Undecidable decision problems are abolished

[olcott]

ZFC was able to reject epistemological antinomies by screening
out the pathological self-reference derived by sets as members
of themselves. Russell's Paradox was eliminated be defining set
theory differently.

In the same way that Russell's Paradox was eliminated we can
get rid of other epistemological antinomies. It is pretty
obvious that epistemological antinomies are simply semantically
unsound.

When we define True(L, x) as (L ⊢ x) provable from the axioms
of L, then epistemological antinomies become simply untrue and
no longer show incompleteness or undecidability.

Since we have already fixed the undecidability issue of Russell's
Paradox by redefining set theory the precedent has already been
set that we can correct these issues by redefining the meaning
of their terms.

Because the undecidability of Russell's Paradox was fixed by changing
the meaning of the term {set theory} we can eliminate incompleteness
and undecidability by redefining meaning of the term {formal system}
as detailed above.



--
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 11/18/23 11:32 AM, olcott wrote:
> ZFC was able to reject epistemological antinomies by screening
> out the pathological self-reference derived by sets as members
> of themselves. Russell's Paradox was eliminated be defining set
> theory differently.
>
> In the same way that Russell's Paradox was eliminated we can
> get rid of other epistemological antinomies. It is pretty
> obvious that epistemological antinomies are simply semantically
> unsound.
>
> When we define True(L, x) as (L ⊢ x) provable from the axioms
> of L, then epistemological antinomies become simply untrue and
> no longer show incompleteness or undecidability.
>
> Since we have already fixed the undecidability issue of Russell's
> Paradox by redefining set theory the precedent has already been
> set that we can correct these issues by redefining the meaning
> of their terms.
>
> Because the undecidability of Russell's Paradox was fixed by changing
> the meaning of the term {set theory} we can eliminate incompleteness
> and undecidability by redefining meaning of the term {formal system}
> as detailed above.
>

But that doesn't abolish ALL "undecideable" decision problems.

Halting is still "Undecidable" by the meaning of the word, and the
actual problem doesn't have the "pathological self-reference" that you
are trying to refer to.

Your problem is that you don't seem to understand what a "reference"
actually is, and thus what a "self-reference" actually means.

Asking H to decide on a program that happens to be built on a copy of
the algorithm that H uses, is NOT a "reference". You only try to show
one, by creating a environment what isn't actually an "equivalent" to
the environment of a Turing Machine deciding on the representation of
another machine, and as such, your "H" isn't actually the equivalent of
any Turing Machine that meets the definition of a Halt Decider.

Yes, if you define that True means Provable, you can get a system that
dosn't have incompleteness, you also can't get the full set of
properties of the Natural Numbers in such a system.

Godel proves that by showing that from the established properties of the
Natural Numbers, you can construct a statement that IS TRUE, but
UNPROVABLE in that system.

Thus, he proves that you your system, must either not be able to show
the needed properties of the Natural Numbers, or it is inconsistant.

If you want to try to prove him wrong, you just need to start from your
logical basis, and then show that you actually CAN derive those
properties, and then prove that you system is still consistant.

This has been pointed out to you many times in the past, but it seems
that you understand that the task is just too great for your little
mind. This just points out that you ideas are actually worthless, as you
are postulating a fundamental change in the nature of logic, but then
are unable to show what that actually does.

Also, you don't understand that this idea isn't actually "new", but is
very similar to ideas that other have come up with, its just they
understand that their ideas are of limited use in restricted fields of
logic, while you don't understand that fact.

[olcott]

On 11/18/2023 10:32 AM, olcott wrote:
> ZFC was able to reject epistemological antinomies by screening
> out the pathological self-reference derived by sets as members
> of themselves. Russell's Paradox was eliminated be defining set
> theory differently.
>
> In the same way that Russell's Paradox was eliminated we can
> get rid of other epistemological antinomies. It is pretty
> obvious that epistemological antinomies are simply semantically
> unsound.
>
> When we define True(L, x) as (L ⊢ x) provable from the axioms
> of L, then epistemological antinomies become simply untrue and
> no longer show incompleteness or undecidability.
>
> Since we have already fixed the undecidability issue of Russell's
> Paradox by redefining set theory the precedent has already been
> set that we can correct these issues by redefining the meaning
> of their terms.
>
> Because the undecidability of Russell's Paradox was fixed by changing
> the meaning of the term {set theory} we can eliminate incompleteness
> and undecidability by redefining meaning of the term {formal system}
> as detailed above.

We can eliminate incompleteness and undecidability derived by
epistemological antinomies by redefining meaning of the term

{formal system} as detailed above.

For the halting problem H(D,D) simply screens out and rejects
input D that is defined to do the opposite of whatever Boolean
value that H returns.

Pathological self-reference {AKA epistemological antinomies}
cannot possibly create incompleteness or undecidability when it
is simply screened out as erroneous.

[Richard Damon]

Yes, you CAN try to redefine them, but then you end up with a very weak
logic system.

Note, The "Halting Problem" doesn't have a Pathological Self-Reference
in its definition, so that isn't the problem. All you doing is limiting
yourself to non-Turing complete computation systems, just like you are
limiting yourself to system that can't actually handle the full
properties of the natural numbers.

You are just showing how little you understand what you are talking about.

[olcott]

When we imagine that every detail of the body of human
knowledge has been formalized as higher order logic then
the only incompleteness are unknowns.

This is the way that human knowledge actually works:

True(L,x) is defined as (L ⊢ x)
False(L,x) is defined as (L ⊢ ~x)

then
epistemological antinomies are simply rejected as not truth
bearers and do not derive incompleteness or undecidability.

[Richard Damon]

And you can't do that in a system that defined True to be provabl.

> This is the way that human knowledge actually works:
>
> True(L,x) is defined as (L ⊢ x)
> False(L,x) is defined as (L ⊢ ~x)

Nope, FALSE statement.

We know there are things that are true that we can not actually prove.

Maybe you don't understand that fact, because your mind is too limited.

>
> then
> epistemological antinomies are simply rejected as not truth
> bearers and do not derive incompleteness or undecidability.
>
>

But that doesn't get rid of "undecidable" cases, as not all of them are
based on epistemological antinomies.

In fact, (almost) no one in classical logic think that epistemolgocial
antinomies are anything other than not a truth bearer. You are just
showing that you don't really understand how those work.

[olcott]

Rebutting things that I did not actually say might seem like a rebuttal
to gullible fools.

People that are paying 100% complete attention will see that such
rebuttals are the strawman error even if unintentional.

People that physically don't have the capacity to pay close attention
may commit the strawman error much of the time and not even know it.

[Mikko]

On 2023-11-18 16:32:10 +0000, olcott said:

> When we define True(L, x) as (L ⊢ x) provable from the axioms
> of L, then epistemological antinomies become simply untrue and
> no longer show incompleteness or undecidability.

That definition does not remove deductive incompleteness of a theory.
E.g., the first order Peano arithmetic is still deductively incomlete.
It only removes the semantic completeness by removing the semantic
concept of truth (and with it all semantics, as the main role of
semantics is to provide a concept of truth).

Mikko

[Richard Damon]

And what did I rebut that you didn't say?

Making false claims is evidence of deceit. This seems to be your basic
method of arguement, claim that someone says something different then
what they actually said by miss-using their words, and building a
strawman argument from it.

>
> People that are paying 100% complete attention will see that such
> rebuttals are the strawman error even if unintentional.
>
> People that physically don't have the capacity to pay close attention
> may commit the strawman error much of the time and not even know it.
>

Yep, which describes yourself. You don't understand what people are
telling you, perhaps because you don't understand that core concepts of
formal logic, so you just presume they are talking non-sense.

That is like how you claim there is a "pathological self-reference" in
the Halting Problem, when you can't even point out where there is an
actual "Reference" (as defined in the field) in the first place.

Look, youi don't even understand the basic rules of argument, that you
respond TO the counter-point and show what is wrong with it.

By just replying to yourself, and just mentioning what you are trying to
"refute", you are just highlighting that your logic can't actually
handle the case, but you need to create a strawman in you description
and fight that,

Maybe I should just start pointing out your errors in ogical argument
form to point out your utter incapability of actually showing what you
claim.

It does seem ironic that someone who wants to claim that Truth only
comes out of proofs, can't actually form a correctly formed proof, but
seems to think that a verbal argument is the same thing.

Maybe that works in the fuzzy field of abstract philosophy, but it
doesn't cut it in actual formal logic, which is why you seem to fall so
flat. Some how you have a blind spot that the rules of logic ARE actual
rules to follow, not merely suggestions.

[olcott]

On 11/19/2023 6:35 AM, Mikko wrote:
> On 2023-11-18 16:32:10 +0000, olcott said:
>
>> When we define True(L, x) as (L ⊢ x) provable from the axioms
>> of L, then epistemological antinomies become simply untrue and
>> no longer show incompleteness or undecidability.
>
> That definition does not remove deductive incompleteness of a theory.

Sure it does, when the criteria that used to prove incompleteness:
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
becomes
¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
Incompleteness cannot possibly exist.

> E.g., the first order Peano arithmetic is still deductively incomlete.
> It only removes the semantic completeness by removing the semantic
> concept of truth (and with it all semantics, as the main role of
> semantics is to provide a concept of truth).
>
> Mikko
>

[olcott]

This includes all human knowledge and excludes unknowns.
Your prior reply only glanced at a few of my words and thus did not
bother to notice that I was talking about the set of human knowledge.

> then
> epistemological antinomies are simply rejected as not truth
> bearers and do not derive incompleteness or undecidability.

Every expression that is neither provable nor refutable is rejected as
not a truth bearer, (within this formal system) thus epistemological
antinomies are excluded and unknowns are excluded and there is nothing
else left over.

[Richard Damon]

But you confuse "Human Knowledge" with actual "Truth", so is just a LIE,

This means that the logic system you are trying to work in is either
inconstant or very weak., as there are statements that can be proven
that they must be either True or False, but we don't (yet) know which it
is, and even understand that it might be actually IMPOSSIBLE to prove
within the system, but your FLAWED systen says they can NOT be true
until proven, and in fact, the statment L ⊢ x needs the proof to be
know, since we need the existance of the proof to be proven for the
statement to be true.

So, either the domain of logic it can handle must be limited to just
that which works under that definition, which excludes many properties
of even the simple Natural Numbers, or it become inconsistant as
statements that can be show must be truth bearers, as they must be True
or False, because they don't allow a middle ground (like the existance
of a number with a computable property) but also, they might not be
either True or False, as we can't actually prove that existance.

