DC Proof is the biggest teaching mistake

[Mostowski Collapse]

That an empty universe of discourse is allowed has
recently become a hiding space for being lazy.
That an empty universe of discourse is allowed prevents

the inference ALL(x):A(x) => EXIST(x):A(x). But why would
this be even important? It doesn't buy you anything. For
example in the Peano "construction" it is proved:

1) ALL(f):[Dedekind(f) => EXIST(w):Peano(w)]

Now DC Proof prevents going to from 1) to 2):

2) EXIST(f):[Dedekind(f) => EXIST(w):Peano(w)]

Does DC Proof think that the enemy 2) could be used to derive 3)?

3) EXIST(f):Dedekind(f)

Hint: It doesn't, 2) does not imply 3). Guess why?

LMAO!

[Dan Christensen]

See my reply to your identical posting at sci.proof just now. Really quite pathetic, even for you, Jan Burse!
.
Dan
.
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On Monday, November 22, 2021 at 5:36:34 AM UTC-5, Mostowski Collapse wrote:
> That an empty universe of discourse is allowed has
> recently become a hiding space for being lazy.
> That an empty universe of discourse is allowed prevents
>

...

[Mostowski Collapse]

I am pretty sure you are voting to flog your dead horse.
So I didn't read what you posted on sci.logic.

Lets see...

[Mostowski Collapse]

Oh your dead horse is a schrödinger horse. Its neither
dead nor alive. How can you prove from no axioms:

EXIST(a):A(a)

When the domain of discourse can be empty? You
must have used an axiom somewhere that

postulates some existence. Usually EXIST(a):A(a)
fails when the domain of discourse U is empty.

[Kip Foh]

Mostowski Collapse wrote:

> I am pretty sure you are voting to flog your dead horse.
> So I didn't read what you posted on sci.logic. Lets see...

Switzerland is gay, war criminals, protecting people from covid_19 in
*2021*, by shooting them, killing with automatic firearms. Pushing a
lethal toxic poison, in people paying their income, which is
*NOT_a_vaccine*. Lying bitches, a shame for europe. They are nothing but
puppets for the global dogmas of idiocies. They are pushing checkpoints
in cities, at groceries, at cinema etc etc, *war_criminals* trying to
genocide humanity. They poison everything, never buy food made in holland.
Arrest them, prosecute and condemn.

Latest - Finally - Medical Proof the Covid Jab is 'Murder'.
https://www.bitchute.com/video/2EIc9fLDcGpX/

AUSTRIA LOCKS DOWN- MANDATORY VACCINES ARE A WAR CRIME AGAINST HUMANITY
https://www.bitchute.com/video/WmX8zol8NSwv/

[Mostowski Collapse]

Hey Luigi, how is mario cart doing?

eurotrip - creepy italian guy
https://www.youtube.com/watch?v=nioOj6FTZQ8

Kip Foh schrieb am Montag, 22. November 2021 um 17:38:47 UTC+1:
> Mostowski Collapse wrote:
>
> > I am pretty sure you are voting to flog your dead horse.
> > So I didn't read what you posted on sci.logic. Lets see...
>

> I am Kip Foh and I am gay, AP brain farto is my bed warmer

[Kip Foh]

Mostowski Collapse wrote:

> Hey Luigi, how is mario cart doing?

Australian Army Rounding People Up
https://www.brighteon.com/92a7a021-f166-468b-adbc-5410c9dc992b

[Mostowski Collapse]

Is this where your gay friends live?

Kip Foh schrieb am Montag, 22. November 2021 um 18:33:58 UTC+1:
> Mostowski Collapse wrote:
>
> > Hey Luigi, how is mario cart doing?
>

> I am Kip Foh and I am gay, AP brain farto is my bed warmer

> Australian kangaroos can jump very high

[Kip Foh]

Mostowski Collapse wrote:

>> I am Kip Foh and I am gay, AP brain farto is my bed warmer Australian
>> kangaroos can jump very high

not sure. How many directions are represented by a 3D tensor?

