A criterion for truth of equi-consistent extensions of ZF?

[Zuhair]

This method tries to solve questions like truth of V=L, or CH, or GCH, choice, etc..

Now the idea is that the axiom that leads to a restriction on the ability to extend ZFC is the faulty axiom. See:

https://math.stackexchange.com/questions/3721227/can-the-question-of-generalized-continuum-hypothesis-be-solved-along-such-lines?

[Rupert]

If "large cardinal property" is defined so as to include "invariant under small forcing", and if we also assume that all large cardinal properties have associated L-like inner models in which they are realised, which are models of GCH (which is not known for all currently known large cardinal axioms but is a plausible conjecture), then in that case both (i) and (ii) would be true.

[Zuhair]

Can the above line of specifications work to prove that V=L is false?

[Rupert]

Assuming consistency of strong enough large cardinals, yes. You could check out Penelope Maddy's thoughts along those lines in "Naturalism in Mathematics", in the section "A Naturalist's case Against V=L".

[Zuhair]

Thank you Rupert. That's been very helpful.

[Ross A. Finlayson]

Goedel, Feferman, ..., V = L? (Constructivist.)

I think it's fair to _leave_ V = L false, in
terms of any construction in terms over infinites,
that though there is still "V = L" for completeness results.

If you're asserting that non-logical axiomatization of
restriction of comprehension introduces quantifier ambiguity
and impredicativity, over the completeness results or that
"this V is not equal to L, the universe", there's still that
in terms, "this (other) V the completeness of algebra of arithmetic",
is no different than "L, the universe of an algebra of arithmetic",
in the case where that is also "the" universe.

I.e., it's fair to prove "V = L" is left false and
that constructivism instead is what results.

I.e., it's conditioned on types under arithmetic.

Arithmetic is fair, abstract, and general, for
what of course that it's any "concrete".

I.e., arithmetizing the concrete is with the
"infinitely many" numbers besides for example
all the models with counting (each).

The "V, the constructible universe", is all that
"L, the universe of things", while "constructible
closures or completions of domains as complete",
has that there are as many copies of that for
counting as there is counting. (Or, "V < L".)

[Ross A. Finlayson]

Tarski will agree for truth V < L, though, not about results in V=L.

I.e., that's, "equi-interpretable with Tarski truth".

It's fair for Tarski truth that independence results are honest,
only allowing for constructivist results (of the intuitive, that,
"honest" might be "wrong").

This is for Cohen independence that for V < L and V <= L, that
also there is "L <= V" where the inequality is not strict, though
of could that both "V NOT > L", where that the constructible is
complete (or sufficient for the model in terms) that what would
otherwise result from the simple strict inequality, in the Bourbaki
sense of course (and even the panel), that "V < L" still has for that
any construction X in V that, X < V < L.

I.e., that's not so where "V in L", except what x in V and x in L,
that "V in L" is "V < L".

Then, for absolute truth for Tarski truth, in foundations, of
course that's fundamental and even a usual general question
for philosophy, truth and logical truth.

[George Greene]

On Thursday, June 18, 2020 at 3:15:06 AM UTC-4, Zuhair wrote:
> This method tries to solve questions
> like truth of V=L, or CH, or GCH, choice, etc..

"Truth" is just plain meaningless in this context.
*E*V*E*R*Y* consistent theory has a model.
EVERY sentence of a theory is true in any model of that theory.
Any of the major independence results are proofs that there is a
model of the thing AND a model of its denial.
They are BOTH "true" -- just never at the same time in the SAME
model.

>
> Now the idea is that the axiom that
> leads to a restriction on the ability to extend ZFC
> is the faulty axiom.

All choices of one or the other truth-value for a sentence independent
of an axiom "restrict" any further extensions -- they just restrict them
in DIFFERENT WAYS -- extensions that are impossible when V=L
may be possible when V=/=L, AND CONVERSELY!!


https://math.stackexchange.com/questions/3721227/can-the-question-of-generalized-continuum-hypothesis-be-solved-along-such-lines?

Why don't you just participate there? There isn't much of anyone left here.

[George Greene]

On Thursday, June 18, 2020 at 3:15:06 AM UTC-4, Zuhair wrote:

> https://math.stackexchange.com/questions/3721227/can-the-question-of-generalized-continuum-hypothesis-be-solved-along-such-lines?

Oh, well you already got -1 over there...
Seriously -- you are not likely to get taken seriously if you persist
in alleging that any independence result (e.g. GCH) might be "true" --
the independence MEANS that it is true in some models and false in others.
You have to come up with some criterion for preferring some kinds of models
to others, and usually that means just adding another axiom.
Adding the denial of that axiom instead was equally legitimate, so you are
never going to get anywhere. You need to phrase the question in terms
of which kinds of models are better, not of whether some independent statement
is or isn't "true" -- it's UP TO YOU whether it's true or not -- you can choose
to investigate models where it's true OR models where it's false.

[George Greene]

On Thursday, June 18, 2020 at 4:13:06 AM UTC-4, Rupert wrote:
> > https://math.stackexchange.com/questions/3721227/can-the-question-of-generalized-continuum-hypothesis-be-solved-along-such-lines?
>
> If "large cardinal property" is defined so as to include "invariant under small forcing", and if we also assume that all large cardinal properties have associated L-like inner models in which they are realised, which are models of GCH (which is not known for all currently known large cardinal axioms but is a plausible conjecture), then in that case both (i) and (ii) would be true.

Rupert, his post at stackexchange tries to define "true" FOR A THEORY.
This is kind of absurd and you know it.

[Zuhair]

It is not! This is well known. YOU seem to prefer the MULTI-VERSE view, that's fine. Joel David Hamkins is one the proponents of that view. Anyhow there is the old fashion "UNI-VERSE" view, i.e. one universe for mathematical realm. And I'm speaking with respect to this veiw, Maddy also seems to speak along the same lines.

[Julio Di Egidio]

On Sunday, 21 June 2020 08:13:41 UTC+2, Zuhair wrote:
> On Sunday, June 21, 2020 at 8:49:29 AM UTC+3, George Greene wrote:

> > Rupert, his post at stackexchange tries to define "true" FOR A THEORY.
> > This is kind of absurd and you know it.
>
> It is not! This is well known. YOU seem to prefer the MULTI-VERSE view,
> that's fine.

No, it is not even fine: the G sentence is *true*hence*unprovable*, just since
it is independent of the axioms they think they can add the negation of G to
the theory, but even if that is consistent, it is of corse *in*correct*, i.e.
the theory now proves a false statement. And that is model theory in a
nutshell, and yet another planetary-wide fraud.

> Joel David Hamkins is one the proponents of that view. Anyhow there is
> the old fashion "UNI-VERSE" view, i.e. one universe for mathematical realm.
> And I'm speaking with respect to this veiw, Maddy also seems to speak along
> the same lines.

Old-fashioned as in *correct*.

Julio

[George Greene]

On Sunday, June 21, 2020 at 2:13:41 AM UTC-4, Zuhair wrote:
> On Sunday, June 21, 2020 at 8:49:29 AM UTC+3, George Greene wrote:
> > On Thursday, June 18, 2020 at 4:13:06 AM UTC-4, Rupert wrote:
> > > > https://math.stackexchange.com/questions/3721227/can-the-question-of-generalized-continuum-hypothesis-be-solved-along-such-lines?
> > >
> > > If "large cardinal property" is defined so as to include "invariant under small forcing", and if we also assume that all large cardinal properties have associated L-like inner models in which they are realised, which are models of GCH (which is not known for all currently known large cardinal axioms but is a plausible conjecture), then in that case both (i) and (ii) would be true.
> >
> > Rupert, his post at stackexchange tries to define "true" FOR A THEORY.
> > This is kind of absurd and you know it.
>
> It is not! This is well known.

You are SO full of shit.

> YOU seem to prefer the MULTI-VERSE view, that's fine.

It's more than "fine" and it's NOT a theory.
THE BURDEN of defense is on the people WHO THINK they can EVEN DEFINE
"truth" in this context. They cannot meet it.

> Joel David Hamkins is one the proponents of that view.

It's not just a view.
People a hell of a lot smarter THAN you, LET ALONE you, CAN'T defend
"truth" in this paradigm FOR A THEORY -- "truth" is EXPLICITLY
semantic/model-DEpendent in this paradigm. If there are models both ways
then there simply IS NO "truth" of the matter.
When people CLAIM "truth", they ARE PRIVILEGING A CERTAIN CLASS of models,
one that THEY CANNOT DEFINE via THE USUAL method of simply adding axioms
to restrict the models to the class.