>
>> then
>> epistemological antinomies are simply rejected as not truth
>> bearers and do not derive incompleteness or undecidability.
>
> Every expression that is neither provable nor refutable is rejected as
> not a truth bearer, (within this formal system) thus epistemological
> antinomies are excluded and unknowns are excluded and there is nothing
> else left over.
>

So, you INCORRECTLY reject as a "Truth Bearer" statements that ACTUALLY
HAVE A TRUTH VALUE, but that value is just not known.

In other words, you don't understand what TRUTH actually is because of
your own stupidity.

This shows that your mind is just a few sizes too small and doesn't (and
perhaps can't) understand the complexity that simple logic can generate.

[olcott]

Reviewers that don't give a rat's ass about truth and only want to stay
in rebuttal mode even if must lie to do it will refuse to acknowledge
that expressions that require infinite proofs to resolve their true
value are necessarily not truth bearers in formal systems that do not
allow infinite proofs.

[Richard Damon]

Is this comment directed at YOURSELF? since that is who you are replying
to. I guess you are admitting you don't give a rat's ass about what
actually is Truth, but just want to stay in your unsubstantiated
"rebuttal" mode, leaving all the errors pointed out in your logic as
accepted.

Note, since your definition of "Truth" isn't actually a definition of
Truth but of Knowledge, YOU are the one making the lies.

Also, your claim that "that expressions that require infinite proofs to

resolve their true value are necessarily not truth bearers in formal

systems that do not allow infinite proofs." is just an INCORRECT STATEMENT.

"Standard" Logic allows statements to establish there truth with
infinite chains even though proofs, being related to knowledge, needs to
be finite.

If you can find any "official" support for your claim, give it or you
are admitting that you are just a stupid liar.

Then, if you want to establish that changed rule as part of your logic,
show what you logic can do. As I have pointed out many times, you are
free to build a new logic system under the rules of formal logic, with
what ever definitions you want, it then just get put on you to establish
what that logic system can do, and you can't just borrow proofs based on
system with a different set of rules. This will mean you will need to
learn enough of "primative logic" to understand what rules get impacted
by this change. My first guess is this is far above your ability.

[Richard Damon]

On 11/19/23 1:08 PM, olcott wrote:
>

> Reviewers that don't give a rat's ass about truth and only want to stay
> in rebuttal mode even if must lie to do it will refuse to acknowledge
> that expressions that require infinite proofs to resolve their true
> value are necessarily not truth bearers in formal systems that do not
> allow infinite proofs.

Simple thought experiment for you on that claim.

Question, does a number exist which satisifies some particular
computable property?

Such a question must be True or False, as either such a number exists or
it doesn't, and thus either assertion is a "Truth Bearer" by definition.

It is at least conceivably possible, that the only proof that such a
number doesn't exist is to test every possible number, and thus require
an "infinite proof" to establish this fact, so either the non-existance
of a number that satisfies some property might not actually be a "Truth
Bearer" by your definition, even though we KNOW, by the form of the
question, that it must be true or false, and thus be a Truth Bearer by
definition.

Also, by your definition, the question of the question about if that
statement was a Truth Bearer might not be a Truth Bearer, as to show
that there does not exist a finite proof of that property might not be
actually provable in a finite number of steps.

In fact, if you COULD actually prove in a finite number of steps that
you can't prove the statement in a finite number of steps, that could be
used as a proof of the statement that such a number doesn't exist (since
the existance of such a number, if one exists, is provable in a finite
number of steps by starting from that number and computing the answer,
showing it has the property.

This means you logic system sometimes can't actually ask questions until
it knows the answer.

[olcott]

When we stipulate that a truthmaker is what-so-ever makes an expression
of language true then we can know by tautology that every truth has a
truthmaker.

When we arbitrarily limit the set of truthmakers then this arbitrarily
limit screws everything up.

To define a proof as a finite set of inference steps creates the
artificial notion of unprovable truths.

[Richard Damon]

So, just because we can't prove the statement true, doesn't mean it
isn't true.

>
> When we arbitrarily limit the set of truthmakers then this arbitrarily
> limit screws everything up.

Right, so you limiting Truthmakership to only things that are provable
is your failing.

>
> To define a proof as a finite set of inference steps creates the
> artificial notion of unprovable truths.
>

So?

Proof is in the domain of knowledge, and since we can only know things
that we can establish by our own finite capabilities, means that proofs
are normally limited to finite operations.

In the same way that "Computable", means we can get the answer in a
finite number of steps, "Provable" means we can demonstrate the truth of
the statement in a finite number of steps.

I guess this goes back to your silly idea that you are a divine being
not bound by the finiteness of mortals, but on the other hand, you
actually are bound by the finiteness of yourself, thus showing that you
can't be divine.

You still don't understand the difference between Knowledge and Truth,
it seems, in part, due to not understanding the properties of the
infinite (or even the unbounded).


Yes, I believe there are logic system that allow for something called a
"Proof" to be unbounded in length, but such systems will have issues
with defining knowledge.

If you want to work in such fields, just say so, and confine yourself to
them, and not assume that you can just transfer information between
fields that have different logical basis. That leads to the same sort of
problems as presuming that trans-finite mathematics holds the same
properties as the mathematics of finite numbers (like the Reals). They
don't.

[olcott]

My most important point of all this is that epistemological antinomies
are finally understood to simply be semantic nonsense that do not
actually prove incompleteness, undecidability or undefinability.

...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)

[Richard Damon]

No, you are just proving that you don't understand what you are talking
about.

Please try to show where Godel actually used an epistemological antinomy
in manner that required it to be anything other than a statement that
could not be logically resolved, and thus not a "Truth Bearer".

Not just this quote, which shows no such thing, but the step in the
proof that used it in a way that invalidates the proof.


The problem seems to be that you read non-technical descriptions of
things and think you understand what is actually being done in the proof.

All your words are just proving how ignorant you are of anything you
talk about.

Yes, Godel used a statement that was a epistemolgical antinomy,
something like "Statement X asserts that Statement X is not True", which
is, and most people understand it, to be such a statement that doesn't
not have a truth value.

He then converted it with a syntatic transformation that totally changes
its meaning into: "Statement X asserts that Statement X is not Provable
in F". Note, this transformed sentence is NOT an epistemological
antinomy in classic logic, as there is a truth value assignment that can
make the statement have a valid truth value, namely that X is a true
statement that is not provable.

Since the final statement that he gets, is one that MUST be a Truth
Beared, a question about the existance of a number that satisfies a
strictly computable property.

Yes, this proof does not work in a system that restricts truth to only
things that are provable, but that is not the logic system that Godel is
working in.

Your problem is that if you want to try to talk about such a logic
system with that limitation, your first step is to show that such a
system can meet the other requirements of the proof, that it supports
those need properties of the Natural Numbers. That is the problem you
are going to run into, the inevitable result of the limits to logic you
propose is that the logic system can not expand to the point of
generating those properties without falling into inconsistency.

[olcott]

Antinomy
...term often used in logic and epistemology, when describing a paradox
or unresolvable contradiction.
https://www.newworldencyclopedia.org/entry/Antinomy

epistemological antinomies are unprovable because they are semantic
nonsense.

"If a formal system cannot prove gibberish nonsense then the formal
system is incomplete" is itself gibberish nonsense.

[Richard Damon]

But that isn't what Godel was doing.

You are just proving you are talking out of ignorance and YOU are the
one speaking "gibberish".

The statement "G" that is shown to be True and unprovable is NOT an
epistemological antinomy, but a statement that most definitely has a
Truth Value, and thus CAN'T be an epistemoligical antinomy.

Again, you seem to like arguing with yourself and not actually answering
the errors pointed out in your arguments, meaning you are accepting the
errors as actual errors, and thus you are accepting that you statements
are in error, and that you are just repeating the errors to show your
ignorance.


Go ahead, keep digging the grave for your reputation. You are just
burying it deeper.

[olcott]

The only fake deceptive rebuttal to the fact that Gödel was definitely
wrong about that is changing the subject to something else.

[Richard Damon]

Which just PROVES you don't understand what you are talking about.

Your failure to answer the question I asked before proves this.

That shows that YOURS is the "fake deceptive rebuttal".

By your own logic, your statement is garbage because you mentioned using
epistemological antinomies.

So again, WHERE did he actually do this in his proof? Show the step
where he did it.

I bet your problem is you can't actually read any of the proof to see
what he is doing.

You are just too stupid to understand that you don't understand what you
are talking about.

[olcott]

[Richard Damon]

Again, talking to yourself showing who you actually think is doing the
"fake deceptive rebuttal".

And, you can't answer the question? You are just admitting that you are
just a stupid liar.

If he was actually wrong, you could show the point in the proof where he
did a wrong thing.

That fact you can't do that show that you are just being a stupid
ignorant liar.

As I said, by your logic, you just proved that your own proof must be
incorrect, as you also mention using epistemological antinomies.

[olcott]

On the other hand honest reviewers would say of course you are right
about this. Expecting a formal system to prove an epistemological
antinomy is ridiculous. How could Gödel make such a huge mistake?

[Richard Damon]

So, again, where in the proof did he do this wrong thing?

You can't show it, because he didn't do what you are claiming.

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

>
> On the other hand honest reviewers would say of course you are right
> about this. Expecting a formal system to prove an epistemological
> antinomy is ridiculous. How could Gödel make such a huge mistake?
>

Except he didn't expect a formal system to prove an epistemological
antinomy, and the fact you imply that he claimed he did shows your
stupidity.

Yes, a real "Honest Reviewer" would see what Godel wrote, and see that
your claim that he was asking the system to prove an epistemological
antinomy is just a stupid lie on your part.

You can't even state the actual proposition that Godel put forward as
the true but unprovable statement in the system, you only see the
statements, in the meta-system, that can be derived from it.

You are just showing you fundamentally don't understand how logic or
truth or proof actually works.

You are just a LYING DISHONEST STUPID CHARLATAN that has been caught in
your lies and trying to fast talk out of your errors.

You have yet to present ANY actual proof of your claims, and have ducked
every request to provide something to actually back your claims,

Of course, since you actually know nothing about what you talk, you
can't do that, but only bluster.

Of course, you are so stupid, you think you are making your point, but
in truth, you are just proving to the world how utterly stupid you are.

If there was something to your ideas, you have buried it in your disgrace.

[olcott]

There is no way to correctly refute that Gödel was definitely wrong
about this.

I would go further and say the the strongest possible rebuttal cannot
do any better than complete nonsense. My reviewer already knows this.

[Richard Damon]

What are you trying to refute?

If you want to claim he actually claimed the need to prove a
epistemological antinomy, then show where he did that.

The above does NOT say that.

When put in context, it points to the fact that you can use the FORM of
an epistemological antinomy to form a similar (but logically valid)
statement about provability that shows that there exist statements which
are true but not provable.

This is based on the key difference between claims of a statements Truth
and it Provability. Provable -> True, but True does not imply Provable.
Not True -> Not Provable, but Not Provable does not imply not True.