--
A fart can be useful, It gives the body ease, It warms the bed in winter,
And suffocates the flees.

-

[Kip Foh]

Mostowski Collapse wrote:

> Oh your dead horse is a schrödinger horse. Its neither dead nor alive.
> How can you prove from no axioms:

sure, but hanging public places are ready to be re-introduced. There are
no alternatives. You can't allow strangers to euthanize you with toxic
lethal poison into your veins.

and pay attention, there are no 90% already vaccinate. It's a lie. An
insult to humanity and the human brain. My bet is there is no more that 5
to 10% "vaccinated". They are lying criminals.

Australia Rounds Up Close Contacts Into Concentration Camps, Germany Eyes
Austria-like Forced Vax https://www.bitchute.com/video/ILfb6oqY6HB4/

--
Farts are made up of the following: Nitrogen, the main ingredient making
up 59%, hydrogen at 21 %, Carbon Dioxide at 9%, methane at 7%, Oxygen at
3% and other stuff at 1%.

[Mostowski Collapse]

Do you see "vaccine" in the title of this thread?

[Michael Moroney]

On 11/22/2021 1:18 PM, Mostowski Collapse wrote:
> Do you see "vaccine" in the title of this thread?
>

That's the nymshifter troll again. It's not here to discuss math, it's
here to disrupt things, under orders from the Kremlin.

[Kip Foh]

Mostowski Collapse wrote:

> Do you see "vaccine" in the title of this thread?
>
> Kip Foh schrieb am Montag, 22. November 2021 um 19:09:19 UTC+1:
>> Mostowski Collapse wrote:
>> > Oh your dead horse is a schrödinger horse. Its neither dead nor
>> > alive. How can you prove from no axioms:

wrong again. It's not dead or alive, but both death and alive. Or implies
making the choice. Which you can't. And there is no traces of *covid_19*
in the history of mankind. Where it came from??

this possessed capitalist big-pharma bitch is saying to the un-vaccinated
"sacrifices has to be made".

She's behaving like she's possessed!
https://www.bitchute.com/video/EzD8vNApC6Lh/

--
Hydrogen Sulphide is the compound that makes farts stink.

[Kip Foh]

You have no idea what the Schrodinger cat paradox stands for. Go back to
Russia. "Michael" is a Russian name.

--
The average person farts about 14 times a day, which produces about half
a litre of fart gas.

[Mostowski Collapse]

I thought he is from mario cart.
Luigi has a little L on his hat:

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

What does the L mean? Looser?

Michael Moroney schrieb:

[Dan Christensen]

On Monday, November 22, 2021 at 11:36:17 AM UTC-5, Mostowski Collapse wrote:
> Oh your dead horse is a schrödinger horse. Its neither
> dead nor alive. How can you prove from no axioms:
>
> EXIST(a):A(a)
>

If you have an active premise A(x) (i.e. one that hasn't been discharged), or you have proven that A(x) for some x, then you can infer EXIST(a):A(a).

Dan

[Kip Foh]

Mostowski Collapse wrote:

> I thought he is from mario cart. Luigi has a little L on his hat

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MAJOR NEW CRACKDOWN ANNOUNCED IN AUSTRALIA...
https://www.bitchute.com/video/wgK7oZaOAm8L/

[Mostowski Collapse]

You cannot prove something that has EXIST(a)
in the root. So you cannot prove:

<no premisses here,
all discharged> |- EXIST(a):A(a)

For any A(a). Since be definition EXIST(a):A(a)
is false in the empty universe of discourse.
If you find such an A(a), then there is somewhere

a bug in your DC Proof dealing with a possibly
empty universe of discourse. Or you invoked
some menu item during the proof,

that assumes non-empty universe.

