truth value can actually be isomorphic with the original (how is it
isomorphic then?)

>
> It is not that the actual Liar Paradox is true it is only when the Liar
> Paradox is applied to itself that it becomes true.

So?

>
> G states that it is unprovable in F and is unprovable in F because it is
> self-contradictory in F.

Nope. And you insistance of that just shows you are a pathological liar,
as it has been pointed out that this is NOT what G says, but you are too
stupid to understand that.

Note, the statement that G is unprovable in F, is a statement that is
DERIVED for G, but ONLY in the Meta Theory.

>
> When we test the same statement in metamathematics then it becomes true
> because it escapes the self-contradiction the same way that the above
> Liar Paradox escaped the self-contradiction.

Which means it never was self-contradictory, showing how stupid you are.

>
> The prerequisite to attaining this much deeper understanding of the
> essence of Gödel's proof is
> (1) Understanding that Gödel's G was intended to be and is isomorphic to
>     {a proposition which asserts its own unprovability [in PM]}

Nope.

>
> (2) Thus making Gödel's G isomorphic to a self-contradictory expression.

So, the basis of your method of attaining deeped understanding of the
essense of a statement is to study a Strawman and derive unsond and
false conclusion,

Sounds about right for you.

>
>> Tarski shows that there are some statements, that have a truth value,
>> that we can not know that truth value, because the mere existance of a
>> "Definition" (deterministic method) to test them with leads to the
>> contradiction that the Liar's Paradox must be True..
>>
>
> When you understand the essence of his proof you will understand that
> Tarski's metatheory is merely applying the Liar Paradox to itself
> outside of the scope of self-contradiction.

Nope, just shows your stupidity.

>
> "This sentence is not true" is not a truth bearer.
> This sentence is not true: "This sentence is not true" is true because
> the inner sentence is not a truth bearer.

Which isn't what he did, so irrelevent.

You are just showing your inability to understand a sound logical arguement.

>
>> The problem isn't that he expects that a system should be able to
>> resolve the Liar's Paradox, but that a "Definition of Truth" leads to
>> a claimed resolution, namely that the Liar's Paradox IS True (which
>> means it also must be False). He shows that a "Definition of Truth"
>> turns the Liar's Paradox from a non-truth-bearer into a Truth Bearer
>> that is True (and thus also False).
>>
>> Your failure to understand this just shows your stupidity.
>>
>
> In this case it was Tarski's failure to understand.
> Only when we boil things down to their barest essence (as I have had to
> do as a software engineer for almost four decades) do we see these
> things in their true light.
>

Nope, it is YOUR failure to understand what any of the words actually mean.

YOU ARE PROVING YOUR OWN STUPIDITY.

[olcott]

In other words you agree that Tarski did "prove" that the notion of
Truth cannot be fully formalized on a fundamental basis directly related
to the Liar Paradox?

[Richard Damon]

On 5/11/23 10:34 PM, olcott wrote:

>> FROM HIS PROOF!
>>
>> He first does a lot of work to establish a number of properties.
> In other words you agree that Tarski did "prove" that the notion of
> Truth cannot be fully formalized on a fundamental basis directly related
> to the Liar Paradox?
>

Only in the sense that since we KNOW the Liar's paradox can't be true,
and a "Definition of Truth" (not a "notion of Truth) would lead to being
able to prove that the Liar's paradox is true,

This isn't what most people would consider proving using as a basis, as
that normally means proving something because the basis is true.

Also, your use of the word "Notion" seems to indicate that you really
don't understand what Tarski was even talking about. It isn't that we
don't know the nature of Truth, and he goes into a lot of explanation of
a lot of the nature of Truth, but we can't come up with a formulaic
"Definition" that we can use to "test" an arbitrary statement to
determine if it is True or not.

This just goes back to your utter lack of understanding of anything that
these people are talking about.

[olcott]

On 5/11/2023 9:54 PM, Richard Damon wrote:
> On 5/11/23 10:34 PM, olcott wrote:
>
>>> FROM HIS PROOF!
>>>
>>> He first does a lot of work to establish a number of properties.
>> In other words you agree that Tarski did "prove" that the notion of
>> Truth cannot be fully formalized on a fundamental basis directly related
>> to the Liar Paradox?
>>
>
> Only in the sense that since we KNOW the Liar's paradox can't be true,
> and a "Definition of Truth" (not a "notion of Truth) would lead to being
> able to prove that the Liar's paradox is true,
>

That is ridiculous.