This asymmetry is important, but apparently unintelligible to you, as
you can't seem to grasp the difference between factual truth and knowledge.

So, since you refuse to even attempt to show where he makes the claim
youy say he is wrong with, you are just plain guilty of using a strawman
arguement (which by your own definitions, makes you a despicable liar).


Now, if you want to refute the claim he ACTUALLY made with those words,
you need to show the error in his actual proof. Since the proof never
actually claims the need to prove an epistemological antinomy, your
argument is proved to be just more of your lies. The statement his base
proof used, was that the statement G was "There does not exist a Natural
Number G that satisfies a (particular primative recursive relationship)"
where that relationship is what most of the paper is spent building up.

Now, such a statement MUST be a truth bearer, as either there does exist
or there doesn't exist some Natural Number that meets that requirement.

Note also, most of the paper is written not working in the Field that
expresses the statement, but in a meta-field of that field, and in that
meta-field, he can prove that G must be True in F, and that G can not be
proven in F.

Again, G is a statement that can not, by definition, be an
epistemological antinomy, as by its nature, it must have a correct
logical answer. If it doesn't, then you are just claiming that all of
mathematics is just wrong, with no more evidence than you saying so.

Since Godel is able to show that G is in fact TRUE, that in itself shows
that G is not an epistemological antinomy, as by definition such a
statement can not be satisfied by either a True or False value.

So, you are just proving your ignorance and stupidity.

> I would go further and say the the strongest possible rebuttal cannot
> do any better than complete nonsense. My reviewer already knows this.
>

So, I guess you are just admitting that your mind is incapable of
understanding the arguement, because everything is just complete
nonsense to you.

That is YOUR problem, not the field of logics. That a total idiot can't
understand how it works is the problem of the idiot, not of logic.

If this is the best arguement you can present, you have just proven you
have wasted your life.

Your are even showing your utter childishness by acting like the mental
giant of a three year old and acting like you are arguing with yourself
because you don't have the strength to face the person you want to argue
with.

[olcott]

It is dead obvious that epistemological antinomies are semantic
nonsense thus anyone saying that any proof can be based on them
is terribly incorrect.

[Richard Damon]

So, you are agreeing that your proof, since it is based on them, i.e.
mentioning them, is terribly incorrect. Thank you for stipulating that.

You haven't shown that Godel used them in any way more than your own
description,

This just shows how ignorant you are of what you are taliking about.

If you want to try to show that the proof is actually based on an
epistemological antinomy has a truth value, show where he does that,
otherwise you are just admitting you don't have a clue and are just puffing.

Also, they aren't "non-sense", they have a lot of semantic meaning, they
just can't be resolved to a truth value, and in fact, can be a great
basis for statements that can be shown to NOT have a truth value, which
is a useful feature in some places.

[olcott]

It is dead obvious that epistemological antinomies are semantic
nonsense thus anyone saying that any proof can be based on them

(such as the above sentence) is terribly incorrect.

[Richard Damon]

So, you are agreeing that your proof, since it is based on them, i.e.
mentioning them, is terribly incorrect. Thank you for stipulating that.

You haven't shown that Godel used them in any way more than your own
description,

This just shows how ignorant you are of what you are taliking about.

If you want to try to show that the proof is actually based on an
epistemological antinomy has a truth value, show where he does that,
otherwise you are just admitting you don't have a clue and are just puffing.

Also, they aren't "non-sense", they have a lot of semantic meaning, they
just can't be resolved to a truth value, and in fact, can be a great
basis for statements that can be shown to NOT have a truth value, which
is a useful feature in some places.

Just repeating your false claim just proves that you have nothing to go on.

Answer the refutation, or you are just admitting you are a liar.

[Richard Damon]

On 11/20/23 9:38 AM, olcott wrote:

Also, this shows that you don't understand how logic works.

For example, the classical logical form of "Proof by Contradiction" is a
proof that in one sense of the word is "based" on an epistemological
antinomy, in that it is based on the fact that if from an "assumed true"
statement, you can prove an epistemological antinomy, then that
statement must be false.

If you want to try to define that such a logical argument is incorrect,
then you need to throw out most of the existing logical systems.

Of course, you have shown historically, that you don't understand how
any of the logic works, so it isn't a surprise that you don't understand
this.

You are just proving your utter ignorance of how any of this sort of
logic works, likely because you don't understand this "foreign" concept
of "Truth".

[olcott]

It is dead obvious that epistemological antinomies are semantic
nonsense thus anyone saying that any proof can be based on them
(such as the above sentence) is terribly incorrect.

Hopefully the one lying about this does not get the eternal
incineration in the Revelation 21:8 lake of fire required for
"all liars" that seems far too harsh.

The Church of Jesus Christ of Latter day saints temporary purgatory
like option seems more appropriate.

[olcott]

Proof by contraction when one begins with a self-contradictory
expression is like trying to make an angel food cake from dog shit.

[Mikko]

On 2023-11-19 15:43:59 +0000, olcott said:

> On 11/19/2023 6:35 AM, Mikko wrote:

>> On 2023-11-18 16:32:10 +0000, olcott said:
>>
>>> When we define True(L, x) as (L ⊢ x) provable from the axioms
>>> of L, then epistemological antinomies become simply untrue and
>>> no longer show incompleteness or undecidability.
>>

>> That definition does not remove deductive incompleteness of a theory.
>
> Sure it does, when the criteria that used to prove incompleteness:
> Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
> becomes
> ¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
> Incompleteness cannot possibly exist.

The OP did not change or remove the defintion of semantic incompleteness,
only of True.

Mikko

[Richard Damon]

Who said you started with a self contradictory expression?

You are just showing that you don't understand what is being talked about

You claim Godel starts with a self-contradictory statement, but you
can't actually show where it is, but need to use "simplification" that
aren't even in the logic system that the original statement was made in,
showing your total ignorance of how logic works

All you are doing is proving you are an ignorant pathological liar. (you
seem to be incapable of understanding the nature of your error, thus
PATHOLOGICAL liar)

[Richard Damon]

Also, this shows that you don't understand how logic works.

For example, the classical logical form of "Proof by Contradiction" is a
proof that in one sense of the word is "based" on an epistemological
antinomy, in that it is based on the fact that if from an "assumed true"
statement, you can prove an epistemological antinomy, then that
statement must be false.

If you want to try to define that such a logical argument is incorrect,
then you need to throw out most of the existing logical systems.

Of course, you have shown historically, that you don't understand how
any of the logic works, so it isn't a surprise that you don't understand
this.

You are just proving your utter ignorance of how any of this sort of
logic works, likely because you don't understand this "foreign" concept
of "Truth".


>

> Hopefully the one lying about this does not get the eternal
> incineration in the Revelation 21:8 lake of fire required for
> "all liars" that seems far too harsh.

I have no fear of that, but you should,

>
> The Church of Jesus Christ of Latter day saints temporary purgatory
> like option seems more appropriate.
>

So, you don't understand what the Bible actually says and go by the
words of "experts" that have been shown to be liars.

(Apologies to any Mormons offended by my remark, but try to take an
honest look at the history of Joseph Smith and see if he passes the
ancient biblical test of a Prophet)

[olcott]

Yes you will get that understanding if you glance at one or two of my
words before artificially contriving a fake rebuttal.

When you actually pay complete attention then what was previously
was Incomplete(L) becomes ¬TruthBearer(L,x) the detection of an
epistemological antinomy.

[olcott]

"Who said you started with a self contradictory expression?"

Gödel

[Richard Damon]

Nope.

You just don't understand what he said.

Show me where he actually started his logic sequence from an actual
self-contradictory statement.

The ACTUAL step of the proof, not just the general statement that the
proof "uses" such a statement.

For instance, the argument by contradiction "uses" a self-contradictory
statement, but it doesn't start with one.

The thing that you don't seem to understand is that it is possible to
start with a sentence, like an epistemological antinomy, and then apply
a semantic and syntactic transformation to it that gives a brand new
statement that isn't logically dependent on the original sentence (and
so your argument fails) but is a way to find a statement with certain
properties.

Your tiny mind seems unable to conceive of this sort of operation, which
is why you are stuck in just low level logic forms.

[olcott]

"Who said you started with a self contradictory expression?"

Gödel just said that in the quote above when you understand that
epistemological antinomies are self-contradictory expressions.

*Antinomy*

...term often used in logic and epistemology, when describing a paradox
or unresolvable contradiction.
https://www.newworldencyclopedia.org/entry/Antinomy

[Richard Damon]

No, he didn't say that he STARTED the logical chain of reasoning from
the statement in his proof. He USED it. But it wasn't as a proposition
in a logical inference, so your statement is itself, NONSENSE.

You are just proving that you don't understand what you are saying, and
your logic applies just as must to your claim as his.

YOUR statement starts with the use of an epistemological statements, and
thus it must be nonsense.

As I have pointed out, the fact that you can't go into the proof and
show where he actually did what you are claiming, and don't even attempt
it, just shows how utterly stupid your argument is and that you, at
least subconsciously understand that fact.

You just don't understand how logic works, what Truth actually is, or
how to do a proof.

The fact that you refuse to actually respond properly shows that you
have the mental age of a three year old.

You KNOW that, but refuse to acknowledge it, because you mind, and your
logic, is based on lie and deceit. You have been called out on this and
seem to be running scared. You are "projecting" your errors on others
that chalange you, reveling the errors that you know are in your logic.

Sorry, you have ruined your reputation, and are destined to be on the
eternal trash heap because that is all you are worth.

[olcott]

When you understand that an epistemological antinomy is a self-
contradictory expression then the above quoted sentence is
understood to be a ridiculous error.

Even gullible fools will know that changing the subject away
from the above quoted sentence is such a lame attempt at deception
that they will reject such attempts as nonsense.

[Richard Damon]

You are just repeating yourself and not answering the questions or
responding to the errors that have been pointed out to you repeatedly.

This shows that you are just an ignorant pathological lying TROLL.

I have not change the subject of the sentence, but gone to the core
meaning of the sentence. The fact you don't understand that shows your
utter ignorance of the topic.

I see just three possibilities.

1) You just don't understand the words being used, because you are just
totally untrained in the field, but then the honest responce would be to
ask about the terms that you seem to not understand. That you don't do
this says the even if this is the case, you are not interested in an
Honest discussion.

2) You honestly think these meen something different that how I am using
it. But in this case, again, you should be responding to specific points
to discuss why you see something different out of them. The fact you
don't, means that even if this is the case, you are not interested in an
Honest discussion.

and that just leaves:

3) You are not interested in an honest discussion, but knowing there are
problems with your arguement you intended to just ignore your errors and
propogate your LIES and FALSEHOODS to try to advance your BIG LIE.