[Dan Christensen]

On Monday, November 22, 2021 at 7:51:25 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Montag, 22. November 2021 um 21:24:41 UTC+1:
> > On Monday, November 22, 2021 at 11:36:17 AM UTC-5, Mostowski Collapse wrote:
> > > Oh your dead horse is a schrödinger horse. Its neither
> > > dead nor alive. How can you prove from no axioms:
> > >
> > > EXIST(a):A(a)
> > >
> > If you have an active premise A(x) (i.e. one that hasn't been discharged), or you have proven that A(x) for some x, then you can infer EXIST(a):A(a).
> >

> You cannot prove something that has EXIST(a)
> in the root. So you cannot prove:
>
> <no premisses here,
> all discharged> |- EXIST(a):A(a)
>

.
True.
.

> For any A(a). Since be definition EXIST(a):A(a)
> is false in the empty universe of discourse.

.
Makes no sense. Sorry.

.
> If you find such an A(a), then there is somewhere
>
> a bug in your DC Proof dealing with a possibly
> empty universe of discourse. Or you invoked
> some menu item during the proof,
>
> that assumes non-empty universe.

.
Wrong again, Jan Burse. The problem of non-empty vs. empty domains of discussion can be easily solved in DC Proof, as in most math textbooks, by restricting each quantifier to some set, e.g. EXIST(a):[a in s & A(a)] or ALL(a):[a in s => A(a)]. Then you can specify EXIST(a):a in s for a non-empty domain s or ~EXIST(a): a in s for an empty domain s. Yes, I know that requires a bit more work. Don't be so lazy, Jan Burse!
.
Dan
.

[Mostowski Collapse]

Proving means showing something generally valid.
If the universe of discourse can be empty, then
a sentence of the form:

EXIST(a):A(a)

Cannot be proved, because it isn't valid when the
universe of discourse is empty. What element
of b of the universe of discourse

should make A(b) make true? When the
universe of discourse is empty then EXIST(a):A(a)
is automatically false.

Similarly for:

EXIST(d):EXIST(a):[a e d & A(a)]

You cannot prove it, when you allow
empty domains, it has also an EXIST in the
root of the formula. So unless you use

axioms or menu items with ontological
commitment, you cannot prove it when
an empty domain is allowed. But you can

prove this here when you allow empty domains:

ALL(d):EXIST(a):[a e d => A(a)]

There are cases where it is provable.

Dan Christensen schrieb am Dienstag, 23. November 2021 um 03:53:03 UTC+1:
> > For any A(a). Since be definition EXIST(a):A(a)
> > is false in the empty universe of discourse.

> Makes no sense. Sorry.

[Mostowski Collapse]

Corr.:

> prove this here when you allow empty domains:

> ALL(d):EXIST(a):[a e d & A(a)]

I guess you can also prove this one,
when d is a free variable:

EXIST(a):[a e d & A(a)]

But DC Proof does then not have the
proof completed,

in some thread in the past there was
even a problem with stopping too early in
a proof, it showed something wrong.

Must be still somewhere on sci.math
or sci.logic.

[Mostowski Collapse]

For example when constructing the reals, its
standard to then prove:

1) EXIST(x):[x e R & x^2 = 2]

To give an example of an irrational number and
that now exists. This is not the same as:

2) ALL(x):[x e Q => x^2 <> 2]

I have never seen a proof of 1) or 2) in DC Proof.

LoL

[Mostowski Collapse]

When constructing the reals bottom up, you will
likely need countable infinite sequences, and to
show the existence of sqrt(2),

you will then need the existence of a particular
countable infinite sequence, but a countable infinite
sequence needs the existence of omega I guess.

Its not enough to have a predicate that somehow
delivers the countable infinite sequence, you then
end up in WM potential infinity, the point

of constructing the reals is to have first clas real numbers
objects, so yes, it is EXTREMLY IMPORTANT, to be be able
to prove things like:

EXIST(f):Dedekind(f)

EXIST(w):Peano(w)

Etc..