> Anyhow there is the old fashion "UNI-VERSE" view,

It's OLD for a REASON and EVEN IT can't make differing models disappear.
All it can do is handwave. And it frankly does a shitty job of that too.

[George Greene]

On Sunday, June 21, 2020 at 9:38:08 AM UTC-4, Julio Di Egidio wrote:
> No, it is not even fine: the G sentence is *true*hence*unprovable*,

Given that there are models of PA in which PA's G-sentence IS FALSE -- that's true NOT JUST of PA but OF EVERY relevant theory for this question -- it is just LYING to say that G "is true".

[Khong Dong]

What's the point of arguing when the construction of PA's G-sentence is invalid
in the first place?

[Julio Di Egidio]

<< (G) F |- G_F <-> ~Prov_F([G_F]). Thus, it can be shown, even inside F,
that G_F is true if and only if it is not provable in F. >>
<https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom>

So, if G were false, the system would be proving a false statement, hence G
*must* be true (a true arithmetical statement!) and, *as such*, unprovable.
Which is GIT in a nutshell. QED.

> > [Then, as I said] ... just since

it is independent of the axioms they think they can add the negation of G to
the theory, but even if that is consistent, it is of corse *in*correct*, i.e.
the theory now proves a false statement. And that is model theory in a
nutshell, and yet another planetary-wide fraud.

(HTH, the innocent reader. You'll just keep snipping and repeating your
mantra, you umpteenth unreasoning asshole... EOD.)

Julio

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Thursday, 2 July 2020 07:58:53 UTC+2, George Greene wrote:
>> On Sunday, June 21, 2020 at 9:38:08 AM UTC-4, Julio Di Egidio wrote:
>> > No, it is not even fine: the G sentence is *true*hence*unprovable*,
>>
>> Given that there are models of PA in which PA's G-sentence IS FALSE
>> -- that's true NOT JUST of PA but OF EVERY relevant theory for this
>> question -- it is just LYING to say that G "is true".
>
> << (G) F |- G_F <-> ~Prov_F([G_F]). Thus, it can be shown, even inside F,
> that G_F is true if and only if it is not provable in F. >>
> <https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom>

This is badly presented in the link;
but you have missed the initial caveat:

"In fact, *in favourable circumstances*, it can be shown that GF is true,
provided that F is indeed consistent." (emphasis added)

> So, if G were false, the system would be proving a false statement, hence G
> *must* be true (a true arithmetical statement!) and, *as such*, unprovable.
> Which is GIT in a nutshell. QED.

Fnobble....

btw, the Stanford entry mentions Peter Milne's article on the issue of
which results in this area require truth of the theory being examined,
and which do not.

I recommend it to anyone interested:
Milne, P., 2007, “On Gödel Sentences and What They Say,” Philosophia
Mathematica, 15: 193–226.

> Julio

--
Alan Smaill

[Ross A. Finlayson]

It is simply a matter over resources for completeness or incompleteness.

Cohen's establishment of independence of CH goes without saying as deft.

[Julio Di Egidio]

On Thursday, 2 July 2020 16:40:02 UTC+2, Alan Smaill wrote:
> Julio Di Egidio <ju...@diegidio.name> writes:
> > On Thursday, 2 July 2020 07:58:53 UTC+2, George Greene wrote:
> >> On Sunday, June 21, 2020 at 9:38:08 AM UTC-4, Julio Di Egidio wrote:
> >> > No, it is not even fine: the G sentence is *true*hence*unprovable*,
> >>
> >> Given that there are models of PA in which PA's G-sentence IS FALSE
> >> -- that's true NOT JUST of PA but OF EVERY relevant theory for this
> >> question -- it is just LYING to say that G "is true".
> >
> > << (G) F |- G_F <-> ~Prov_F([G_F]). Thus, it can be shown, even inside F,
> > that G_F is true if and only if it is not provable in F. >>
> > <https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom>
>
> This is badly presented in the link;
> but you have missed the initial caveat:
>
> "In fact, *in favourable circumstances*, it can be shown that GF is true,
> provided that F is indeed consistent." (emphasis added)

No, it's you who pick the parts that are irrelevant or badly broken: I have
a thesis and a reasoning.

> > So, if G were false, the system would be proving a false statement, hence G
> > *must* be true (a true arithmetical statement!) and, *as such*, unprovable.
> > Which is GIT in a nutshell. QED.
>
> Fnobble....

As said, you guys just do not know what reasoning or discussion even is:
only authorities and parrots. It's the end of the world...

*Plonk*

Julio

[Julio Di Egidio]

On Thursday, 2 July 2020 18:59:14 UTC+2, Julio Di Egidio wrote:
> On Thursday, 2 July 2020 16:40:02 UTC+2, Alan Smaill wrote:
> > Julio Di Egidio <ju...@diegidio.name> writes:
> > > On Thursday, 2 July 2020 07:58:53 UTC+2, George Greene wrote:
> > >> On Sunday, June 21, 2020 at 9:38:08 AM UTC-4, Julio Di Egidio wrote:
> > >> > No, it is not even fine: the G sentence is *true*hence*unprovable*,
> > >>
> > >> Given that there are models of PA in which PA's G-sentence IS FALSE
> > >> -- that's true NOT JUST of PA but OF EVERY relevant theory for this
> > >> question -- it is just LYING to say that G "is true".
> > >
> > > << (G) F |- G_F <-> ~Prov_F([G_F]). Thus, it can be shown, even inside F,
> > > that G_F is true if and only if it is not provable in F. >>
> > > <https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom>
> >
> > This is badly presented in the link;
> > but you have missed the initial caveat:
> >
> > "In fact, *in favourable circumstances*, it can be shown that GF is true,
> > provided that F is indeed consistent." (emphasis added)
>
> No, it's you who pick the parts that are irrelevant or badly broken: I have
> a thesis and a reasoning.
>
> > > So, if G were false, the system would be proving a false statement, hence G
> > > *must* be true (a true arithmetical statement!) and, *as such*, unprovable.
> > > Which is GIT in a nutshell. QED.
> >
> > Fnobble....

BTW, notice that the above was just the definition of G, this is the actual
derivation of the thesis: not even that you get right...

> As said, you guys just do not know what reasoning or discussion even is:
> only authorities and parrots. It's the end of the world...

And the most fucking pathetic one at that.

(EOD.)

*Plonk*

Julio

[Alan Smaill]

Julio Di Egidio <ju...@diegidio.name> writes:

> On Thursday, 2 July 2020 16:40:02 UTC+2, Alan Smaill wrote:
>> Julio Di Egidio <ju...@diegidio.name> writes:
>> > On Thursday, 2 July 2020 07:58:53 UTC+2, George Greene wrote:
>> >> On Sunday, June 21, 2020 at 9:38:08 AM UTC-4, Julio Di Egidio wrote:
>> >> > No, it is not even fine: the G sentence is *true*hence*unprovable*,
>> >>
>> >> Given that there are models of PA in which PA's G-sentence IS FALSE
>> >> -- that's true NOT JUST of PA but OF EVERY relevant theory for this
>> >> question -- it is just LYING to say that G "is true".
>> >
>> > << (G) F |- G_F <-> ~Prov_F([G_F]).
>> > Thus, it can be shown, even inside F,
>> > that G_F is true if and only if it is not provable in F. >>
>> > <https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom>
>>
>> This is badly presented in the link;
>> but you have missed the initial caveat:
>>
>> "In fact, *in favourable circumstances*, it can be shown that GF is true,
>> provided that F is indeed consistent." (emphasis added)
>
> No, it's you who pick the parts that are irrelevant or badly broken: I have
> a thesis and a reasoning.

Then make yourself clear.

>> > So, if G were false, the system would be proving a false statement,
>> > hence G *must* be true (a true arithmetical statement!) and, *as
>> > such*, unprovable. Which is GIT in a nutshell. QED.
>>
>> Fnobble....

For example, I really cannot tell how you intend others to take
your paragraph above -- a parody of nonsense is my first reaction
(rather than simple nonsense). But I really cannot tell.

> As said, you guys just do not know what reasoning or discussion even is:
> only authorities and parrots. It's the end of the world...

So, give us some response of substance, something constructive;
otherwise any sort of reasoned debate becomes more and more
difficult.