Face it, you have lost, your plan has been ripped apart and shown to be
worthless. All you are doing it killing and buring your reputation, and
and small positive things that might be hiding in your ideas.

By doing this, you are just proving yourself to be the sort of person
described in the chapter of Revelation you like to quote, and that the
eternal burning trash heap is your destination, because that is all you
life is worth.

This does seem to match up with your previous cases of claiming it was
ok to have child pornograph, because "you were God", and your mental
derangement where you thought that somehow you were God, but were still
dying of cancer. (Hows that going for you, or was that just more lies),

[olcott]

...14 Every epistemological antinomy
AKA every self-contradictory expression

can likewise be used for a similar undecidability proof...

AKA can likewise be used to provide a sequence of
inference steps proving that self-contradictory
expressions cannot be proven.

[Richard Damon]

So, you STILL don't understand what you are saying,

By this logic, any proof that mentions epistemological antinomies are
invalid, thus YOUR arguement that mentions them as a grounds to call
proofs invalid is also invalid.

You are just proving yourself to be an ignorant troll.

Try to answer the questions put to you, or just be labeled the troll you
are.


Note, he doesn't say that the sequence of inference steps actually used
the epistemological antinomy, but that concept seems above your
understanding, because you are just too stupid.

[olcott]

"By this logic, any proof that mentions epistemological
antinomies are invalid"

Not at all. I didn't say anything like that.

...14 Every epistemological antinomy can likewise be used
for a similar undecidability proof...(Gödel 1931:43-44)

*The above sentence proves that the above sentence is incorrect*

[Richard Damon]

Only under the interpretation of the words that says that "using" an
epistemolgical antinomy in "some" manner makes a proof invalid.

Your arguement "uses" an epistemological antinomy, so is thus invalid.

Note, As I have pointed out, Godel isn't saying that he is using an
epistemological antinomy as a PREMISE to his proof, so your argument
doesn't apply to it.

Your AKA is in INCORRECT inference.

While a "Proof" is a sequence of inference steps, not every statment
"used" by the proof is a premise to the proof.

You seem to have a too simple understanding of a proof.

If you want to disagree with me, point out where in Godel's proof he
actually used an epistemological antinomy as a PREMISE to a logical step
in the proof.

Until you do, you are just shown to be the ignorant pathological lying
troll that you are.

Since you just refuse to actually answer the errors pointed out in your
statements, you are shown to not be discussing in good faith, and are
thus just a troll, and your ideas turn to stone by the light of truth,
so you need to keep your ideas under the darkness of deceit and description.

[olcott]

"Note, As I have pointed out, Godel isn't saying that he is using an
epistemological antinomy as a PREMISE to his proof, so your argument
doesn't apply to it."

*Since incompleteness already has a precise definition*
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))

then the epistemological antinomy cannot possibly be correctly
construed as anything besides x in the above expression.

[Richard Damon]

So, you are presuming (INCORRECTLY) that the x in this formula is an
epistemological antinomy in Godel's Proof

It isn't.

But since you don't seem to be able to understand what Godel's G is, and
are too arogent to learn, you are doomed to just being ignorantly wrong.

Note, G is NOT the statement, even in "effect", that G asserts that it
can not be proven.

G is the statement that there does not exist a Natural Number g, that
meets a specifically defined Primitive Recursive Relationship.

And that is all that G is in the field F.

Such a statement MUST be a "Truth Bearer" as either a number g exists
that meets the requirement or it doesn't. That is a basic fact of the
mathematics of Natural Numbers, either numbers exist that meet a
computable property, or they don't, there is no "fuzzy" state between or
outside.

The key point of the proof, is that the specific PRR was built in a
meta-F that has the enumeration of all the axioms of F, assigning them
to numbers, and an encoding system that can express ANY statement, or
series of statements in F as a number, and the PRR is constructed so
that a number that satisfies it WILL be an encoding of a proof of the
statement G, and any proof that might exist in F, will have a number.

And the complicated paper is the proof that such a PRR can be constructed.

Given that we can construct such a PRR, and ask about a number
satisfying it, we can then show in meta-F that the existance of a number
that satisfies the PRR has an identical truth value to the provability
of the statement G in F. Thus the existance of the number g has the same
truth value as the provability of G in F, or the non-existance of the
number g has the same truth value as the unprovability of G in F.

Thus since G asserts that there is no number g, that means we can
logically derive from G the statements that G is true if, and only if, G
is unprovable, thus it is this DERIVED statement that is the statement
of a statement that asserts its own unprovability, and this statement is
in meta-F, not F.

Despite what you try to claim, this is NOT the statement G, but a
statement provable to have a logical equivalence (in meta-F), and since
G (in F) was a truth bearer, so must this derived statement.

Note, that the form of this equivalent statement has a similar mophology
to the liar, the liar is L asserts that L is not True, while this one is
that G asserts that G is not Provable in F. Same form, but different
predicate function referred to. This is what Godel was refering to,
given any epistemological antinomy, with a similar change of predicate,
you could do a similar derivation to find a PRR that is its equivalent.

Note, this means the epistemological antinomy itself, was never used as
a premise of any logical deduction, so the "non-sense" of them never
mattered. The morphical transformation turns that "non-sense" into a
Truth Bearer that can show that some statements are not provable in any
system rich enough to perform the proof in, which just requires a number
of the basic properties of the Natural Numbers.

[olcott]

"So, you are presuming (INCORRECTLY) that the x in this
formula is an epistemological antinomy in Godel's Proof"

*I am presuming nothing* There is no possible other place
to correctly insert the epistemological antinomy in the
definition of incompleteness besides x.

[olcott]

"So, you are presuming (INCORRECTLY) that the x in this
formula is an epistemological antinomy in Godel's Proof"

*I am presuming nothing* There is no possible other place
to correctly insert the epistemological antinomy in the
definition of incompleteness besides x.

Also again and again and again I have only been talking
about this one freaking quote in everyone of my last
very many messages:

...14 Every epistemological antinomy can likewise be used
for a similar undecidability proof...(Gödel 1931:43-44)

[olcott]

*I have only been referring to this one quote*
*I have only been referring to this one quote*
*I have only been referring to this one quote*

...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)

*I have not been referring to anything else*
*I have not been referring to anything else*
*I have not been referring to anything else*

"So, you are presuming (INCORRECTLY) that the x in this
formula is an epistemological antinomy in Godel's Proof"

*I am presuming nothing* There is no possible other place

to correctly insert the above quoted epistemological antinomy

in the definition of incompleteness besides x.

[Richard Damon]

And why does he need to insert it in there? What is the rest of the
proof for then?

You just don't seem to understand what you are talking about.

This is likely because you are nothing more than a ignorant,
pathologically lying troll.

[Richard Damon]

Which doesn't mean what you think it does.

And your instance that it does, even after it has been explained
otherwise, just shows that you are just an ignorant lying troll.

You seem to think that the only way you can use something is as a
predicate of a logical operation.

Your imagination is defective.

[Richard Damon]

Who said he inserted the epistemological antinomy in the definition of
incompleteness.

Incompleteness is just defined as there exists a statement in the system
tha is True but not Provable.

An epistemological antinomy can't be such a statement, as it isn't true.

The incompleteness PROOF, (which is diffent than an definition) just
creates a statement, which can be shown to be true in the system, but
not provable. That statement is NOT an epistemological antinomy, as that
couldn't be shown to be true.

Your problem seems to be that your limited imagination can't handle the
logic of the proof.

Yes, the proof has, in effect, the structure of a given epistemological
antinomy included within it, but only a structural basis, the
epistemological antinomy having been changed by making it assert about
provability instead of truthfulness, which yields a statement which has
a valid solution.

[Richard Damon]

On 11/20/23 11:10 AM, olcott wrote:
> On 11/20/2023 9:40 AM, Mikko wrote:
>> On 2023-11-19 15:43:59 +0000, olcott said:
>>
>>> On 11/19/2023 6:35 AM, Mikko wrote:
>>>> On 2023-11-18 16:32:10 +0000, olcott said:
>>>>
>>>>> When we define True(L, x) as (L ⊢ x) provable from the axioms
>>>>> of L, then epistemological antinomies become simply untrue and
>>>>> no longer show incompleteness or undecidability.
>>>>
>>>> That definition does not remove deductive incompleteness of a theory.
>>>
>>> Sure it does, when the criteria that used to prove incompleteness:
>>> Incomplete(L) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
>>> becomes
>>> ¬TruthBearer(L,x) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
>>> Incompleteness cannot possibly exist.
>>
>> The OP did not change or remove the defintion of semantic incompleteness,
>> only of True.
>>
>> Mikko
>>
>
>
> Yes you will get that understanding if you glance at one or two of my
> words before artificially contriving a fake rebuttal.
>
> When you actually pay complete attention then what was previously
> was Incomplete(L) becomes ¬TruthBearer(L,x) the detection of an
> epistemological antinomy.
>
>

The problem is you don't get to change the definition of "Incompleteness".

Note, that for most systems L, there does not exist an x ∈ Language(L)
that is not a TruthBearer in L.

I expalined this to you elsewhere. (in your Godel's huge mistake).

Your inability to undestand what these people are saying doesn't give
you the right to change the meaning of their words try to show they are
saying something that is non-sense.

THe fact that you don't understand the statement that Godel defines as
G, doesn't mean that G is actually a epistemological antinmomy, even if
he uses the term elsewhere in his paper (and not even part of the actual
proof).

So, even when you do exclude non-truthbeares as allowable statments (as
most normal logic systems do), it turns out that if they met the
requirements listed in Godel's proof (mainly being consistant and
support the needed properties of Natural Numbers) then there exist TRUE
statements (and thus CAN'T be non-truthbeares) that are elements of the
Language of the system that are not provable in that system.

[Mikko]

On 2023-11-20 16:10:44 +0000, olcott said:

> On 11/20/2023 9:40 AM, Mikko wrote:
>> On 2023-11-19 15:43:59 +0000, olcott said:
>>
>>> On 11/19/2023 6:35 AM, Mikko wrote:
>>>> On 2023-11-18 16:32:10 +0000, olcott said:
>>>>
>>>>> When we define True(L, x) as (L ⊢ x) provable from the axioms
>>>>> of L, then epistemological antinomies become simply untrue and
>>>>> no longer show incompleteness or undecidability.
>>>>
>>>> That definition does not remove deductive incompleteness of a theory.
>>>
>>> Sure it does, when the criteria that used to prove incompleteness:
>>> Incomplete(L) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
>>> becomes
>>> ¬TruthBearer(L,x) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
>>> Incompleteness cannot possibly exist.
>>
>> The OP did not change or remove the defintion of semantic incompleteness,
>> only of True.
>>
>> Mikko
>>
>
>
> Yes you will get that understanding if you glance at one or two of my words

Instead of carefully reading all of them? Sorry, too late.