Its the bread and butter of theorem proving, otherwise
your domain of discourse can be empty, in the case of DC
Proof, but when its non-empty one still doesn't

know whether it might contain an sqrt(2) object in the
form of an existence proof.

LMAO!

[Mostowski Collapse]

If you don't have sqrt(2) as a real number object, you might
have it as a class. But you refuse to work with classes.
Dedekind cuts, can be also viewed as classes. Assume

Q is some WM potential infinite class, then this Dedekind
cut is also a class:

{ x e Q | x^2 < 2 }

But class have the drawback that you cannot put them
into the scope of quantifiers,

usually if they are only FOL syntactic suggar, you hardly
see them in quantifier scopes. You can check yourself
how many analysis theorems are existence theorems:

- Mean Value Theorem?
- Rolle's theorem
- Etc.. etc..

They are all existence theorems. Proving them the same
way as you did prove Peano, like here:

ALL(f):[Dedekind(f) => EXIST(w):Peano(w)]

Is a little unsatisfactory, because the proof leaves open
whether Dedekind(f) exists, and what the foundation
for it would be.

[Dan Christensen]

On Tuesday, November 23, 2021 at 2:45:39 AM UTC-5, Mostowski Collapse wrote:
> Corr.:
> > prove this here when you allow empty domains:
> > ALL(d):EXIST(a):[a e d & A(a)]
>
> I guess you can also prove this one,
> when d is a free variable:
> EXIST(a):[a e d & A(a)]
> But DC Proof does then not have the
> proof completed,
>

You would have to formally introduce an axiom of the form EXIST(x): x in d at the beginning of your proof. Then you could have a theorem of the form EXIST(a):[a in d & A(a)]. As you would expect, axioms cannot be discharged.

Dan

[Mostowski Collapse]

Dan-O-Matik was talking nonsense (on sci.logic):
> And while Dedekind infinite sets are often mentioned in math
textbooks, the ZFC axioms are seldom if ever mentioned other
than in passing, as I recall.

An example of a simple theorem where you need some ZF axiom is:

Mostowski Collapse schrieb am Mittwoch, 24. November 2021 um 02:56:37 UTC+1:
> Or formally, s≠0 stands for EXIST(c):[c e s]:
> ALL(a):[a≠0 => EXIST(b):[ALL(x):[x e a => f(x) e b] & b≠0]]
https://groups.google.com/g/sci.logic/c/tVUtBSQUhiE/m/bLEb_i6BBAAJ

The above is provable in FOL+ZF.
It is not provable in DC Proof alone.

[Mostowski Collapse]

Additionally assuming f : x -> y for some x,y is
not required to prove it in FOL+ZF. It would be also
nonsense, since the preimage a is not supposed

to be a subset of x. It can be any set a from the
whole universe of discourse. How would this theorem
then work? Well you need something from ZF,

but FOL itself also helps, since one can prove:

ALL(x):EXIST(y):f(x)=y

[Mostowski Collapse]

I am not suggesting Borel determinancy as an exercise:
“Skolem needed a snack, so he ate enough for an entire wedding party”.
https://golem.ph.utexas.edu/category/2021/07/borel_determinacy_does_not_require_replacement.html

I am myself a little astonished that the basic math
problem needs some axiom? Right? If I have time
I will provide a proof of it using the axiom I have

in mind. I don't know yet how to show that it cannot
be proved without the axiom or a weaker form of it.
But I am assuming I will be not in a hurry since DC

Proof will not be able to prove the same basic math.