> *Plonk*
>
> Julio

--
Alan Smaill

[Alan Smaill]

OK, so that is one confusion on my part, your use of
"thesis" here misled me.

AIUI (you will correct me):
* George Greene objected to a claim that G can simply be taken as true;
* You objected by quoting that

"G_F is true if and only if it is not provable in F."

(my immediate objection was to that claim)
* You then say: if G were false, the system would be proving
a false statement.

If you mean (G) above: (G) F |- G_F <-> ~Prov_F([G_F]),
then I am lost.
I assume you mean the G from
"the G sentence is *true*hence*unprovable*"

Now you say:

So, if G were false, the system would be proving a false

statement.
-- the false statement being G (I presume??).

When the conditions for the theorem are met, G is not provable
(regardless of whether it is true or false).

I am confused here.

> (EOD.)

[Jim Burns]

On 7/2/2020 3:36 AM, Julio Di Egidio wrote:
> On Thursday, 2 July 2020 07:58:53 UTC+2,
> George Greene wrote:
>> On Sunday, June 21, 2020 at 9:38:08 AM UTC-4,
>> Julio Di Egidio wrote:

>>> No, it is not even fine: the G sentence is
>>> *true*hence*unprovable*,
>>
>> Given that there are models of PA in which PA's G-sentence
>> IS FALSE -- that's true NOT JUST of PA but OF EVERY relevant
>> theory for this question -- it is just LYING to say that G
>> "is true".
>
> << (G) F |- G_F <-> ~Prov_F([G_F]).
> Thus, it can be shown, even inside F, that
> G_F is true if and only if it is not provable in F. >>
> <https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom>
>
> So, if G were false, the system would be proving a false
> statement, hence G *must* be true (a true arithmetical
> statement!) and, *as such*, unprovable. Which is GIT
> in a nutshell. QED.

If G is false in a model K of F, then something exists in the
domain of K, some object p, for which Proves_G( p ) is true in K.

Our first take on what p is that p is the Godelization of
some proof Phi in F of G. However, p can't be that, since, if
there were a proof Phi in F of G, G would be true in K (as it
would be in all models of F). So, p is not that.

The existence of p in K implies G is false in K, but Proves_G( p )
does not imply the existence of Phi that proves G.

How can that be?
The Godelization process is not bijective.
Moving from language to domain, for any Phi, Godelization will
map it to some unique object p in K.
However, moving from domain to language, there could well exist
objects p in K that Godelization does not map any Phi to.

If F is consistent and G is false in K, then there must be
such p in K.

[Khong Dong]

Decent parroting.

[Khong Dong]

Kind of reminding us the situation in some country where visiting tourists
must necessarily be accompanied by a government provided tour guide, who would
recite - without thinking - "historical" "facts" on some monuments. The poor
guide has never seen any real truth, for since birth he/she has always been
brainwashed with fabricated "truths".

What would "Godelization [will map it]" do, when even a simple and correct
definition of prime numbers can't be had?

[Ross A. Finlayson]

The usual idea of using arithmetic for consistency with
model theory, that there is a "model", and, a "standard model",
here gets into whether a standard model is "complete", the
non-standard or extra-standard or generic model "incomplete",
with respect to it, the model, structurally witnessing all
the pair-wise facts about the objects of arithmetic.

I.e. "that there is a model" (theory's consistent) and
"that there is a standard model" (theory's consistent
and it's ordinary), are usually assumed to be the same
thing, because the theory (one of Goedel's "theory-theories")
under consideration, is itself ordinary.

But, that the theory interpreting a model of ZF's model of
arithmetic, is ordinary, isn't for arithmetic where it
is for ZF, about the independence of a standard model of
ZF's (in)completeness with respect to arithmetic. (And,
about the deciding either way of the independence in
terms of the ordinary fragments or "extra-ordinary fragments",
which are more covers than parts, for where the ordinary is
regular both in foundation, and regular in partition.)

It seems natural to read Goedel's unproven theorem as
what would have to have the same placement as the ordinal
of Burali-Forti (ORD, greater then itself, "after itself
in succession" much like zero is "after itself", or "limit"),
or the "Russell set" or "Set of All Sets that don't contain
themself", as containing itself or here again "after itself".


I.e., for the model to exist it's an extension and then it's
independent about arithmetic, whether or not there exists
infinity, the number.

This then becomes a statement from the ordinary that
witnesses infinity exists, at least.

I.e., going forward in your development that way,
it seems you could prove that mathematical infinity exists,
deriving its existence, besides whether or not it's ordinary.

(From independence of theory about arithmetic.)

[George Greene]

On Thursday, July 2, 2020 at 2:33:28 AM UTC-4, Khong Dong wrote:
> What's the point of arguing when the construction of
> PA's G-sentence is invalid
> in the first place?

"Valid" does NOT APPLY to CONSTRUCTIONS, YOU IDIOT.
The sentence is well-defined. You know what sentence IT IS.
You know it's THAT sentence and not some other whose "validity" you are debating.
That is all the "validity" its "construction" CAN POSSIBLY NEED.
Your objection is something else, if ITS "construction" is "valid",
which of course it probably isn't.

[George Greene]

On Thursday, July 2, 2020 at 3:37:02 AM UTC-4, Julio Di Egidio wrote:
> So, if G were false, the system would be proving a false statement

There is categorically NO SUCH THING as "proving a false statement".
All theorems are TRUE IN ALL models of the axioms from which they are proved,
so for anybody to say they are false simpliciter is just ridiculous.

[George Greene]

On Thursday, July 2, 2020 at 10:40:02 AM UTC-4, Alan Smaill wrote:
> btw, the Stanford entry mentions Peter Milne's article on the issue of
> which results in this area require truth of the theory being examined,
> and which do not.

I actually have a philosophy degree (undergrad/A.B.) from Stanford,
IN this subfield, with no academic distinction whatever, but despite
the fact that the authorities are the authorities and I am an amateur,
THIS LOCUTION IS BULLSHIT. Just *bullshit*. The conventional wisdom
should just be ashamed of itself. There is no such thing as a false theory.
There is not such thing as a theory that "proves a false statement".
Everybody ACTUALLY KNOWS this and just CHOOSES TO SPEAK AS THOUGH
this were somehow a reasonable thing to say.
It really isn't.

> I recommend it to anyone interested:
> Milne, P., 2007, “On Gödel Sentences and What They Say,” Philosophia
> Mathematica, 15: 193–226.

Or you could just ASK ME. Seriously, WHEN YOU READ THAT, it will clarify
some things that will make it clear why people would do well to stop talking
like that, IF they take standard classical 1st-order logic seriously (which
of course they are free not to, in light of its many limitations, but all those
limitations are relevant to tractability, which is obviously important).

[George Greene]

On Thursday, July 2, 2020 at 12:59:14 PM UTC-4, Julio Di Egidio wrote:

> > > << (G) F |- G_F <-> ~Prov_F([G_F]).
> > > Thus, it can be shown, even inside F,
> > > that G_F is true if and only if it is not provable in F. >>

Of course it can, but that DOESN'T HELP.
The <-> is true EITHER when both sides are true XOR when both sides are false!
Proving this does NOT tell you WHICH of those is the case, therefore it
does NOT tell you that G_F is true!

Of course, there is NO POINT in BOTHERING to prove ANYthing in F
AT ALL, if F is inconsistent, so ASSUMING that F is consistent is
obviously reasonable -- even more so once you consider that the completeness
theorem entails that if F is INconsistent, THAT is NECESSARILY provable.
EVEN if your assumption of consistency is wrong, it does not MIRE you in error.
It just becomes a working hypothesis that can be reversed as soon
as it is provably refuted, which IT ALWAYS CAN be, if it is false.

But WHAT IS REALLY GOING ON here involves 1) the convention that the kinds of proofs being talked about in this paradigm HAVE to be finite (every inference
rule has to have a finite number of inputs/premises and there can only be finitely many of them in the chain/tree of them that constitutes the proof), and
2) the standard model being THE ONLY one in which EVERY element is finite and has a numeral. The fact that this model is all three of identifiable, inner, and intended DOES NOT suffice to make its truths "true" and their unrefuted denials "false" -- the models that disagree are rightly deprecated as non-
standard but they are NOT SUB-standard JUST because they happen to have anonymous "infinite"/supernatural elements!

[George Greene]

On Thursday, July 2, 2020 at 8:35:02 PM UTC-4, Alan Smaill wrote:
> Julio Di Egidio <ju...@diegidio.name> writes:
> > No, it's you who pick the parts that are irrelevant or badly broken: I have
> > a thesis and a reasoning.