> When you actually pay complete attention then what was previously
> was Incomplete(L) becomes ¬TruthBearer(L,x) the detection of an
> epistemological antinomy.

Only one meaning of Incomplete is mentioned above.

TruthBearer as presented above is of different type so not a possible
replacement: Incomplete is a property of a theory but TruthBearer is
a relation of a theory and a sentence.

One can also say that Incomplete(L) ≡  ∃x ∈ Language(L) (¬TruthBearer(L,x)).

Mikko

[olcott]

The function bodies have their function name switched
from Incomplete(L) to ¬TruthBearer(L,x).

> TruthBearer as presented above is of different type so not a possible
> replacement: Incomplete is a property of a theory but TruthBearer is
> a relation of a theory and a sentence.
>
> One can also say that Incomplete(L) ≡  ∃x ∈ Language(L)
> (¬TruthBearer(L,x)).
>
> Mikko
>
>

[Mikko]

On 2023-11-29 04:56:08 +0000, olcott said:

> ¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))

Don't use the same xymbol x for two different meanings
(like above, where it is used both for a free variable
and a bound variable), you only confuse yourself.

Mikko

[olcott]

I am trying to say that when-so-ever an x in the Language of L is
neither provable nor refutable in L then x is not a truth bearer in L.

∀x ∈ Language(L) (¬TruthBearer(L,x) ≡ ((L ⊬ x) ∧ (L ⊬ ¬x)))
This construes every x that would otherwise prove that L is incomplete
as a faulty x that must be excluded from any bivalent formal system.

https://www.liarparadox.org/Wittgenstein.pdf
∀x ∈ Language(L) (True(L,x) ≡ (L ⊢ x))
∀x ∈ Language(L) (False(L,x) ≡ (L ⊢ ¬x))

[Mikko]

On 2023-11-29 15:10:28 +0000, olcott said:

> On 11/29/2023 4:15 AM, Mikko wrote:
>> On 2023-11-29 04:56:08 +0000, olcott said:
>>
>>> ¬TruthBearer(L,x) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
>>
>> Don't use the same xymbol x for two different meanings
>> (like above, where it is used both for a free variable
>> and a bound variable), you only confuse yourself.
>>
>> Mikko
>>
>
> I am trying to say that when-so-ever an x in the Language of L is
> neither provable nor refutable in L then x is not a truth bearer in L.

You could say: TruthBearer(L, x) <-> ((L ⊢ x) ∨ (L ⊢ ¬x)).

> ∀x ∈ Language(L) (¬TruthBearer(L,x) ≡ ((L ⊬ x) ∧ (L ⊬ ¬x)))

That is not a definition but nearly the same. Perhaps one should add
that if x is not in Languabe(L) then ¬TruthBearer(L,x).

> This construes every x that would otherwise prove that L is incomplete
> as a faulty x that must be excluded from any bivalent formal system.

There is no otherwise. It is still true that, with your symbols,
Incomplete(L) <-> ∃x ∈ Language(L) (¬TruthBearer(L,x)).

Mikko

[olcott]

I am redefining the criteria that previously detected Incomplete(L)
to detect Incorrect(x) instead.

https://www.liarparadox.org/Wittgenstein.pdf
∀x ∈ Language(L) (True(L,x) ≡ (L ⊢ x))
∀x ∈ Language(L) (False(L,x) ≡ (L ⊢ ¬x))

The key issue that this solves is that formal systems are no longer
determined to be incomplete on the basis that they cannot determine
whether or not a self-contradictory sentence is true or false.

[Richard Damon]

On 11/29/23 10:10 AM, olcott wrote:
> On 11/29/2023 4:15 AM, Mikko wrote:
>> On 2023-11-29 04:56:08 +0000, olcott said:
>>
>>> ¬TruthBearer(L,x) ≡  ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
>>
>> Don't use the same xymbol x for two different meanings
>> (like above, where it is used both for a free variable
>> and a bound variable), you only confuse yourself.
>>
>> Mikko
>>
>
> I am trying to say that when-so-ever an x in the Language of L is
> neither provable nor refutable in L then x is not a truth bearer in L.

You may be CLAIMING that, but you can't prove that.

In fact, Godel shows that there exist a statement G that IS a truth
bearing, and is in fact TRUE in F but can't be proven in F

>
> ∀x ∈ Language(L) (¬TruthBearer(L,x) ≡ ((L ⊬ x) ∧ (L ⊬ ¬x)))
> This construes every x that would otherwise prove that L is incomplete
> as a faulty x that must be excluded from any bivalent formal system.

Except that is a false statement, and claiming it makes your system
inconsistent, as there are not statements that you call ~Truthbearer,
that do in fact have a truth value.

>
> https://www.liarparadox.org/Wittgenstein.pdf
> ∀x ∈ Language(L) (True(L,x) ≡ (L ⊢ x))
> ∀x ∈ Language(L) (False(L,x) ≡ (L ⊢ ¬x))
>

yes, you can define such a rule, but then MUST limit the axioms of your
system to not allow the creation of the Natural Numbers in it, or your
system becomes inconsistant.

You just don't seem to understand that property,

[Richard Damon]

And no one actually was trying to show that.

It is shown that there exist a TRUE statement (and thus can't be
"self-contradictory) that can't be proven in the system.

You are just showing that you have been tilting at strawmen for years,
because you don't actually understand what you are talking about.

Rmember, the statement G that Godel showed was unprovable is a statement
that states: There does not exist a natural number g, that satisfies a
particular Primative Recursive Relastionship (that was derived in the
proof).

The existance of a number that satisfies a computable criteria is ALWAYS
a truth bearer, as either such a number WILL exist, or WILL NOT exist,
and thus the statement must be True or False.

You inability to understand the statement (or the proof in general)
doesn't change that fact.

[Jim Burns]

On 11/29/2023 10:10 AM, olcott wrote:
> On 11/29/2023 4:15 AM, Mikko wrote:

>> [...]

>
> I am trying to say that
> when-so-ever an x in the Language of L is
> neither provable nor refutable in L
> then x is not a truth bearer in L.

It follows from
what you're trying to say
that various definitions should change.

That would be less effective than you'd like,
it seems to me.

There are these technical terms we've defined:
definientia, singular definiens.
There are these phrases, formulas, etc,
which the definientia represent:
definienda, singular definiendum.

A definition defines a definiens to represent
a definiendum.

In practice, there is nowhere to go and
no one to stand before to argue for
these changes. (sci.logic surely isn't.)
But ignore that.

Hypothetically,
we make an extremely radical change to
these definitions.
We throw out all the offending definientia.
Stop using them. Completely.

Nothing changes.
What was true about
formal systems, arithmetic and
incompleteness
remains true about
formal systems, arithmetic and
incompleteness.
Now, we can't say it,
at least, not the way we have been,
but the truth of
what we're not saying
hasn't changed.

Consider a less fraught example.
The Pythagorean theorem expresses
a fact about right triangles.
Throw out the definiens "right triangle".
It remains true for the definiendum,
a right triangle, that the square of
its longest side is equal to the sum of
the squares of the two other sides.
It's just that we can't say that.

The Pythagorean theorem, Gödel's theorems,
theorems in general aren't edicts.
They aren't authorizing truth by
virtue of their being expressed.

Theorems are recognitions of truths.
If, for any reason, we do not recognize
their truth, they are true anyway.

> ∀x ∈ Language(L)
> (¬TruthBearer(L,x) ≡ ((L ⊬ x) ∧ (L ⊬ ¬x)))

_ ⊢ x
is where sentences go which
you are considering true,
at least for the length of a proof of x
where your hypotheses go.

Your use of 'L' suggests that
you're putting the language there,
both its true and false sentences.
That wouldn't make sense.

Perhaps you're intending to say
IsaTheory(T)
L = LanguageOf(T)
∀x ∈ L
(¬TruthBearer(L,T,x) ≡ ((T ⊬ x) ∧ (T ⊬ ¬x)))

If {x e L|¬TruthBearer(L,T,x)} is not empty
then there are sentences T cannot decide.
You (PO) seem to assign more moral weight
to this than is really warranted.

[olcott]

*The part that you ignored was the important part*
For the entire body of analytic truth True(x) generically means that a
set of inference steps exists from expressions of language that had been
stipulated to be true.

Math and logic are a subset of analytic truth, thus are not actually
allowed to change the way that True(x) generically works.

The lack of this set of inference steps simply means untrue.

>> This construes every x that would
>> otherwise prove that L is incomplete
>> as a faulty x that must be excluded from
>> any bivalent formal system.
>>
>> https://www.liarparadox.org/Wittgenstein.pdf
>> ∀x ∈ Language(L) (True(L,x) ≡ (L ⊢ x))
>> ∀x ∈ Language(L) (False(L,x) ≡ (L ⊢ ¬x))
>
>
>

[olcott]

On 11/30/2023 9:54 AM, Jim Burns wrote:

Every element of the body of analytic truth is only true on the basis of
a set of inference steps derived from a set of expressions of language
that have been stipulated to be true. This is the way that True(x)
actually works. When we are only examining the subset of analytic truth
that is contained within formal system L then we have True(L, x). In
this case unprovable in L simply means untrue in L.

The notion of True in math is not allowed to override the generic way
that True(x) actually works.

>> This construes every x that would
>> otherwise prove that L is incomplete
>> as a faulty x that must be excluded from
>> any bivalent formal system.
>>
>> https://www.liarparadox.org/Wittgenstein.pdf
>> ∀x ∈ Language(L) (True(L,x) ≡ (L ⊢ x))
>> ∀x ∈ Language(L) (False(L,x) ≡ (L ⊢ ¬x))
>
>
>

[Jim Burns]

On 11/30/2023 1:16 PM, olcott wrote:
> On 11/30/2023 9:54 AM, Jim Burns wrote:

>> The Pythagorean theorem, Gödel's theorems,
>> theorems in general aren't edicts.
>> They aren't authorizing truth by
>> virtue of their being expressed.
>>
>> Theorems are recognitions of truths.
>> If, for any reason, we do not recognize
>> their truth, they are true anyway.

> For the entire body of analytic truth
> True(x) generically means that
> a set of inference steps exists from
> expressions of language that
> had been stipulated to be true.

The inference steps allowed
are also not declared by edict.

For a finite sequence of claims,
if there is a false claim,
then there is a first false claim.

That has very little to do with claims or
truth, and much more to do with finite
sequences. For finite sequence of playing
cards, if it has a club, it has a first club.
Etc, etc etc.

Equivalently,
if a finite sequence of playing cards
doesn't have a first club, then
it doesn't have any club.

Also equivalently,
if a finite sequence of claims
doesn't have a first falsehood, then
it doesn't have any falsehood.