[Mostowski Collapse]

Now this is like fukushima. I guess Dan-O-Matiks
brain is now melting. We finally found it:

∀x(Px → Qf(x)) → (∀x∀y(f(x)=y → (Px ∧ Qy)) ↔ ∀xPx) is valid.
A Dan-O-Matik function is a binary relation iff the domain is the universal class
https://www.umsu.de/trees/#~6x%28Px~5Qf%28x%29%29~5%28~6x~6y%28f%28x%29=y~5Px~1Qy%29~4~6xPx%29

We could only chill fukushima, if Dan-O-Matik were
as cool as this man here, who sings many languages:

Alpha Blondy - Jérusalem
https://www.youtube.com/watch?v=WcqK9Ls7Eos

But Dan-O-Matik isn't cool, he doesn't grasp that a FOL
function symbol and a set-like function are not the same.

[Mostowski Collapse]

Actually learning 3 languages would be helpful:
- FOL function symbol
- class-like functions
- set-like functions

[Mostowski Collapse]

But there is cure, Dan-O-Matik could exercise and get fluent
in all 3 languages by this little booklet:

Basic Set Theory - Azriel Levy
https://www.amazon.de/dp/0486420795

Only 20 bucks or so.

[Mostowski Collapse]

Unfortunately Dan-O-Matik is only genuinely stupid:

Two things are infinite: the universe and human
stupidity; and I’m not sure about th’universe!
- Albert Einstein

Dan-O-Matik even tops that. He still doesn't understand
FOL, and how a FOL function symbol works. A definition def
for A -> B would be WELL DEFINED if we can prove the folllowing:

(*) def(f) & def(g) => f=g

Dan-O-Matik doesn't understand that this is impossible
for FOL function symbols with his serial formula. From
computer science there are ways to fix it.

For example one fix introduces a new symbol undefined ⊥.
We have to assume the universe of discourse is non-empty.
And then uses Dan-O-Matik serial formula:

ALL(x):[x in A => f(x) in B]

And adds this axiom:

ALL(x):[~x in A => f(x)=⊥]

Now you can prove (*). But most mathematicians do it more
easy, they don't use FOL function symbols, but set like function
symbols, since the domain A is already set like:

ALL(x):[~x in A => ~EXIST(y):(x,y) e f]

[Mostowski Collapse]

For example JavaScript implements undefined ⊥:

Legts get back to the 10 booleans array problem.
Just open any browser, switch to development view,
and enter in the console:

> a=[0,0,0,0,0,0,0,0,0,0];a[10]
undefined

In JavaScript accessing out of bound gives undefined.
There is a bound check, but there is no exception but a
special value.

This is closer to some logical modelling of automated
verification of programs which uses undefined ⊥.

[Mostowski Collapse]

Corr.: The WELL DEFINED issue arises when for example
A={0} and B={0}, we then want from def(h) <=> h : A -> B:

(*) def(f) & def(g) => f=g

Otherwise issue arises with the collection of functions,
they just become too big if we don't constrain FOL function

symbols outside of the domain.

[Mostowski Collapse]

How would one formulate in DC Proof, which is supposed to
be the tool for mathematics textbooks, the following:

if (f + g)′(x) is undefined then f′(x) or g′(x) are undefined

See also:

Definedness, Solomon Feferman - Dedicated to Alonzo Church
https://math.stanford.edu/~feferman/papers/definedness.pdf

Ha Ha, on page 7:

The axioms DES are of clear philosophical interest, providing one solution
of the familiar puzzles as to how to handle non-referring definite descriptions,
such as in ‘the present king of France is bald’. But they also serve to regulate
the use of mathematical definitions such a lim xn

[Mostowski Collapse]

But Fefi was sloppy, when defining his Impredicative Theory of
Operations and Classes (IOC), at least in this paper in sec 7
there is a blooper, when he says:

Now write: (i) f:(A→B)for∀x∈A(fx∈B)

Thats Dan-O-Matik nonsense! Did he fix that in another
publication? Same problem in Gerhard Jäger, these things
were never really used? Page 6:

(f :a→b) := (∀x∈a)(fx∈b)
https://home.inf.unibe.ch/ltg/publications/2009/jae09b.pdf

But maybe harmless for what they do with it? Fefi only
uses it to express totality over the natural numbers for
his gcd toy example.