EVERYbody has a thesis and a reasoning.
Even the ones from the respected authorities often have obvious
flaws. It's almost a matter of personal temperament which flaws
BOTHER you enough to motivate typing this hard, especially given
the irrelevance of this PARTICULAR forum.

>
> Then make yourself clear.
>
> >> > So, if G were false, the system would be proving a false statement,
> >> > hence G *must* be true (a true arithmetical statement!) and, *as
> >> > such*, unprovable. Which is GIT in a nutshell. QED.
> >>
> >> Fnobble....
>
> For example, I really cannot tell how you intend others to take
> your paragraph above

I don't either, but you (and the rest of the world too) really need to
quit talking about "proving a false statement". The statements in these
theories are NOT specifically "about" any PARTICULAR things beyond
their own terms (except for
the ones with existential axioms requiring the existence of things
from which terms/names are perversely withheld -- in my subfield we
routinely "skolemized" those existentials just to prove to the rest of
the world how utterly and deplorably irrelevant they were).
The IMPORTANT part of all these models is THE FUNCTIONS, NOT THE ELEMENTS
of the domain!
Acting like you care about the domain and privileging some one model with
some particular domain is just embarrassing, frankly.

[George Greene]

On Thursday, July 2, 2020 at 9:25:01 PM UTC-4, Alan Smaill wrote:
> When the conditions for the theorem are met, G is not provable
> (regardless of whether it is true or false).

You CAN'T SAY that!!
When the conditions of the theorem are met, and G is not provable, then
SINCE G *SAYS* "G is not provable", G *MUST*BE* true!
You CAN'T SAY "whether it is true or false"!
In the case you are talking about, IT'S TRUE! There CAN'T BE any "whether"!!


Except it isn't, because, precisely as you pointed out, there's a whole article
“On Gödel Sentences and What They Say”, and that article wouldn't be important if G *really*only* said "G is not provable" -- it's an article's worth MORE SUBTLE than that.

The TL;DR version is that the provability predicate is only guaranteed TO MEAN "provable" IN THE STANDARD MODEL -- NON-standard models have infinite/"supernatural" elements encoding infinitary proofs of statements that have no finite proofs -- yet one-and-the-same G and the same provability-predicate are being used in them, since both of those are defined syntactically and DON'T vary by model (even though the truth-values of undecidable statements constructed from them DO so vary).

[Khong Dong]

On Friday, 3 July 2020 23:06:51 UTC-6, George Greene wrote:
> On Thursday, July 2, 2020 at 2:33:28 AM UTC-4, Khong Dong wrote:
> > What's the point of arguing when the construction of
> > PA's G-sentence is invalid
> > in the first place?
>
> "Valid" does NOT APPLY to CONSTRUCTIONS, YOU IDIOT.
> The sentence is well-defined. You know what sentence IT IS.

Jeazus! Listen, stupid. On constructing a building, a software program, a
network, a language structure, a syntactical string, a proof, ... if you
don't follow the permissible guidelines, specifications then you'd factually
construct something all right: it's just whatever you've constructed isn't
valid.

If the police tells you you don't a valid driver license, what do you think
the police is telling you? Idiot.

> You know it's THAT sentence and not some other whose "validity" you are debating.

Idiot. "THAT sentence" is supposed to be a validly constructed well-formed-
formula sentence which isn't the case. That Goedel, you, and many
mathematicians incorrectly think that it's a wff-"sentence" doesn't make it
a valid formula sentence. Got it?

> That is all the "validity" its "construction" CAN POSSIBLY NEED.
> Your objection is something else, if ITS "construction" is "valid",
> which of course it probably isn't.

No idiot. The construction might or might not be valid but what I exactly
pointed out is this: it's not a valid formula-sentence just like you might
not have a valid driver license!

[Khong Dong]

But yes I did/do mean to say both the construction of G-"sentence" and the
"sentence" itself are invalid. The former is an invalid process, the later is
an invalid artifact.

[Alan Smaill]

George Greene <gre...@email.unc.edu> writes:

> On Thursday, July 2, 2020 at 10:40:02 AM UTC-4, Alan Smaill wrote:
>> btw, the Stanford entry mentions Peter Milne's article on the issue of
>> which results in this area require truth of the theory being examined,
>> and which do not.
>
> I actually have a philosophy degree (undergrad/A.B.) from Stanford,
> IN this subfield, with no academic distinction whatever, but despite
> the fact that the authorities are the authorities and I am an amateur,
> THIS LOCUTION IS BULLSHIT. Just *bullshit*. The conventional wisdom
> should just be ashamed of itself. There is no such thing as a false theory.
> There is not such thing as a theory that "proves a false statement".
> Everybody ACTUALLY KNOWS this and just CHOOSES TO SPEAK AS THOUGH
> this were somehow a reasonable thing to say.
> It really isn't.

I agree that there is a problem in flinging around "true" as though
there is no issue here -- there is an issue.

Nevertheless, saying "true in the standard model", which I imagine is
what you would prefer(?), has its own drawbacks. As Franzen says,
when talking about arithmetic statements like Goldbach, they mean
just what we thought they meant before model theory came into play.

Yes, careful exposition is needed.

>> I recommend it to anyone interested:
>> Milne, P., 2007, “On Gödel Sentences and What They Say,” Philosophia
>> Mathematica, 15: 193–226.
>
> Or you could just ASK ME. Seriously, WHEN YOU READ THAT,

I had read that, I hope others will also.
I add that eg Smullyan's "Goedel's Incompleteness Theorems" uses
terminology of "true" in his abstract presentation in Ch 1,
and later, being careful of course. I don't think "standard
model" gets a mention -- not in index, anyway.

> it will clarify some things that will make it clear why people would
> do well to stop talking like that, IF they take standard classical
> 1st-order logic seriously (which of course they are free not to, in
> light of its many limitations, but all those limitations are relevant
> to tractability, which is obviously important).

Milne is working in the usual first-order setting.

abstract:

Proofs of Gödel's First Incompleteness Theorem are often accompanied by
claims such as that the Gödel sentence constructed in the course of the
proof says of itself that it is unprovable and that it is true. The
validity of such claims depends closely on how the sentence is
constructed. Only by tightly constraining the means of construction can
one obtain Gödel sentences of which it is correct, without further ado,
to say that they say of themselves that they are unprovable and that
they are true; otherwise a false theory can yield false Gödel sentences

--
Alan Smaill

[Alan Smaill]

George Greene <gre...@email.unc.edu> writes:

> On Thursday, July 2, 2020 at 9:25:01 PM UTC-4, Alan Smaill wrote:
>> When the conditions for the theorem are met, G is not provable
>> (regardless of whether it is true or false).
>
> You CAN'T SAY that!!
> When the conditions of the theorem are met, and G is not provable, then
> SINCE G *SAYS* "G is not provable", G *MUST*BE* true!

I thought you objected to using "true" simpliciter --
but you clarify below.

I've attenmpted to clarify my position upthread.

> You CAN'T SAY "whether it is true or false"!
> In the case you are talking about, IT'S TRUE! There CAN'T BE any "whether"!!
>
> Except it isn't, because, precisely as you pointed out, there's a
> whole article “On Gödel Sentences and What They Say”, and that article
> wouldn't be important if G *really*only* said "G is not provable" --
> it's an article's worth MORE SUBTLE than that.

Right.

> The TL;DR version is that the provability predicate is only guaranteed
> TO MEAN "provable" IN THE STANDARD MODEL -- NON-standard models have
> infinite/"supernatural" elements encoding infinitary proofs of
> statements that have no finite proofs -- yet one-and-the-same G and
> the same provability-predicate are being used in them, since both of
> those are defined syntactically and DON'T vary by model (even though
> the truth-values of undecidable statements constructed from them DO so
> vary).

That's one way of understanding what's going on.
Whichever way, clarification is needed.

--
Alan Smaill

[Julio Di Egidio]

Do you know how correctness is defined? What you say, there and elsewhere, is
so fucking ridiculously wrong that I really have nothing to add, the essential
problem being that you guys just cannot think, period.