What makes this observation useful is that,
for some claims in some sequences,
we can look at them and know that
they are not-first-false,
even if we don't know what the claims mean.

For example,
Q in ⟨... P∨Q ¬P Q ...⟩ is
not-first-false in ⟨... P∨Q ¬P Q ...⟩
Either P∨Q ¬P Q are all true
or one of P∨Q or ¬P is false before Q
We have no meaning for Q
Nonetheless, Q is not-first-false.

In a finite sequence of claims in which
we can see that each is not-first-false,
we can see that each is true,
even if we don't know what the claims mean.

> For the entire body of analytic truth
> True(x) generically means that
> a set of inference steps exists from
> expressions of language that
> had been stipulated to be true.

The inference steps allowed,
such as P∨Q,¬P ⊢ Q
have visibly not-first-false conclusions.
Visibly not-first-false claims will be
visibly not-first-false, whatever we say
or we don't say about the matter.

> ... stipulated to be true.

Usually, these finite sequences of
not-first-false claims include descriptions
of whatever the topic of the day is.

These descriptions will be specific to
the topic of the day, and don't need to be
(are unlikely to be) true of all things,
on-topic or off-topic.

When, in the proof of the Pythagorean theorem,
we _stipulate_ that the geometric figure which
has our attention is a right triangle, what
we do is narrow the topic of the day down to
right triangle.

We are justified in being certain that
our stipulation is true because it describes
the topic of the day, and we know what
the topic of the day is.

> Math and logic are a subset of analytic truth,
> thus are not actually allowed to change
> the way that True(x) generically works.

My Modest Proposal is that
we don't change the way in which,
in a finite sequence of claims,
if any claim is false,
then some claim is first-false,
and that we don't change the way in which
visibly not-first-false claims are
visibly not-first-false,
and the way in which,
if we describe something,
that description is true of what's described,
but it might not be true of things which
aren't described.
YMMV.

> The lack of this set of inference steps
> simply means untrue.

The lack of this set of inference steps

simply means unknown.

In some cases, for the formally undecidable,
they are known to be unknowable.

Other things which are unknowable:
| Triangle ABC is a right triangle.
|
That's unknowable because some triangles
are right, some aren't right, and we don't
have a way here to tell which is referred to.

For the formally undecidable in theory T,
we know
(by a finite all-not-first-false sequence)
that,
in some models of T it's true, and
in some models of T it's false.

We (nearly all of us) are pleased that
formally undecidables are formally undecidable.
The alternative would be
the metamathematical equivalent of
all right triangles vanishing, poof!
It would be very worrying.

[olcott]

On 11/30/2023 3:17 PM, Jim Burns wrote:
> On 11/30/2023 1:16 PM, olcott wrote:
>> On 11/30/2023 9:54 AM, Jim Burns wrote:
>
>>> The Pythagorean theorem, Gödel's theorems,
>>> theorems in general aren't edicts.
>>> They aren't authorizing truth by
>>> virtue of their being expressed.
>>>
>>> Theorems are recognitions of truths.
>>> If, for any reason, we do not recognize
>>> their truth, they are true anyway.
>
>> For the entire body of analytic truth
>> True(x) generically means that
>> a set of inference steps exists from
>> expressions of language that
>> had been stipulated to be true.
>
> The inference steps allowed
> are also not declared by edict.
>

I have no idea how this applies to what I said.
When we know that a cat is an animal and we know
that a cat is a living thing then we only know
these things because they are stipulated to be
true thus providing semantic meanings to otherwise
totally meaningless finite strings.

> For a finite sequence of claims,
> if there is a false claim,
> then there is a first false claim.
>
> That has very little to do with claims or
> truth, and much more to do with finite
> sequences. For finite sequence of playing
> cards, if it has a club, it has a first club.
> Etc, etc etc.
>

I am not assuming that the inference steps must be
a finite sequence. The only way that we can know that
the Goldbach conjecture must be true or false is that
we know that testing every element of the set of natural
numbers would determine this.

I am only referring to analytic truth.
Analytic expressions of language either have a sequence
of inference steps from expressions of language stipulated
to be true or they are untrue.

> In some cases, for the formally undecidable,
> they are known to be unknowable.
>

Formally undecidable is abolished by correcting the
misconception regarding True(L,x).

> Other things which are unknowable:
> | Triangle ABC is a right triangle.
> |
> That's unknowable because some triangles
> are right, some aren't right, and we don't
> have a way here to tell which is referred to.
>

So if we knew that a triangle does have one 90 degree angle
then how does this fail to meet the definition of a right triangle?

> For the formally undecidable in theory T,
> we know
> (by a finite all-not-first-false sequence)
> that,
> in some models of T  it's true, and
> in some models of T  it's false.
>
> We (nearly all of us) are pleased that
> formally undecidables are formally undecidable.

Sure that is the way that learn-by-rote of math and logic works.
On the other hand philosophy of logic does not take every detail
of logic as necessarily coherent. Logicians and mathematicians
take textbooks as their infallible gospel. That is not the way
that truth really works.

> The alternative would be
> the metamathematical equivalent of
> all right triangles vanishing, poof!
> It would be very worrying.
>
>

Every triangle X having a 90 degree angle conclusively proves
X is a right triangle.

[Richard Damon]

But the key point is that True allows for an INFINTE set of inference
steps, but classical logic requires a proof to have only a finite number
of steps.

Thus, what is shown is that a system is incomplete if the only sequence
of steps for some truth in it is an infinite sequence.

If you want to limit truth to only finite length sequences, then you can
not get the properties of the Natural Numbers in a consistent logic system.

If you want to allow proofs to be of infinite length, then provable no
longer means knowable, as knowledge, by definition, is limited to what
can be shown with finite work since we are finite.

[Jim Burns]

On 11/30/2023 5:09 PM, olcott wrote:
> On 11/30/2023 3:17 PM, Jim Burns wrote:

>> [...]

>
> The only way that we can know that
> the Goldbach conjecture must be true or false
> is that we know that testing every element of
> the set of natural numbers would determine this.

No.

First, we aren't able to find out that way.
We can't perform infinitely-many checks.
We are finite.

Second, we don't need to find out that way.

We can state arguments that depend upon
our topic of the day being
a natural number,
but which also _don't_ depend upon
our topic of the day being
a particular natural number.

Using this method,
whether infinitely-many or finitely-many exist,
the conclusion is true of each described.

The _natural numbers_ are infinitely-many.
The _statements about them_ are
finitely-many and finite-length.

That, we can do. Maybe.
But, if we can't, it won't be our inability
to perform supertasks which stops us.

[olcott]

So then not knowing what a truthmaker is you claim that
the Goldbach conjecture doesn't have one.

[olcott]

I have over-ruled and redefined Incomplete so that it ceases to exist.
This same change also eliminates all undecidability. The purpose of
this change is to force True(L,x) to work consistently across every
element of human analytical knowledge.

[olcott]

On 11/30/2023 9:55 PM, Jim Burns wrote:

I have over-ruled and redefined Incomplete so that it ceases to exist.
This same change also eliminates all undecidability. The purpose of
this change is to force True(L,x) to work consistently across every
element of human analytical knowledge.

[Richard Damon]

Which means you haven't actually done anything about actual
"Incompleteness" or "Truth", but only Olcott-Incompleteness and
Olcott-Truth, which no one cares about because your versions don't
actually let us do what we need to.


This has always been your problem, you redefine things to put yourself
in your own fantasy world, and you think you have done something about
reality.

[Richard Damon]

[olcott]

This corrects the divergence of modern logic from the syllogism
so that True(x) works the same way that it works for the entire
body of analytic knowledge: a sequence of inference steps from
expressions of language that have been stipulated to be true makes
x true. The absence of these steps makes x untrue.

[Jim Burns]

On 12/1/2023 12:55 AM, olcott wrote:
> On 11/30/2023 9:55 PM, Jim Burns wrote:

>> We can state arguments that depend upon
>> our topic of the day being
>> a natural number,
>> but which also _don't_ depend upon
>> our topic of the day being
>> a particular natural number.
>>
>> Using this method,
>> whether infinitely-many or finitely-many exist,
>> the conclusion is true of each described.
>>
>> The _natural numbers_ are infinitely-many.
>> The _statements about them_ are
>> finitely-many and finite-length.
>>
>> That, we can do. Maybe.
>> But, if we can't, it won't be our inability
>> to perform supertasks which stops us.
>
> I have over-ruled

That isn't an authority which you have.
That isn't an authority which exists.
Math doesn't work like that.

Math laughs at "authority", guffaws at it.
2+2=4 despite all thrones, dominions and powers.

> I have over-ruled and redefined

Definitions govern how words are used.
No more than that.

If
I define a FISON to be
| an ordered set
| 2.ended, starting from 0
| and, for each of its splits, i‖i⁺¹ exists
| which is last.before‖first.after that split
then
you would be well-advised to believe me
and use that definition to understand me.

Suppose you don't, though.
Suppose you over-rule and redefine "FISON"

You will misunderstand me, and
probably have me misunderstand you.

Other than that? Nothing.

Definitions govern how words are used.
No more than that.

The FISONs, the definienda, are unaffected.
If I was right, I will be right.
If I was wrong, I will be wrong.

And misunderstood.
But that's words.

> I have over-ruled and redefined Incomplete
> so that it ceases to exist.

To review:
A formal system above
a certain low level of expressiveness
is incomplete or is inconsistent.

Along with 2+2=4, that's not something which
you have the power to change.

_At best_
you can create confusion around
what you're saying
(not a typical use of "best").

You have the expressiveness
You have chosen "complete".
You get "inconsistent", at no extra charge.
Enjoy.

> I have over-ruled and redefined Incomplete
> so that it ceases to exist.

Here is ST a tiny, little system which
has enough expressiveness to be incomplete.
| The empty set ∅ exists.
| For each x and y, their adjunct x∪{y} exists.
| Sets with the same elements are equal.

What is there in ST which
you over-rule and redefine?

Or, are you over-ruling and redefining
Q not-first-false in ⟨… P∨Q ¬P Q …>⟩ ?

Over-ruling and redefining
a finite sequence with a false claim
having a first-false claim?

[olcott]

When provable from the axioms of L means true in L, and unprovable in L
means untrue in L then incompleteness and undecidability cannot exist.
When we try to prove that a kitten <is> a 15 story office building the
proof fails. When we try to prove the Liar Paradox the proof fails.
Everything that is unprovable simply becomes untrue. It works this same
way for the entire body of human analytical knowledge.

Every element of the entire body of analytical knowledge is provable
from expressions of language that have been stipulated to be true such
as {cats are animals} and {animals are living things}. These axioms
of natural language provide the true premises for sound deductive
inference: https://iep.utm.edu/val-snd/ Essentially the same thing as
the syllogism.