[Richard Damon]

On 5/11/23 11:30 PM, olcott wrote:
> On 5/11/2023 9:54 PM, Richard Damon wrote:
>> On 5/11/23 10:34 PM, olcott wrote:
>>
>>>> FROM HIS PROOF!
>>>>
>>>> He first does a lot of work to establish a number of properties.
>>> In other words you agree that Tarski did "prove" that the notion of
>>> Truth cannot be fully formalized on a fundamental basis directly related
>>> to the Liar Paradox?
>>>
>>
>> Only in the sense that since we KNOW the Liar's paradox can't be true,
>> and a "Definition of Truth" (not a "notion of Truth) would lead to
>> being able to prove that the Liar's paradox is true,
>>
>
> That is ridiculous.
>
>

Why? Do you think the Liar's Paradox should be provable to be True?

You seem to want to put him down for "basing" his proof on a
contradiction, but he isn't basing it in the way you want to do so.

You are just stuck trying to push a LIE, but can't quite figure out how
to do it.

Sorry, you are just too stupid to handle logic.

[olcott]

On 5/11/2023 10:45 PM, Richard Damon wrote:
> On 5/11/23 11:30 PM, olcott wrote:
>> On 5/11/2023 9:54 PM, Richard Damon wrote:
>>> On 5/11/23 10:34 PM, olcott wrote:
>>>
>>>>> FROM HIS PROOF!
>>>>>
>>>>> He first does a lot of work to establish a number of properties.
>>>> In other words you agree that Tarski did "prove" that the notion of
>>>> Truth cannot be fully formalized on a fundamental basis directly
>>>> related
>>>> to the Liar Paradox?
>>>>
>>>
>>> Only in the sense that since we KNOW the Liar's paradox can't be
>>> true, and a "Definition of Truth" (not a "notion of Truth) would lead
>>> to being able to prove that the Liar's paradox is true,
>>>
>>
>> That is ridiculous.
>>
>>
>
> Why? Do you think the Liar's Paradox should be provable to be True?
>

The Liar Paradox is not a truth bearer, END-OF-STORY !!!

> You seem to want to put him down for "basing" his proof on a
> contradiction, but he isn't basing it in the way you want to do so.
>
> You are just stuck trying to push a LIE, but can't quite figure out how
> to do it.
>
> Sorry, you are just too stupid to handle logic.

[Richard Damon]

On 5/12/23 12:51 AM, olcott wrote:
> On 5/11/2023 10:45 PM, Richard Damon wrote:
>> On 5/11/23 11:30 PM, olcott wrote:
>>> On 5/11/2023 9:54 PM, Richard Damon wrote:
>>>> On 5/11/23 10:34 PM, olcott wrote:
>>>>
>>>>>> FROM HIS PROOF!
>>>>>>
>>>>>> He first does a lot of work to establish a number of properties.
>>>>> In other words you agree that Tarski did "prove" that the notion of
>>>>> Truth cannot be fully formalized on a fundamental basis directly
>>>>> related
>>>>> to the Liar Paradox?
>>>>>
>>>>
>>>> Only in the sense that since we KNOW the Liar's paradox can't be
>>>> true, and a "Definition of Truth" (not a "notion of Truth) would
>>>> lead to being able to prove that the Liar's paradox is true,
>>>>
>>>
>>> That is ridiculous.
>>>
>>>
>>
>> Why? Do you think the Liar's Paradox should be provable to be True?
>>
>
> The Liar Paradox is not a truth bearer, END-OF-STORY !!!

Right, so why do you fault Tarski for saying that?

His proof shows that if a "Definition of Truth" existed, it provides a
way to prove the Liar's Paradox is True.

Therefore, there can be no "Definition of Truth".

I've explained that to you many times, but you say that is invalid logic.

The only way it is invalid is if you think it is possible to actually
prove the Liar's paradox.

You are just showing your stupidity.

[olcott]

On 5/12/2023 9:06 AM, Richard Damon wrote:
> On 5/12/23 12:51 AM, olcott wrote:
>> On 5/11/2023 10:45 PM, Richard Damon wrote:
>>> On 5/11/23 11:30 PM, olcott wrote:
>>>> On 5/11/2023 9:54 PM, Richard Damon wrote:
>>>>> On 5/11/23 10:34 PM, olcott wrote:
>>>>>
>>>>>>> FROM HIS PROOF!
>>>>>>>
>>>>>>> He first does a lot of work to establish a number of properties.
>>>>>> In other words you agree that Tarski did "prove" that the notion of
>>>>>> Truth cannot be fully formalized on a fundamental basis directly
>>>>>> related
>>>>>> to the Liar Paradox?
>>>>>>
>>>>>
>>>>> Only in the sense that since we KNOW the Liar's paradox can't be
>>>>> true, and a "Definition of Truth" (not a "notion of Truth) would
>>>>> lead to being able to prove that the Liar's paradox is true,
>>>>>
>>>>
>>>> That is ridiculous.
>>>>
>>>>
>>>
>>> Why? Do you think the Liar's Paradox should be provable to be True?
>>>
>>
>> The Liar Paradox is not a truth bearer, END-OF-STORY !!!
>
> Right, so why do you fault Tarski for saying that?
>
> His proof shows that if a "Definition of Truth" existed, it provides a
> way to prove the Liar's Paradox is True.