*Plonk*

Julio

[Jim Burns]

On 7/4/2020 1:38 AM, Khong Dong wrote:
> On Friday, 3 July 2020 23:06:51 UTC-6,
> George Greene wrote:
>> On Thursday, July 2, 2020 at 2:33:28 AM UTC-4,
>> Khong Dong wrote:

>>> What's the point of arguing when the construction of
>>> PA's G-sentence is invalid
>>> in the first place?
>>
>> "Valid" does NOT APPLY to CONSTRUCTIONS, YOU IDIOT.
>> The sentence is well-defined. You know what sentence IT IS.
>
> Jeazus! Listen, stupid. On constructing a building, a software
> program, a network, a language structure, a syntactical string,
> a proof, ... if you don't follow the permissible guidelines,
> specifications then you'd factually construct something all
> right: it's just whatever you've constructed isn't valid.
>
> If the police tells you you don't a valid driver license,
> what do you think the police is telling you?

A valid statement or a valid proof are _valid_ in a sense
which is different from a valid driver's license.

Formal statements have multiple interpretations.

A _valid_ statement is true under each of its multiple
interpretations.

A _valid_ proof always has a true conclusion whenever
its hypotheses are true.

Nam, you really ought to pay attention to what "valid"
means _here_ You (NN) keep asking, in various ways,
How can we KNOW that this statement is true?
It could mean this, it could mean that.

Your own answer to the question you pose is that it is
_impossible to know_ unless we can stop it from having
multiple interpretations.

However, _if a statement is valid_ it is true regardless of
which interpretation it has. _This is how we know it's true_

This is why so much of logic and mathematics is concerned
-- NOT about whether a certain statement is TRUE --
but about whether that statement is VALID:
in order for us to be able to say it's true,
_even without knowing what it means_

[Khong Dong]

On Saturday, 4 July 2020 10:20:22 UTC-6, Jim Burns wrote:
> On 7/4/2020 1:38 AM, Khong Dong wrote:
> > On Friday, 3 July 2020 23:06:51 UTC-6,
> > George Greene wrote:
> >> On Thursday, July 2, 2020 at 2:33:28 AM UTC-4,
> >> Khong Dong wrote:
>
> >>> What's the point of arguing when the construction of
> >>> PA's G-sentence is invalid
> >>> in the first place?
> >>
> >> "Valid" does NOT APPLY to CONSTRUCTIONS, YOU IDIOT.
> >> The sentence is well-defined. You know what sentence IT IS.
> >
> > Jeazus! Listen, stupid. On constructing a building, a software
> > program, a network, a language structure, a syntactical string,
> > a proof, ... if you don't follow the permissible guidelines,
> > specifications then you'd factually construct something all
> > right: it's just whatever you've constructed isn't valid.
> >
> > If the police tells you you don't a valid driver license,
> > what do you think the police is telling you?
>

> A [language structure theoretically] valid [*wff*] statement [...]

> is different from a valid driver's license.

Exactly, parrot Jim Burns! And I was/am talking about *that* _different_
*kind* of the word " _valid_ " when referring to the so-called G-"sentence"!

Iow, the so-called G-"sentence" is _NOT_ validly constructed (defined) as a
FOL language wff sentence. Hence you simply can't attribute the G-"sentence"
to being true or false, because it's _NOT_ - again - a validly formed (well
formed) sentence.

Hope that your severe confusion has been helped.

[Khong Dong]

Again, Jim, the _forging_ of the (existence of the) string G-"sentence" is
so invalid a process that Goedel's 1931 paper can't have a valid conclusion
that _that_ non-sentence string - labelled as G-"sentence" - is true or false.

So to speak, Goedel - without valid authorization - can _forge_ a passport but
that doesn't make him have a valid passport!

[Peter]

Gödel's 1931 paper didn't use a FO language.

Gödel specified a sentence that is true in some models but false in
others. That was its very point.

[Khong Dong]

On Saturday, 4 July 2020 11:22:50 UTC-6, Peter wrote:
> Khong Dong wrote:
> > On Saturday, 4 July 2020 10:47:12 UTC-6, Khong Dong wrote:
> >> On Saturday, 4 July 2020 10:20:22 UTC-6, Jim Burns wrote:
> >>> On 7/4/2020 1:38 AM, Khong Dong wrote:
>
>
> >> Iow, the so-called G-"sentence" is _NOT_ validly constructed (defined) as a
> >> FOL language wff sentence. Hence you simply can't attribute the G-"sentence"
> >> to being true or false, because it's _NOT_ - again - a validly formed (well
> >> formed) sentence.
> >>
> >> Hope that your severe confusion has been helped.
> >
> > Again, Jim, the _forging_ of the (existence of the) string G-"sentence" is
> > so invalid a process that Goedel's 1931 paper can't have a valid conclusion
> > that _that_ non-sentence string - labelled as G-"sentence" - is true or false.
>
> Gödel's 1931 paper didn't use a FO language.

Exactly. That's why Gödel's 1931 paper is a meta mathematics argument.

>
> Gödel specified a sentence that is true in some models but false in
> others. That was its very point.

But Gödel simply didn't have any relevant FOL language sentence to be true
in some models - provided Gödel had validly defined "true", "model", "true in
some models" in the first place (which he had not)!

All that aside, do you acknowledge now Gödel's 1931 paper doesn't have the
intended validly constructed sentence - where a truth-value can be assigned?

[Jim Burns]

1.
"Valid", when applied to drivers' licenses, does not mean
"well-formed". A fake driver's license could be perfectly
well-formed and still fake, if the issuing authority (BMV)
doesn't accept it.

2.
If it were anyone else speaking, it would be worth remarking on
how odd it is to say
"G is not valid"
instead of
"G is not well-formed"
in order to say
"G is not a wff [a well-formed formula]".

3.
G is a well-formed sentence [closed formula].

4.
For a theory F, there are multiple models of F to which the
language of F can be taken to be referring.
In some models, G is true.
In some models, G is false.
Hence you can't simply attribute to G being true or false.
G is true or false _depending on the model_

On the other hand, for a theorem P for F, there are _still_
multiple models of F, but you CAN simply attribute to P
being true, because P is true in every model of F.
Because P is a theorem, it is valid.

For example.
Consider the sentence ~(SS0 < SS0) in Robinson arithmetic Q.
There are different models of Q.
Nevertheless, we know that ~(SS0 < SS0) is simply true in Q
because we know a proof of ~(SS0 < SS0) in Q.
Because of that proof, we know that ~(SS0 < SS0) is _valid_
in Q: that is, ~(SS0 < SS0) is true in every model of Q.

[Khong Dong]

And yes, Gödel's "didn't use a FO language" sentence in a FOL formal proof
to prove his thesis, but his Gödel-"sentence" is _supposed to be_ a FOL
language sentence to be provable/un-provable in some FOL formal theory!

Except, again, Gödel-"sentence" is not a FOL language sentence for it to be
true in some models or provable/undecidable in some FOL theories.

[Khong Dong]

Utterly wrong, incompetent. If fake driver's license is not well-formed if
it misses some photo, signature - or is a blank piece of document!

> A fake driver's license could be perfectly
> well-formed and still fake, if the issuing authority (BMV)
> doesn't accept it.

Take your forged "well-formed" driver's license without your photo and see if
the insurance company would let you take your new car out of the dealer's
parking lot.

OMG! This is the best mind of the education-industry to defend the alleged
1931 theorem?

[Khong Dong]

That is: "A fake driver's license is not well-formed if it misses some photo,

[Peter]

Khong Dong wrote:
> On Saturday, 4 July 2020 11:22:50 UTC-6, Peter wrote:
>> Khong Dong wrote:
>>> On Saturday, 4 July 2020 10:47:12 UTC-6, Khong Dong wrote:
>>>> On Saturday, 4 July 2020 10:20:22 UTC-6, Jim Burns wrote:
>>>>> On 7/4/2020 1:38 AM, Khong Dong wrote:
>>
>>
>>>> Iow, the so-called G-"sentence" is _NOT_ validly constructed (defined) as a
>>>> FOL language wff sentence. Hence you simply can't attribute the G-"sentence"
>>>> to being true or false, because it's _NOT_ - again - a validly formed (well
>>>> formed) sentence.
>>>>
>>>> Hope that your severe confusion has been helped.
>>>
>>> Again, Jim, the _forging_ of the (existence of the) string G-"sentence" is
>>> so invalid a process that Goedel's 1931 paper can't have a valid conclusion
>>> that _that_ non-sentence string - labelled as G-"sentence" - is true or false.
>>
>> Gödel's 1931 paper didn't use a FO language.
>
> Exactly. That's why Gödel's 1931 paper is a meta mathematics argument.

The incompleteness theorem is a metatheorem whether it is about FO
arithmetic of higher order arithmetic.