[Dan Christensen]

On Monday, December 13, 2021 at 4:25:27 PM UTC-5, Mostowski Collapse wrote:
> How would one formulate in DC Proof, which is supposed to
> be the tool for mathematics textbooks, the following:
>
> if (f + g)′(x) is undefined then f′(x) or g′(x) are undefined
>

There is no formal notion undefined. You look at the relevant definition and determine whether or not it can be applied to the statement in question. It not, then we say that that it is UNDEFINED (with respect to that definition). Get it, Jan Burse??? Didn't think so. Oh, well...

[Mostowski Collapse]

Why can Terence Tao then prove:

Theorem 10.1.13 (Differential calculus)
(Sum rule) If f and g are differentiable at x0, then f + g is also
differentiable at x0, and (f + g)′(x0) = f′(x0) + g′(x0).
https://lms.umb.sk/pluginfile.php/111477/mod_page/content/5/TerenceTao_Analysis.I.Third.Edition.pdf

You say this textbook mathematics is not formalizable?

[Mostowski Collapse]

With defined operator ↓, a corollar from the Sum rule:

f′(x) ↓ v g′(x) ↓ => (f + g)′(x) ↓

Or taking contraposition, now talking about undefined:

~ (f + g)′(x) ↓ => ~ f′(x) ↓ v ~ g′(x) ↓

[Dan Christensen]

On Monday, December 13, 2021 at 5:37:56 PM UTC-5, Mostowski Collapse wrote:

> You say this textbook mathematics is not formalizable?
> Dan Christensen schrieb am Montag, 13. Dezember 2021 um 23:33:32 UTC+1:
> > On Monday, December 13, 2021 at 4:25:27 PM UTC-5, Mostowski Collapse wrote:
> > > How would one formulate in DC Proof, which is supposed to
> > > be the tool for mathematics textbooks, the following:
> > >
> > > if (f + g)′(x) is undefined then f′(x) or g′(x) are undefined
> > >
> > There is no formal notion undefined. You look at the relevant definition and determine whether or not it can be applied to the statement in question. It not, then we say that that it is UNDEFINED (with respect to that definition). Get it, Jan Burse??? Didn't think so. Oh, well...

> Why can Terence Tao then prove:
>
> Theorem 10.1.13 (Differential calculus)
> (Sum rule) If f and g are differentiable at x0, then f + g is also
> differentiable at x0, and (f + g)′(x0) = f′(x0) + g′(x0).
> https://lms.umb.sk/pluginfile.php/111477/mod_page/content/5/TerenceTao_Analysis.I.Third.Edition.pdf
>

More grasping at straws, Jan Burse??? What has this got to do functions outside of there domain? Hint: Nothing.

[Mostowski Collapse]

Thats not the correct approach in logic. Since Peano, we find something else:

"Function application. "(𝐹‘𝑥)" should be read "the value of function 𝐹 at 𝑥" and
has the same meaning as the more familiar but ambiguous notation F(x).
For example, (cos‘0) = 1 (see cos0 14824). The left apostrophe notation
originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell]
p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26.
See df-fv 5865. In the ASCII (input) representation there are spaces around the
grave accent; there is a single accent when it is used directly, and it is
doubled within comments."
http://us.metamath.org/mpeuni/conventions.html

Metamath can use the same symbol for function F and the function graph F,
in the above F is the same referent. Metamath uses then the empty set as
undefined marker, in case an argument is not in dom(F).

There are even aspects of definite description, F need not be a function graph
per see. They can therefore prove the following definite descriptions theorem.
It is not like in Russell, where we get false, its more we get the defined marker:

/* Provable in Metamath */
~ EXISTUNIQUE(x) (a,x) e F => (F'a) = {}

I don't think DC Proof/Donnie Darko has such a maker u:

/* Not Provable in DC Proof with Donnie Darke Function approach */
~ EXISTUNIQUE(x) (a,x) e F => F(a) = u

The Peano apostroph was later adopted by for example Withhead Russell,
Quine, Takeuti Zaring. It only needs some set theory like ZFC. So yes, DC
Proof with Donnie Darke Function approach is not what is common in logic.