That is a nutty idea.
Any system that proves that a self-contradictory expression is true is a
broken system.

Analytic truth is derived from applying truth preserving operations to
expressions of language that have been stipulated to be true.
This cannot possibly derive non-truth bearers as true.

Prolog uses the exact same system that I just specified expressions that
are stipulated to be true are Prolog facts with Prolog rules as a set of
truth preserving operations.

Prolog is smart enough to reject the Liar Paradox.

?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

The above test shows that LP is infinitely recursive never resolving to
a truth value.

[Richard Damon]

Right, and the proof shows that would be any system with a "definition
of Truth", so you AGREE with Tarski.

>
> Analytic truth is derived from applying truth preserving operations to
> expressions of language that have been stipulated to be true.
> This cannot possibly derive non-truth bearers as true.

Right, so you agre with Tarksi that there can not be a "Definition of
Truth".

>
> Prolog uses the exact same system that I just specified expressions that
> are stipulated to be true are Prolog facts with Prolog rules as a set of
> truth preserving operations.

Prolog is limited in the logic it can do,

But, so it seems are you.

>
> Prolog is smart enough to reject the Liar Paradox.
>
> ?- LP = not(true(LP)).
> LP = not(true(LP)).
>
> ?- unify_with_occurs_check(LP, not(true(LP))).
> false.
>
> The above test shows that LP is infinitely recursive never resolving to
> a truth value.
>
>

So?

[olcott]

Not at all, Tarski's system is incorrect. All of analytical truth1 is a
body of semantic tautologies that excludes the liar paradox.

1 It is commonly known that analytical truth includes all of math and
all of logic. My new idea is that it also includes the model of the
world.

>>
>> Analytic truth is derived from applying truth preserving operations to
>> expressions of language that have been stipulated to be true.
>> This cannot possibly derive non-truth bearers as true.
>
> Right, so you agre with Tarksi that there can not be a "Definition of
> Truth".
>

Not at all.

>>
>> Prolog uses the exact same system that I just specified expressions that
>> are stipulated to be true are Prolog facts with Prolog rules as a set of
>> truth preserving operations.
>
> Prolog is limited in the logic it can do,
>
> But, so it seems are you.
>
>>
>> Prolog is smart enough to reject the Liar Paradox.
>>
>> ?- LP = not(true(LP)).
>> LP = not(true(LP)).
>>
>> ?- unify_with_occurs_check(LP, not(true(LP))).
>> false.
>>
>> The above test shows that LP is infinitely recursive never resolving
>> to a truth value.
>>
>>
>
> So?

Tarski was too stupid (on this one issue) to understand that the Liar
Paradox is excluded from the body of truth.

[olcott]

Not at all, Tarski's system is incorrect. All of analytical truth1 is a
body of semantic tautologies that excludes the liar paradox.

1 It is commonly known that analytical truth includes all of math and
all of logic. My new idea is that it also includes the model of the
world.

Prolog is smart enough to reject the Liar Paradox because it uses the
same system that I use, only expressions of language that have been
derived by applying truth preserving operations [Prolog rules] to
expressions of language that have been stipulated to be true [Prolog
facts] are true. Everything else [Prolog's negation as failure] counts
as untrue.

?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

The above test shows that LP is infinitely recursive never resolving to
a truth value.

[Richard Damon]

Then why does that "Definition of Truth" PROVE the Liar's Paradox?

If his system is "incorrect" what SPECIFIC step did he do that was
improper? (not conclusion, what STEP).

Your problem is you don't actually understand how logic works.

>
> 1 It is commonly known that analytical truth includes all of math and
> all of logic. My new idea is that it also includes the model of the
> world.

Except since your model doesn't work at all, you have a problem.

>
> Prolog is smart enough to reject the Liar Paradox because it uses the
> same system that I use, only expressions of language that have been
> derived by applying truth preserving operations [Prolog rules] to
> expressions of language that have been stipulated to be true [Prolog
> facts] are true. Everything else [Prolog's negation as failure] counts
> as untrue.

And Prolog is too limited to handle the logic of these proofs.

The fact that you can't understand that just shows how stupid you are.

[Richard Damon]

Why do you say that? WHere does he say what you say he is saying?