[Jim Burns]

On 12/1/2023 1:42 PM, olcott wrote:
> On 12/1/2023 11:38 AM, Jim Burns wrote:

>> [...]

>
> When
> provable from the axioms of L means
> true in L, and
> unprovable in L means
> untrue in L
> then
> incompleteness and undecidability
> cannot exist.

When

the objects in the language of L
can be represented by
objects in the domain of L
then
one object represents
"x can't be proved in L"

Consider
| "x can't be proved in L" can't be proved in L

If it's true,
it can't be proved, and
L is incomplete.

If it's false,
it can be proved,
but it's false! and
L is inconsistent.

When
the objects in the language of L
can be represented by
objects in the domain of L
then
the choice is between
incomplete and inconsistent.

> When we try to prove that a kitten <is>
> a 15 story office building
> the proof fails.

In some contexts,
that failure might be unacceptable.

In quantum mechanics and in cosmology,
kittens and 15-story office buildings are
pretty much indistinguishable.

Have you heard the definition of a topologist as
someone who can't distinguish between
a doughnut and a coffee cup?

The reason that's funny and "true"
(for certain values of true),
is that
what topologists study puts
doughnuts and coffee cups in
the same class.

In topology, we _want_ to prove that
a doughnut "is" a coffee cup,
in all the _relevant_ ways,
"relevant" carrying a lot of weight here.

I can imagine that other contexts exist
in which we _want_ to prove that
a kitten "is" a 15-story office building.

[olcott]

On 12/1/2023 2:14 PM, Jim Burns wrote:
> On 12/1/2023 1:42 PM, olcott wrote:
>> On 12/1/2023 11:38 AM, Jim Burns wrote:
>
>>> [...]
>>
>> When
>> provable from the axioms of L means
>> true in L, and
>> unprovable in L means
>> untrue in L
>> then
>> incompleteness and undecidability
>> cannot exist.
>
> When
> the objects in the language of L
> can be represented by
> objects in the domain of L
> then
> one object represents
> "x can't be proved in L"
>
> Consider
> | "x can't be proved in L" can't be proved in L
>
> If it's true,
> it can't be proved, and
> L is incomplete.

I stipulate that this means that x is simply untrue in L.
This <is> the way that the entire body of analytic truth
really works. That math diverges from this is its error.

> If it's false,
> it can be proved,
> but it's false! and
> L is inconsistent.
>
> When
> the objects in the language of L
> can be represented by
> objects in the domain of L
> then
> the choice is between
> incomplete and inconsistent.
>

False dichotomy.
It is perfectly consistent to say that G is untrue in F.
When unprovable means untrue then it does not mean incomplete.

>> When we try to prove that a kitten <is>
>> a 15 story office building
>> the proof fails.
>
> In some contexts,
> that failure might be unacceptable.
>

People with a psychotic break from reality may insist
that we must be able to prove that kittens <are> 15
story office buildings. The coherence theory of truth
screens out such claims.

My purpose in defining True(L, x) as provable from the
axioms of L is to override Tarski undefinability so that
automated reasoning has a consistently sound basis.

Also that <is> the way that correct reasoning actually
works within the entire body of human knowledge thus
making it much more clear that when math diverges from
this that math is incorrect.

The axioms of natural language stipulate that {cats are animals}
thus giving semantic meaning to that otherwise totally meaningless
finite string.

[Jim Burns]

On 12/1/2023 3:46 PM, olcott wrote:
> On 12/1/2023 2:14 PM, Jim Burns wrote:

>> [...}

>
> People with a psychotic break from reality
> may insist that we must be able to prove that
> kittens <are> 15 story office buildings.

Some people are concerned with topology.
They are said, jokingly, to believe that
doughnuts are coffee cups.

Are you (PO) concerned with topology?
If you aren't, that's fine.
Nearly everyone else on the planet isn't.

However,
if you pretend that no one is concerned
with topology, that doesn't speak well of you.

[olcott]

I have a single-minded focus and distractions away from this
point are construed as the strawman deception.

[Richard Damon]

And Godel proves that if your system has such a definition then your
system must either be:

Inconsistent

or

Unable to represent in its logic the properties of the Natuaral Numbers.

>
>> If it's false,
>> it can be proved,
>> but it's false! and
>> L is inconsistent.
>>
>> When
>> the objects in the language of L
>> can be represented by
>> objects in the domain of L
>> then
>> the choice is between
>> incomplete and inconsistent.
>>
>
> False dichotomy.
> It is perfectly consistent to say that G is untrue in F.
> When unprovable means untrue then it does not mean incomplete.

IF "untrue" is different than false, you have just made a statement that
is illogical in most normal logic system, as non-truth bearers are NOT
elements of the language of the system, and Godels G was shown to be a
statement that WAS in the language of F by its construction.

If untrue is the same as false, you now have a statement that is false,
but can also be proven to be true, so your system is inconsistent.

>
>>> When we try to prove that a kitten <is>
>>> a 15 story office building
>>> the proof fails.
>>
>> In some contexts,
>> that failure might be unacceptable.
>>
>
> People with a psychotic break from reality may insist
> that we must be able to prove that kittens <are> 15
> story office buildings. The coherence theory of truth
> screens out such claims.

No, you are the only one that claims that.

>
> My purpose in defining True(L, x) as provable from the
> axioms of L is to override Tarski undefinability so that
> automated reasoning has a consistently sound basis.
>

Which, since that ISN'T the definition of true is normal logic, means it
is exactly the same sort of claim as "Kittens are 15 story office
buildings".

If you want to work in a non-standard logic, go ahead, but don't assume
that things developed with normal logic apply. This has been pointed out
before, but you seem to be admitting you don't know what to do with
that, so you will continue to just spout off unsound logic statements.

> Also that <is> the way that correct reasoning actually
> works within the entire body of human knowledge thus
> making it much more clear that when math diverges from
> this that math is incorrect.

Nope. You are conflating True and Known, showing you don't understand
either.

>
> The axioms of natural language stipulate that {cats are animals}
> thus giving semantic meaning to that otherwise totally meaningless
> finite string.
>

Right, and True means it is ACTUALLY TRUE, and doesn't imply that we do
(or even can) know it. So, the basis of your logic system seems to be a lie.

[Richard Damon]

[Richard Damon]

Provable n L implies True in L.

Unprovable does NOT imply False in logic.

If you want to work in a limited Formal System with such a rule, go
ahead, just be aware that this will strictly limit what you can do in
that system without making your system inconsistent.

The fact that you don't understand this doesn't make it not true.

>
> Every element of the entire body of analytical knowledge is provable
> from expressions of language that have been stipulated to be true such
> as {cats are animals} and {animals are living things}. These axioms
> of natural language provide the true premises for sound deductive
> inference: https://iep.utm.edu/val-snd/ Essentially the same thing as
> the syllogism.
>

Right, KNOWLEDGE, as analytical knowledge is provable.

But there are things that are True, that may never be known, and your
failure to understand that shows the limitations of your thought process.

[Richard Damon]

On 12/1/23 11:43 AM, olcott wrote:

> This corrects the divergence of modern logic from the syllogism
> so that True(x) works the same way that it works for the entire
> body of analytic knowledge: a sequence of inference steps from
> expressions of language that have been stipulated to be true makes
> x true. The absence of these steps makes x untrue.
>
>
>

Nope, and just shows that you don't understand a word of what you say.

Yes, a (potentially infinite) sequence of inference steps from
expression of language that have been stipulated to be true makes x true.

Your INCORRECT assumption that the sequence needs to be finite, shows
the limitation of your thinking, and why you are wrong.

[Richard Damon]

On 12/1/23 6:41 PM, olcott wrote:
> On 12/1/2023 4:26 PM, Jim Burns wrote:
>> On 12/1/2023 3:46 PM, olcott wrote:
>>> On 12/1/2023 2:14 PM, Jim Burns wrote:
>>
>>>> [...}
>>>
>>> People with a psychotic break from reality
>>> may insist that we must be able to prove that
>>> kittens <are> 15 story office buildings.
>>
>> Some people are concerned with topology.
>> They are said, jokingly, to believe that
>> doughnuts are coffee cups.
>>
>> Are you (PO) concerned with topology?
>> If you aren't, that's fine.
>> Nearly everyone else on the planet isn't.
>>
>> However,
>> if you pretend that no one is concerned
>> with topology, that doesn't speak well of you.
>
>
> I have a single-minded focus and distractions away from this
> point are construed as the strawman deception.
>
> My purpose in defining True(L, x) as provable from the
> axioms of L is to override Tarski undefinability so that
> automated reasoning has a consistently sound basis.

Except you only do so by limiting the domain of your logic to things
that very simple.

>
> Also that <is> the way that correct reasoning actually
> works within the entire body of human knowledge thus
> making it much more clear that when math diverges from
> this that math is incorrect.
>
> The axioms of natural language stipulate that {cats are animals}
> thus giving semantic meaning to that otherwise totally meaningless
> finite string.
>

And they also stipulate that there either exists a natural number with a
given computable property or their doesn't, thus a statement about the
existance of such a number has a truth value, (perhaps not a knowable
value) even if we can't prove or disprove its existance.

Only by banning that question, which means banning mathematics, can you
get around that problem.

[Jim Burns]

On 12/1/2023 6:41 PM, olcott wrote:
> On 12/1/2023 4:26 PM, Jim Burns wrote:
>> On 12/1/2023 3:46 PM, olcott wrote:

>>> People with a psychotic break from reality
>>> may insist that we must be able to prove that
>>> kittens <are> 15 story office buildings.
>>
>> Some people are concerned with topology.
>> They are said, jokingly, to believe that
>> doughnuts are coffee cups.
>>
>> Are you (PO) concerned with topology?
>> If you aren't, that's fine.
>> Nearly everyone else on the planet isn't.
>>
>> However,
>> if you pretend that no one is concerned
>> with topology, that doesn't speak well of you.
>
> I have a single-minded focus
> and distractions away from this point
> are construed as the strawman deception.

You think that
a single-minded focus away from your mistakes
will save you from making mistakes.

Spoiler alert!

It won't.

[olcott]

Ludwig Wittgenstein one of the most famous philosophers
of logic perfectly agrees with me. He was a leader of the
logical positivists.

I know that he is correct because I figured out every single
detail of his view and why these details are correct before I
ever heard of him.
https://www.liarparadox.org/Wittgenstein.pdf
Formalized as:
True(L,x) ≡ (T ⊢ x)
False(L,x) ≡ (T ⊢ ¬x)

Logicians only memorize the rules of logic and take them
as inherently infallible have no actual understanding of
these things.

Rote memorization is the complete depth of their knowledge.
If some of the rules of logic don't fit together coherently
they don't have the ability or the inclination to notice this.