>>
>> Gödel specified a sentence that is true in some models but false in
>> others. That was its very point.
>
> But Gödel simply didn't have any relevant FOL language sentence to be true

Again - Gödel's version of Gödel's incompleteness theorem wasn't about a
FO language.

> in some models - provided Gödel had validly defined "true", "model", "true in
> some models" in the first place (which he had not)!
>
> All that aside, do you acknowledge now Gödel's 1931 paper doesn't have the
> intended validly constructed sentence - where a truth-value can be assigned?

I am not aware of any errors in Gödel's 1931 paper, but I have only read
an English translation.

>
>

[Peter]

No it isn't. Gödel's 1931 paper is a about a theory of types.

[Peter]

That doesn't mean that a fake driver's license is a driver's license
that is missing something. But in any case "invalid" as a logician
would use the word when talking about formulae or proofs is not the same
as "invalid" as a police officer would use the word when talking about
driver's licenses.

[Khong Dong]

On Saturday, 4 July 2020 12:42:31 UTC-6, Peter wrote:

> Again - Gödel's version of Gödel's incompleteness theorem wasn't about a
> FO language.

What an utterance of ignorance and bs!

Is Gödel's incompleteness theorem _about_ some "Formally Undecidable" sentence
written in some _FO language_ *to you*?

[Khong Dong]

On Saturday, 4 July 2020 12:49:22 UTC-6, Peter wrote:
> [snipped ... idiotic trolling]

[Peter]

Note that I wrote *Gödel's version* of Gödel's incompleteness theorem...
I've added the emphasis. And it is clear that P isn't a FO theory.

Perhaps we could establish which version of GIT you wish to discuss?

[Jim Burns]

Because of our liquor laws here in the United States,
it is not unheard of for persons too young to legally buy
liquor to somehow get hold of fake drivers' licenses.

I don't have much experience in this area, but there wouldn't
be much point to showing a fake license which was obviously
fake -- which was at least well-formed, not missing photos,
or signatures, or was blank. The point of a fake is to deceive.

[Peter]

Khong Dong wrote:
> On Saturday, 4 July 2020 12:49:22 UTC-6, Peter wrote:
>> [snipped ... idiotic trolling]

Since you mentioned driver's licenses, my post -

But in any case "invalid" as a logician would
use the word when talking about formulae or
proofs is not the same as "invalid" as a police
officer would use the word when talking about
driver's licenses.

was relevant rather than idiotic or trolling.

[Khong Dong]

[Khong Dong]

On Saturday, 4 July 2020 12:56:02 UTC-6, Jim Burns wrote:
> On 7/4/2020 2:15 PM, Khong Dong wrote:

> > That is: "A fake driver's license is not well-formed if it
> > misses some photo, signature - or is a blank piece of document!"

> "Valid", when applied to drivers' licenses, does not mean
> "well-formed".

You're an educated poster with a Bachelor Degree, right? Your post is a clear
reason why I don't feel regretted at all for saying that since 1931 the
education-industry has been so bankrupted in producing brainwash arguers like
you, Peter, ... (I think Grothendieck might have had the same "feeling" here).

If you cleanly erase the year of birth in your driver license and cleanly type
"googleplex" in lieu, would your driver license still be valid because it's
still "well-formed" or because "well-formed-ness" is not applicable to driver
licenses? Ignoramus.

[Peter]

Khong Dong wrote:
> On Saturday, 4 July 2020 12:54:36 UTC-6, Peter wrote:
>> [snipped ... idiotic trolling]

You have often posted here about GIT but you have yet to realize that
Gödel's original paper was not about a FO theory.

You write about complex numbers and transcendental numbers without
knowing what either is.

You lie about having a degree in mathematics, but clearly you don't have
enough intelligence to graduate from high school.

[Khong Dong]

[Peter]

Khong Dong wrote:
> On Saturday, 4 July 2020 13:41:53 UTC-6, Peter wrote:
>> [snipped ... idiotic trolling]

Has it never occurred to you that your lie about having a degree in
mathematics is quite unconvincing? How stupid do you have to be to
claim to have a degree in a subject that you know nothing about?

[Peter]

Khong Dong wrote:
> On Saturday, 4 July 2020 13:41:53 UTC-6, Peter wrote:
>> [snipped ... idiotic trolling]

How are you getting on with Me's suggestion that you prove

|z_1 + z_2| <= |z_1| + |z_2| for all complex numbers z_1, z_2?

Do you even know what |z_1 + z_2| <= |z_1| + |z_2| says?

[Khong Dong]

[Khong Dong]

On Saturday, 4 July 2020 13:54:18 UTC-6, Peter wrote:
> [snipped ... idiotic ranting of the racist liar Peter Percival]

[Jim Burns]

On 7/4/2020 3:26 PM, Khong Dong wrote:
> On Saturday, 4 July 2020 12:56:02 UTC-6,
> Jim Burns wrote:
>> On 7/4/2020 2:15 PM, Khong Dong wrote:

>>> That is: "A fake driver's license is not well-formed if it
>>> misses some photo, signature - or is a blank piece of document!"
>
>> "Valid", when applied to drivers' licenses, does not mean
>> "well-formed".
>
> You're an educated poster with a Bachelor Degree, right?

Yes. I'm sure I didn't take any classes on drivers' licenses,
though.

Given your laser-like focus on drivers' licenses, someone
might be forgiven for thinking that they were your major.

But you say otherwise. Mathematics, right?

> Your post is a clear reason why I don't feel regretted at all
> for saying that since 1931 the education-industry has been

> so bankrupted in producing brainwash arguers like you.

> Peter, ... (I think Grothendieck might have had the same
> "feeling" here).
>
> If you cleanly erase the year of birth in your driver license
> and cleanly type "googleplex" in lieu, would your driver license
> still be valid because it's still "well-formed" or because
> "well-formed-ness" is not applicable to driver licenses? Ignoramus.

A valid statement is true under all interpretation of its language,
given the axioms of its theory.

A valid proof has a true conclusion whenever its hypotheses
are true.

A valid driver's license is not valid the way a valid statement
is valid, nor is it valid the way a valid proof is valid.

[Khong Dong]

What a fucking idiotic trolling.

[Jim Burns]

Q.
Suppose, for a statement P with multiple meanings, that we don't
know which of its possible meanings is its actual meaning.
How can it be possible to know if P is true or false?

A.
If P is known to be _valid_ or if it is known to follow from
_valid premises_ by a _valid argument_ then we know that
all of its possible meanings are true, including the actual
meaning of P, whichever that is.

[Khong Dong]

Q.
Is the string:

primeFunction(n) = Ep[prime(p) & p=nth-prime]

a syntactically valid FOL wff?

A.
No.

Why is Jim Burns such an idiotic, ignorant troll?

[Khong Dong]

The answer is that under the current education system,
those who might happen to believe in some kind of
crank-matheology would have more knowledge and honesty
than the Bachelor Degree in Physics JB.

[Julio Di Egidio]

On Sunday, 5 July 2020 01:26:29 UTC+2, Jim Burns wrote:

> Q.
> Suppose, for a statement P with multiple meanings, that we don't
> know which of its possible meanings is its actual meaning.
> How can it be possible to know if P is true or false?
>
> A.
> If P is known to be _valid_ or if it is known to follow from
> _valid premises_ by a _valid argument_ then we know that
> all of its possible meanings are true, including the actual
> meaning of P, whichever that is.

Which is still as wrong as I have been telling all along: namely, that is
true only if you assume, or somehow know, that the system is *correct*, which
means the system only proves true statements and does not prove false ones.
Read Smullyan GITs for reference.

Conflating truth with provability, hence dismantling truth proper, in favour
of literally what you make up, the lying with numbers of mathematical logic,
and of course under "suitable" circumstances, i.e. under who's got the biggest
guns: that is what model theory essentially is for, as part of an overall so
evident plan that only fucking braindead idiots would be able to still miss it.
Think about that...

Anyway, I am out of this shithole: Usenet is dead, congratulations.

Julio

[Peter]

Hadn't you better sort out what 'valid' means first? (Without a
time-wasting detour via driver's licenses.)

having done that, you'll need to define which FO language you're talking
about. There is more than one.

[Julio Di Egidio]

The piece of shit is back, for another round of crapping over anything that
isn't completely dead already. And so we go...

ESAD.