[Mostowski Collapse]

So only ALL(a):[a e x => f(x) e b] , the Donnie Darko approach, like
dark numbers in WM, is just WM with extra steps, not saying
what is f outside of the domain a is not what we do in logic.

Thats just too dark for logic, these dark booleans. LoL

[Mostowski Collapse]

The only new development in logic might be these partial function
logics, where we dont need f'x can go back to f(x), but we have
axioms and inference rules for the operator ↓ .

But there is even a simpler approach than using an operator ↓,
one can also use a special value ⊥. DC Proof didn't make any
choice so far, its Function action

postulates same referent f for both the function graph
and the function, but the Function axiom is quite weak:

- It only applies when f is a function graph
- It doesn't say what f(x) should be outside of the domain

This is unlike the f'x approach which allows both. Then the
f(x) approach is less subject to the first problem, but the second
problem is solved in the new f(x) apprach either through

an operator ↓ or a special value ⊥ .

[Mostowski Collapse]

Then there is a whole field where the underlying logic is changed,
and where the underlying logic is not anymore classical logic,
but where for example 3-valued logic is used.

A prominent example is the SQL database system query language,
which has even an ISO standard. In SQL there is a special value ⊥,
written NULL. But there is also a third truth value.

"Since Null is not a member of any data domain, it is not considered
a "value", but rather a marker (or placeholder) indicating the
undefined value. [...] SQL implements three logical results, so SQL
implementations must provide for a specialized three-valued logic (3VL)."
https://en.wikipedia.org/wiki/Null_%28SQL%29#Comparisons_with_NULL_and_the_three-valued_logic_%283VL%29

[Dan Christensen]

On Tuesday, December 14, 2021 at 2:50:36 AM UTC-5, Mostowski Collapse wrote:

> Proof with Donnie Darke Function approach is not what is common in logic.
> Dan Christensen schrieb am Montag, 13. Dezember 2021 um 23:52:45 UTC+1:
> > On Monday, December 13, 2021 at 5:37:56 PM UTC-5, Mostowski Collapse wrote:
> >
> > > You say this textbook mathematics is not formalizable?
> > > Dan Christensen schrieb am Montag, 13. Dezember 2021 um 23:33:32 UTC+1:
> > > > On Monday, December 13, 2021 at 4:25:27 PM UTC-5, Mostowski Collapse wrote:
> > > > > How would one formulate in DC Proof, which is supposed to
> > > > > be the tool for mathematics textbooks, the following:
> > > > >
> > > > > if (f + g)′(x) is undefined then f′(x) or g′(x) are undefined
> > > > >
> > > > There is no formal notion undefined. You look at the relevant definition and determine whether or not it can be applied to the statement in question. It not, then we say that that it is UNDEFINED (with respect to that definition). Get it, Jan Burse??? Didn't think so. Oh, well...
> >
> > > Why can Terence Tao then prove:
> > >
> > > Theorem 10.1.13 (Differential calculus)
> > > (Sum rule) If f and g are differentiable at x0, then f + g is also
> > > differentiable at x0, and (f + g)′(x0) = f′(x0) + g′(x0).
> > > https://lms.umb.sk/pluginfile.php/111477/mod_page/content/5/TerenceTao_Analysis.I.Third.Edition.pdf
> > >
> > More grasping at straws, Jan Burse??? What has this got to do functions outside of there domain? Hint: Nothing.