[Richard Damon]

So, you are just admitting to your use of the logical fallicy of proof
by authority. It must be true because "one of the most famous
philosophers of logic" says so. By that logic is is also FALSE because
many more people of that classification say so, and then an ad-hominem
attack on anyone who disagrres with you.

Also, if I remember right about the paper you are quoting, it is a
publication, after death, of note that he himself nevver attempted to
publish, and thus, you don't actually have an actual indication that he
finally agreed with it, he may well have figured out the error and just
dropped the line.

Also, you are ignoring the errors pointed out by many other famous
philosophers who have seen the work.

Note, by your statement that "Logicias only meorize the rules..." you
are effecgtively admitting that your own logic doesn't follow those
rules, which is fine, but that means you need to see what your logic can
actually do, which you don't seem capable of doing, making your
observations really worthless.

Many of you ideas are NOT "new" but I have seen brought up before, but
those people understood that they were branching off into new territory
and observed what are the actual limitations of such a logical system.
You, who seems to have ignored any actual study of the history of the
field, have just fulfilled the saying and doomed yourself to repeating
all the mistakes that were made and discovered.

If you want to try to show that some rules don't fit together, then try
to do a ACTUAL FORMAL PROOF that shows the contradiction. Note, that
means a contradiction as defined by that system. The fact that you don't
like the FACT that most formal systems come up incomplete is NOT a
contractidiction, but must your own limitition in understand.

All you have done is proved your own self-imposed ignorance.

[Ross Finlayson]

But, Goedel's incompleteness theorems, have that
thusly those theorems, are bereft again, simply
via "artifacts or sputniks of quantifications".

Of course then some associate that with extra-ordinary
theories, and that ZF is a piffling fragment of the universe
of objects that are sets (or classes, as about class/set distinction,
about predicates, about predicates being relations, about that
really it's relations that are primary not predicates, that propositions
are subject class/set distinction, that there are non-Cartesian functions,
about continuous domains, ...).

Some have that ZF doesn't have a standard model.
ZF's still great for what it is after finite combinatorics,
and also makes for what happens when censoring
comprehension to make a (... false) axiomatization of the
definition of an inductive set as ordinary, well-founded, ...,
but, just saying, there's a bigger world, "universe" of sets.

[Ross Finlayson]

"Logical, positivists", are usually enough, neither.

Logical positivism ideally though is of course , ..., "true".

[Ross Finlayson]

Now Richard I want to compliment you because I appreciate your fervor
and much of what you say, but, "All you have done is proved your own
self-imposed ignorance. "

I.e., you provide a defense of a certain formalism, but, it's closed,
and Goedel broke it open.

It's like "why did Cohen prove CH independent ZFC", well, because otherwise
either way it would disprove either Cantor/Schroeder/Bernstein because
there'd be an infinitude of cardinals between any two, or, it would disprove
Cantor/Schroeder/Bernstein because there wouldn't. Do you know _how_ Cohen
proved CH independent ZFC? He introduced an axiom that there's a maximal ordinal,
then uses it both ways.

Most people failed to read the memo even assuming they should infer one exists,
and unfortunately they don't much know where their food comes from nor to
thoroughly chew it.

[olcott]

When we define the measure of analytical true that way that it
consistently works for the whole body of analytical knowledge
then the only actual incompleteness are unknown truths such as
the Goldbach conjecture. Such as system as Wittgenstein's and
mine simply determines that epistemological antinomies are simply
untrue.

...14 Every epistemological antinomy can likewise be used for a
similar undecidability proof...(Gödel 1931:43-44)

The conventional mathematical notion of undecidability incorrectly
assumes that self-contradictory sentences must be provably true or
false. That is so ridiculously stupid that I can imagine how this
mistake was not discovered back in 1931.

[Jim Burns]

On 12/1/2023 3:46 PM, olcott wrote:
> On 12/1/2023 2:14 PM, Jim Burns wrote:
>> On 12/1/2023 1:42 PM, olcott wrote:

>>> When
>>> provable from the axioms of L means
>>> true in L, and
>>> unprovable in L means
>>> untrue in L
>>> then
>>> incompleteness and undecidability
>>> cannot exist.
>>
>> When
>> the objects in the language of L
>> can be represented by
>> objects in the domain of L
>> then
>> one object represents
>> "x can't be proved in L"
>>
>> Consider
>> | "x can't be proved in L" can't be proved in L

Oops.
Better:
"preceded by its quotation can't be proved in L"
preceded by its quotation can't be proved in L.

Not a self-reference, but
a self-description.

>> If it's true,
>> it can't be proved, and
>> L is incomplete.
>
> I stipulate that
> this means that x is simply untrue in L.

1/2. You introduce a private meaning.

> This <is> the way that
> the entire body of analytic truth really works.

2/2. You claim (stipulate?) that
everyone is using your private meaning.


You think that stipulating is
your dragon-slaying sword.

Stipulating is almost always inappropriate.
But, yes, in those instances in which
it is appropriate, it is a dragon-slayer.

I can stipulate that ABC is a right triangle.
By doing that, I chalk the outline of
the conversation. _I_ am talking about
a right triangle. If you join me, _you_ are
talking about a right triangle. If you reject
my stipulation, you haven't joined me.
Stipulation slays non-right-triangle dragon.

I can't (I really, really shouldn't) stipulate
that the square of the hypotenuse of ABC is

equal to the sum of the squares of

the two remaining sides of ABC.

The goal should be to convince you of that.
A stipulation does no such thing.
Used in this way,
a stipulation does not slay dragons,
it stomps off the field, crying about how
the dragon unfairly didn't lay down and die.

> That math diverges from this is its error.

A finite sequence of claims with
no first-false claim
has no false claim.

Q in ⟨… P∨Q ¬P Q …⟩ is not first-false.

If you stipulate otherwise,
you'd be better off with 15-story kittens.

>> If it's false,
>> it can be proved,
>> but it's false! and
>> L is inconsistent.
>>
>> When
>> the objects in the language of L
>> can be represented by
>> objects in the domain of L
>> then
>> the choice is between
>> incomplete and inconsistent.
>
> False dichotomy.

Theorem.
Convincing everyone the choices are
incomplete or inconsistent
is what the proof of the theorem is for.

Ignoring a proof does not make it wrong.

> It is perfectly consistent to say that
> G is untrue in F.
> When unprovable means untrue
> then it does not mean incomplete.

Words have meanings.
Remove the words, and the meanings remain,
(silently now)
with the same nature they always had.

----
Consider the system ST [Boolos] with
empty set, adjunct, and extensionality.
| ∃x∀u: u∉x
| ∀x∀y∃z: ∀u(u∈z ⟺ u∈x ∨ u=y)
| ∀x∀y: x=y ⟺ ∀u(u∈x ⇔ u∈y)

I stipulate that ST is
what we're taking about, right now.
Reject my stipulation. Go ahead.
Now we two are aren't talking about anything.
No wins, no losses, but no progress.

I stipulate that

"z is the adjunct of y to x" means
z = x†y ⟺ ∀u(u∈z ⟺ u∈x ∋ u=y)

x†y = x∪{y}
x†y†z = (x†y)†z

"0 is the empty set" means
∀u: u∉0

"y is the successor of x" means
y = x⁺¹ ⟺ y = x†x

"x is the predecessor of y" means
x = y⁻¹ ⟺ y = x⁺¹

"x is less than y" means
x < y ⟺ x ∈ y

"x is a natural number" means
ℕ∋(x) ⟺
x = 0 ∨
(x⁻¹<x ∧ ∀u<x:(u=0 v u⁻¹<x))

"z is the ordered pair ⟨x,y⟩" means
z = ⟨x,y⟩ ⟺ z = 0†(0†x)†(0†x†y)

⟨x,y⟩ = {{x},{x,y}} [Kuratowski]
⟨x,y,z⟩ = ⟨⟨x,y⟩,z⟩
⟨x,…,y,z⟩ = ⟨⟨x,…,y⟩,z⟩

For ℕ∋(x) ℕ∋(y) ℕ∋(z)
"z is the sum of x and y" means
x + y = z ⟺
⟨ ⟨x,0,x⟩ ⟨x,0⁺¹,x⁺¹⟩ … ⟨x,y,z⟩ ⟩ exists
such that
for each of its splits Fᣔ<ᣔH
some ⟨x,i,j⟩‖⟨x,i⁺¹,j⁺¹⟩ is last‖first in F‖H

For ℕ∋(x) ℕ∋(y) ℕ∋(z)
"z is the product of x and y" means
x × y = z ⟺
⟨ ⟨x,0,0⟩ ⟨x,0⁺¹,0+x⟩ … ⟨x,y,z⟩ ⟩ exists
such that
for each of its splits Fᣔ<ᣔH
some ⟨x,i,j⟩‖⟨x,i⁺¹,j+x⟩ is last‖first in F‖H

For ℕ∋(n)
"z is f applied to x recursively n times" means
z = f⁽ⁿ⁾(x) ⟺
⟨ ⟨0,x⟩ ⟨0⁺¹,f(x)⟩ … ⟨n,z⟩ ⟩ exists
such that
for each of its splits Fᣔ<ᣔH
some ⟨i,y⟩‖⟨i⁺¹,f(y)⟩ is last‖first in F‖H

That is the usual natural number arithmetic,
given here as definitions in and theorems of ST

----
Apart from
| ∃x∀u: u∉x
| ∀x∀y∃z: ∀u(u∈z ⟺ u∈x ∨ u=y)
| ∀x∀y: x=y ⟺ ∀u(u∈x ⇔ u∈y)
|
all those stipulations are optional.
For the rest, they are
definiens ⟺ definiendum

Remove the definiens and
the definiendum remains.

Those stipulations are essential for humans
communicating to other humans
how to represent arithmetic in ST,
but their truth or falsity is unaffected
by those stipulations.

Those are dragon-slayer stipulations, but
only because rejecting them has no consequences.

[olcott]

*Think it through*
Everything that you know is true on the basis of its meaning
AKA the entire body of analytical knowledge <is> only known
to be true on the basis of its meaning.

(a) Cats <are> Animals is true on the basis of the meaning
of {cats} and the meaning of {animals}.

(b) Animals <are> living things is true on the basis of the
meaning of {animals} and the meaning of {living things}.

(c) That {cats} <are> {living things} is sound deductive
inference on the basis of true premises (a) and (b)

The entire body of analytic knowledge is proven to work this
same way in that counter-examples are categorically impossible.

If you diligently try to find a counter-example you will find
that none can possibly exist because analytic truth is defined
to depend on its meanings. This means that its proof can always
be traced back to its meanings or it is not analytic truth.

True(L,x) ≡ (T ⊢ x) traces x back to its meanings in L.
In formal proofs these semantic meanings would be syntactically
formalized.

It is also common knowledge that all of math and all of
logic are subsets of the body of analytic truth.