*Plonk*

Julio

[Peter]

No. What lies to the right of the equals sign above looks like a
formula. But what should lie either side of an equals sign are terms.
"Looks like" because I can make no sense of "p=nth-prime".

>>
>> Hadn't you better sort out what 'valid' means first? (Without a
>> time-wasting detour via driver's licenses.)
>>
>> having done that, you'll need to define which FO language you're talking
>> about. There is more than one.
>>
>>>
>>> A.
>>> No.
>>>
>>> Why is Jim Burns such an idiotic, ignorant troll?
>
> The piece of shit is back, for another round of crapping over anything that
> isn't completely dead already. And so we go...

I've changed my mind. See above.
>
> ESAD.
>
> *Plonk*
>
> Julio
>

[Rupert]

On Sunday, June 21, 2020 at 7:49:29 AM UTC+2, George Greene wrote:
> On Thursday, June 18, 2020 at 4:13:06 AM UTC-4, Rupert wrote:
> > > https://math.stackexchange.com/questions/3721227/can-the-question-of-generalized-continuum-hypothesis-be-solved-along-such-lines?
> >
> > If "large cardinal property" is defined so as to include "invariant under small forcing", and if we also assume that all large cardinal properties have associated L-like inner models in which they are realised, which are models of GCH (which is not known for all currently known large cardinal axioms but is a plausible conjecture), then in that case both (i) and (ii) would be true.
>
> Rupert, his post at stackexchange tries to define "true" FOR A THEORY.
> This is kind of absurd and you know it.

I'm not endorsing it as an attempted truth definition, no. Just commenting on when his suggested criteria would be satisfied, as far as I'm able to tell.

[Khong Dong]

I didn't say but the language is an extension of L(PA) having
''nth-prime' as the new individual constant symbol. So there you
have it. Is the equation-string a syntactically valid wff _to you_ now?

If not, can you explain why there's _that_ syntactical invalidity?

[Peter]

No, for the reason given before. Apart from that, "nth-prime" looks
like a function, or the value of a function for an indeterminate
argument n, so your notation is confusing.

[Khong Dong]

On Sunday, 5 July 2020 15:28:24 UTC-6, Peter wrote:
> Khong Dong wrote:
> > On Sunday, 5 July 2020 13:00:03 UTC-6, Peter wrote:
> >> Julio Di Egidio wrote:
> >>> On Sunday, 5 July 2020 16:57:26 UTC+2, Peter wrote:
> >>>> Khong Dong wrote:
> >>>>> On Saturday, 4 July 2020 17:26:29 UTC-6, Jim Burns wrote:
> >>>>>> On 7/4/2020 6:25 PM, Khong Dong wrote:
> >>>>>>> On Saturday, 4 July 2020 16:19:20 UTC-6,
> >>>>>>> Jim Burns wrote:
> >>>>>>
> >>>>>>>> A valid statement is true under all interpretation of its
> >>>>>>>> language, given the axioms of its theory.
> >>>>>>>>

> >>>>>>>> A valid proof has a true conclusions whenever its hypotheses

> >>>>>>>> are true.
> >>>>>>>>
> >>>>>>>> A valid driver's license is not valid the way a valid statement
> >>>>>>>> is valid, nor is it valid the way a valid proof is valid.
> >>>>>>>
> >>>>>>> What a fucking idiotic trolling.
> >>>>>>
> >>>>>> Q.
> >>>>>> Suppose, for a statement P with multiple meanings, that we don't
> >>>>>> know which of its possible meanings is its actual meaning.
> >>>>>> How can it be possible to know if P is true or false?
> >>>>>>
> >>>>>> A.
> >>>>>> If P is known to be _valid_ or if it is known to follow from
> >>>>>> _valid premises_ by a _valid argument_ then we know that
> >>>>>> all of its possible meanings are true, including the actual
> >>>>>> meaning of P, whichever that is.
> >>>>>
> >>>>> Q.
> >>>>> Is the string:
> >>>>>
> >>>>> primeFunction(n) = Ep[prime(p) & p=nth-prime]
> >>>>>
> >>>>> a syntactically valid FOL wff?
> >>
> >> No. What lies to the right of the equals sign above looks like a
> >> formula. But what should lie either side of an equals sign are terms.
> >> "Looks like" because I can make no sense of "p=nth-prime".
> >
> > I didn't say but the language is an extension of L(PA) having
> > ''nth-prime' as the new individual constant symbol. So there you
> > have it. Is the equation-string a syntactically valid wff _to you_ now?
>
> No, for the reason given before.

You're not telling sci.logic the truth: your "the reason given before"
has been explained to be no longer.

> Apart from that, "nth-prime" looks
> like a function, or the value of a function for an indeterminate
> argument n, so your notation is confusing.

That's an ignorance given the formal language having been informed.
You should really learn trivial knowgede somehow, but I give you a
hint: the quantifier symbol 'E' is the problem - independent of whatever
language you care to choose.

Do you get it now?

[Peter]

Ok, here's my reason: What lies to the right of the equals sign above
looks like a formula (It's "Ep[prime(p) & p=nth-prime]" that I'm
referring to.) But what should lie either side of an equals sign are terms.

What is "nth-prime"? Is it a predicate symbol, a function symbol or an
individual constant?

[Khong Dong]

As the other poster has alluded to, what's in your mind, coming out of
your mouth is basically a just piece of ignorant trolling shit.

[Me]

On Sunday, July 5, 2020 at 1:58:40 AM UTC+2, Khong Dong wrote:

> Is the string:
>
> primeFunction(n) = Ep[prime(p) & p=nth-prime]
>

> a [...] FOL wff?

Well, "primeFunction(n)" looks like a function expression with argument "n". In other words, it looks like a /term/. But at the right hand side of the "=" there seems to be a /formula/. Hence in a "standard system" of /FOPL with identity/ the expression most likely wouldn't be considered a wff. The usual formation rule concerning "=" is:

If /s/ is s a term and /t/ is a term, then /s = t/ is a wff.

[Me]

On Monday, July 6, 2020 at 12:05:10 AM UTC+2, Khong Dong wrote:
> On Sunday, 5 July 2020 15:58:22 UTC-6, Peter wrote:
>> Khong Dong wrote:
>>>

>>> Is the equation-string

>>>
>>> primeFunction(n) = Ep[prime(p) & p=nth-prime]
>>>

>>> a syntactically valid wff _to you_ now?
>>

>> No, for the [following] reason:

>>
>> What lies to the right of the equals sign above looks like a formula
>> (It's "Ep[prime(p) & p=nth-prime]" that I'm referring to.) But what
>> should lie either side of an equals sign are terms.
>>
>> What is "nth-prime"? Is it a predicate symbol, a function symbol or an
>> individual constant?
>>

> As the other poster has alluded to <bla>

Huh?! Read the fucking answer, silly!

[Khong Dong]

My fuller question is "a syntactically valid FOL wff?". Is there any _honest_
reason that _you_ _deliberately_ snipped it, FF? I guess not.

[Me]

On Monday, July 6, 2020 at 5:29:07 AM UTC+2, Khong Dong wrote:

> My fuller question is "a syntactically valid FOL wff?". Is there any _honest_
> reason that _you_ _deliberately_ snipped it, FF?

Is there a wff in your book which is not "syntactically valid"?

[Khong Dong]

Your "fucking" is entirely ill-advised.

As the other poster has alluded to, you, PP, JB are full of shit.

The contention is never about something not being a term but is supposed to
be a term. The contention is whether or not there is _a different sense_ for
the phrase "valid"/"validly" in mathematical usage, and the full-of-shit PP
and JB (and now you) have incorrectly insisted there's none other when being
used in truth-validity (as in truth validity theorem).

Of course it's trivial there's a different sense - syntactical sense at least:
e.g. the above FOL expression is not valid wff *syntactically*. What a waste
of time you three have done.

"Valid"/"invalid" can be also used in proof too (whether or not you'd
_validly_ conform with permissible definitions and rules of reasoning).

If you three full-of-shit idiots (FF, PP, JB) couldn't know where the below
comes from, let me know:

<quote>

The validity of an inference depends on the form of the inference. That is,
the word "valid" does not refer to the truth of the premises or the
conclusion, but rather to the form of the inference.

<quote>

Fucking idiot.

[Khong Dong]

So you don't have any clue what the point of contention between NN, JB, PP is.
Why jumping into the argument and tried to defend PP, JB?