Please point out the pont where he is ACCEPTING the Liar's paradox.

The point where the Liar coms up, he uses that fact to point out that
the intial assumption MUST be incorrect, as it lead to proving a
non-true statement.

I think your problem is that you just don't understand the proof you are
reading and reading into it the errors that you yourself make.

[olcott]

The step where he used the Liar Paradox as the basis of his proof.

> Your problem is you don't actually understand how logic works.
>

My problem is that others do not understand the philosophical
foundations of logic as deeply as I do, they merely follow what they
read in a textbook as if it was the infallible word of God.

>>
>> 1 It is commonly known that analytical truth includes all of math and
>> all of logic. My new idea is that it also includes the model of the
>> world.
>
> Except since your model doesn't work at all, you have a problem.

When-so-ever an expression of language is derived by applying only truth
preserving operations to expressions of language that have been
stipulated to be true we are guaranteed that this expression is true.

>
>>
>> Prolog is smart enough to reject the Liar Paradox because it uses the
>> same system that I use, only expressions of language that have been
>> derived by applying truth preserving operations [Prolog rules] to
>> expressions of language that have been stipulated to be true [Prolog
>> facts] are true. Everything else [Prolog's negation as failure] counts
>> as untrue.
>
> And Prolog is too limited to handle the logic of these proofs.
>

It is not in fact too limited to handle these proofs as I have
concretely proved. I invented Minimal Type Theory that translates logic
expressions into directed graphs and a cycle in the graph indicates the
expression never resolves to a truth value.

Prolog simply does this same thing.

> The fact that you can't understand that just shows how stupid you are.
>
>>
>> ?- LP = not(true(LP)).
>> LP = not(true(LP)).
>>
>> ?- unify_with_occurs_check(LP, not(true(LP))).
>> false.
>>
>> The above test shows that LP is infinitely recursive never resolving
>> to a truth value.
>>
>>
>
>

[Richard Damon]

Then your logic is too simple to handle the needed logic.

Prolog only does FIRST order logic, and not all of it.

The logic used here is at least Second order, so out of the reach of Prolog.

If you want to claim differently, show how Prolog verifies a proof of
the Pythagorean Theorem.

In fact, almost all of the examples you actaully try to run with are
down at the simple level of Categorical logic which only handles things
of one super-class divided into sub-classes. That logic is way to simple
to handle the things the theorems have been talking about. My guess is
that is as complicated of logic that you can understand, so you try to
force everything into it, and FAIL.

[olcott]

Please explain this in English:

?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

ChatGPT
The first statement in Prolog is an assignment where LP is defined as
the negation of the truth value of LP itself. This means that LP is the
logical negation of LP, indicating that LP is false if it is true, and
true if it is false.

The second statement is a query that checks if there exists a
unification between LP and not(true(LP)). It attempts to find a
consistent value for LP that satisfies the equation.

In this case, the result of the query is false, indicating that there is
no valid unification between LP and not(true(LP)). In other words, there
is no consistent value that can simultaneously satisfy the equation LP =
not(true(LP)). This suggests that there is a contradiction in the logic,
as the equation cannot hold true for any value of LP.

[Richard Damon]

So?

the Liar's paradox is built on very simple logic.

I don't diagree that the Liar's Paradox is a non-truth-bearer, and the
fact that you keep arguing about it just shows how little you understand
about the conversation you are having.

Prolog is incapable of handling the level of logic needed to handle
Tarski or Godel, and since your understanding of Logic seems to be no
better than Prolog, you can't handle them either.

You are just confirming your utter stupidity and ignorance about all of
this.

What did any of what ChatGPT say that negates my comments.

You are just proving your stupidity.

[olcott]

Anyone that uses the Liar Paradox as any basis for showing the
properties of truth has committed a category error1, the Liar Paradox is
excluded from the category of truth.

1 Flibble's key insight

[Richard Damon]

So, you jut don't understand the concept of a proof by contradiction?

I guess your understanding of logic is too primative.

You yourself said that any system that accepts the Liar's Paradox as a
truth bearer must be broken.

Since what Tarski shows is that any system with a "Definition of Truth"
will accept the Liar's Paradox as a True Statement, you must agree that
he is correct.

That, or you are admitting that you speak with forked tounge and nothin
gyou say actually makes sense.

[Richard Damon]

On 5/13/23 4:28 PM, olcott wrote:

> Anyone that uses the Liar Paradox as any basis for showing the
> properties of truth has committed a category error1, the Liar Paradox is
> excluded from the category of truth.

So, since YOUR arguement uses the Liar's Paradox as a basis of
determining true, YOU have committed a category error?