[Me]

On Monday, July 6, 2020 at 6:07:24 AM UTC+2, Khong Dong wrote:
> On Sunday, 5 July 2020 21:38:39 UTC-6, Me wrote:
> > On Monday, July 6, 2020 at 5:29:07 AM UTC+2, Khong Dong wrote:
> > >
> > > My fuller question is "a syntactically valid FOL wff?". Is there
> > > any _honest_ reason that _you_ _deliberately_ snipped it, FF?
> > >
> > Is there a wff in your book which is not "syntactically valid"?
> >

> So you <bla>

Could you please answer the question, dumbo?

Again, you asked "Is there [a] reason [why] _you_ snipped ['syntactically valid'], FF?"

Sure. I told you that the "equation-string" you presented isn't a wff (most likely). Hence there's no need to mention any additional "attribute" concerning that sting.

See: https://en.wikipedia.org/wiki/Occam%27s_razor

[Peter]

It's not easy to say. Is "nth-prime" a predicate symbol, a function

symbol or an individual constant?

>

[Peter]

Is there any _honest_ reason that _you_ _deliberately_ won't address
this question of what is on the RHS of the equals sign? It looks like a
formula but needs to be a term.

[Me]

On Monday, July 6, 2020 at 3:20:49 PM UTC+2, Peter wrote:
> Khong Dong wrote:
> >

> > Is the string:
> >
> > primeFunction(n) = Ep[prime(p) & p=nth-prime]
> >
> > a syntactically valid FOL wff?
> >
> It's not easy to say. Is "nth-prime" a predicate symbol, a function
> symbol or an individual constant?

On the other hand, no matter what, in the context of "FOPL with identity" the string

primeFunction(n) = Ep[prime(p) & p=nth-prime]

most likely would't be a wff. After all, the string "Ep[prime(p) & p=nth-prime]" (most likely) wouldn't be a term in this system (for all 3 possibilities concerning "nth-prime" metioned by you).

[Me]

On Monday, July 6, 2020 at 5:55:22 AM UTC+2, Khong Dong wrote:

> "Valid"/"invalid" [...] whether or not you'd [...] conform with permissible

> definitions and rules of reasoning).

I remember a professor who used that terminology n his lecture ("Introduction to logic for philosophers").

He wrote "|= A" (and read: "A is semantically valid") when <...> and wrote "|- A" (and read: "A is syntactically valid") when <...>. Then (of course) Gödel's completeness theorem can be formulated the following way:

|= A iff |- A .

[Peter]

Me wrote:
> On Monday, July 6, 2020 at 3:20:49 PM UTC+2, Peter wrote:
>> Khong Dong wrote:
>>>
>>> Is the string:
>>>
>>> primeFunction(n) = Ep[prime(p) & p=nth-prime]
>>>
>>> a syntactically valid FOL wff?
>>>
>> It's not easy to say. Is "nth-prime" a predicate symbol, a function
>> symbol or an individual constant?
>
> On the other hand, no matter what, in the context of "FOPL with identity" the string
>
> primeFunction(n) = Ep[prime(p) & p=nth-prime]
>
> most likely would't be a wff. After all, the string "Ep[prime(p) & p=nth-prime]" (most likely) wouldn't be a term in this system

I made the same point on 05/07/2020 at 20:00 BST.

[Khong Dong]

I'm sorry but ((|= A) <=> (|- A)) needs to be proven as a meta mathematical
theorem - but it's not possible to prove it so. Hint: there are FOPL sentences
that are both truth and proof undecidable.

[Peter]

If |= and |- are referring to first order logic then proofs of |= A iff
|- A are in many textbooks. Gödel found the first proof around 1930 but
Henkin's proof of around 1949 is easier to follow.

[Me]

On Monday, July 6, 2020 at 5:37:25 PM UTC+2, Khong Dong wrote:
> On Monday, 6 July 2020 07:54:25 UTC-6, Me wrote:
> > On Monday, July 6, 2020 at 5:55:22 AM UTC+2, Khong Dong wrote:
> > >
> > > "Valid"/"invalid" [...] whether or not you'd [...] conform with
> > > permissible definitions and rules of reasoning).
> > >

> > I remember a professor who used that terminology in his lecture

> > ("Introduction to logic for philosophers").
> >
> > He wrote "|= A" (and read: "A is semantically valid") when <...> and wrote
> > "|- A" (and read: "A is syntactically valid") when <...>. Then (of course)
> > Gödel's completeness theorem can be formulated the following way:
> >
> > |= A iff |- A .
> >
> I'm sorry but ((|= A) <=> (|- A)) needs to be proven as a meta mathematical
> theorem - but it's not possible to prove it so.

Nonsense. Learn some logic, dumbo.

See: https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem#G%C3%B6del's_original_formulation

Hint:

"Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.

It was first proved by Kurt Gödel in 1929. It was then simplified in 1947, when Leon Henkin observed in his Ph.D. thesis that the hard part of the proof can be presented as the Model Existence Theorem (published in 1949). Henkin's proof was simplified by Gisbert Hasenjaeger in 1953."

Source: https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem

[Khong Dong]

Listen, dumbo. I've clearly proved (to Rupert et al.) GIT is invalid on
multiple counts - with clear elaborated details on each. If you are ignorant
enough and couldn't understand it that's your issue but parroting isn't a
good way for you to learn.

(Sure. If you don't believe in SR then learn and parrot the other physics-
textbooks of centuries ago. Good for you.)

[Peter]

Khong Dong wrote:
> On Monday, 6 July 2020 10:38:25 UTC-6, Me wrote:
>> On Monday, July 6, 2020 at 5:37:25 PM UTC+2, Khong Dong wrote:
>>> On Monday, 6 July 2020 07:54:25 UTC-6, Me wrote:
>>>> On Monday, July 6, 2020 at 5:55:22 AM UTC+2, Khong Dong wrote:
>>>>>
>>>>> "Valid"/"invalid" [...] whether or not you'd [...] conform with
>>>>> permissible definitions and rules of reasoning).
>>>>>
>>>> I remember a professor who used that terminology in his lecture
>>>> ("Introduction to logic for philosophers").
>>>>
>>>> He wrote "|= A" (and read: "A is semantically valid") when <...> and wrote
>>>> "|- A" (and read: "A is syntactically valid") when <...>. Then (of course)
>>>> Gödel's completeness theorem can be formulated the following way:
>>>>
>>>> |= A iff |- A .
>>>>
>>> I'm sorry but ((|= A) <=> (|- A)) needs to be proven as a meta mathematical
>>> theorem - but it's not possible to prove it so.
>>
>> Nonsense. Learn some logic, dumbo.
>>
>> See: https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem#G%C3%B6del's_original_formulation
>>
>> Hint:
>>
>> "Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.
>>
>> It was first proved by Kurt Gödel in 1929. It was then simplified in 1947, when Leon Henkin observed in his Ph.D. thesis that the hard part of the proof can be presented as the Model Existence Theorem (published in 1949). Henkin's proof was simplified by Gisbert Hasenjaeger in 1953."
>>
>> Source: https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem
>
> Listen, dumbo.

Do calm down. You gain nothing by using such childish language.

> I've clearly proved (to Rupert et al.) GIT is invalid on

It is the completeness theorem that is being discussed, not the
incompleteness theorem.

[Khong Dong]

On Monday, 6 July 2020 11:00:06 UTC-6, Peter wrote:
> Khong Dong wrote:

> > I've clearly proved (to Rupert et al.) GIT is invalid on
>
> It is the completeness theorem that is being discussed, not the
> incompleteness theorem.

Yep. Way back I also showed (to Rupert et al.) Completeness is invalid way back.
At any rate, undecide(cGC) => invalid(Completeness).

Idiotic Troll.

[Peter]

You claimed some such thing, but showed nothing. There are proofs of
both the completeness theorem and the incompleteness theorem in Shoenfield.

> Idiotic Troll.
>

[Khong Dong]

On Monday, 6 July 2020 11:14:00 UTC-6, Peter wrote:
> [snipped .. idiotic trolling rants]

[Me]

On Monday, July 6, 2020 at 6:57:00 PM UTC+2, Khong Dong wrote:

> I've clearly proved [...] GIT is invalid on multiple counts

No, you haven't. You are delusional!

[Peter]

Khong Dong wrote:

> At any rate, undecide(cGC) => invalid(Completeness).

You haven't proved undecide(cGC). You haven't even given a coherent
account of what it means.

[Khong Dong]

On Monday, 6 July 2020 11:23:59 UTC-6, Peter wrote:
> [snipped ... idiotic ranting]