> Thats not the correct approach in logic. Since Peano, we find something else:
>
> "Function application. "(𝐹‘𝑥)" should be read "the value of function 𝐹 at 𝑥" and
> has the same meaning as the more familiar but ambiguous notation F(x).
> For example, (cos‘0) = 1 (see cos0 14824). The left apostrophe notation
> originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell]
> p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26.
> See df-fv 5865. In the ASCII (input) representation there are spaces around the
> grave accent; there is a single accent when it is used directly, and it is
> doubled within comments."
> http://us.metamath.org/mpeuni/conventions.html
>

We still have f(x) being undefined (or meaningless) for x outside the domain of function f. See, for example, Tao's "Analysis I" (quoted here).

[Mostowski Collapse]

Thats exactly why EXISTUNIQUE doesn't work. I 100'% agree
with Terence Tao. When you use a FOL function symbol,
and you only prove:

ALL(a):[x e a => ALL(y):[y e b => [(x,y) e F <=> f(x)=y]]]

Then obviously f(x) for ~(x e a) isn't pinned down. And then
you can for example not replace the EXIST quantfier in the
below, by an EXISTUNIQUE quantifier:

/* Doesn't work with EXISTUNIQUE also as per Terence Tao */
ALL(a):[Set(a) => EXIST(f):ALL(b):[b in a => f(b)=b]]

I am only trying to explain you the basic facts of beginner
mathematics. The metamath page does something different,
it doesn't use two symbols F and f, as in Terence Tao.

With the metamath approach you can prove EXISTUNIQUE.
With the Terence Tao approach you cannot prove
EXISTUNIQUE.

Whats wrong with you?

[Mostowski Collapse]

With the metamath approach you can prove EXISTUNIQUE

when the undefined marker is not in the codomain. Its like
here, the undefined marker is supposed to usually

not be used in codomains:

Mostowski Collapse schrieb am Dienstag, 14. Dezember 2021 um 10:07:32 UTC+1:
> "Since Null is not a member of any data domain, it is not considered
> a "value", but rather a marker (or placeholder) indicating the
> undefined value. [...]

> https://en.wikipedia.org/wiki/Null_%28SQL%29#Comparisons_with_NULL_and_the_three-valued_logic_%283VL%29
https://groups.google.com/g/sci.math/c/SOIECV0E-Dc/m/JYKAbTdhCAAJ

But since you deal with identity functions, EXISTUNIQUE
will be provable when the undefined marker is not
in the domain, I guess you only look at identity

functions where dom=codom.

[Mostowski Collapse]

So maybe what Fefi (Solom Feferman) explains is the more
cleaner approach. You don't have to fiddle with domains and
codomains, there is only new operator ↓. In the metamath

approach you would require that functions are kind of
normalized, for example that a function f = { ... (x, ⊥) ...}
would be forbidden, whereas a function that doesn't

have the ordered pair (x, ⊥) would be tolerated.

[Dan Christensen]

On Tuesday, December 14, 2021 at 1:34:07 PM UTC-5, Mostowski Collapse wrote:
> Thats exactly why EXISTUNIQUE doesn't work. I 100'% agree
> with Terence Tao. When you use a FOL function symbol,
> and you only prove:
>
> ALL(a):[x e a => ALL(y):[y e b => [(x,y) e F <=> f(x)=y]]]
>

Tao actually wrote, as I think every working mathematician would agree, that if you have a binary relation P such such that ALL(a):[a in X => EXIST(b):[b in Y & P(a,b) & ALL(c):[c in Y => [P(a,c) => c=b]]]] then we will have a UNIQUE function f: X --> Y such that ALL(a):ALL(b):[a in X & b in Y => [f(a)=b <=> P(a,b)].

This is consistent with the handling of functions in DC Proof.

[Mostowski Collapse]

The deeper problem is not Terrence Tao versus Dan-O-Matik
versus metamath. In all 3 approaches this here then fails:

/* Not Provable */
ALL(a):[Set(a) => EXISTUNIQUE(f):ALL(b):[b in a => f(b)=b]]

Thats very easy to see. Just chew on the downgrade Lemma:

/* Downgrade Lemma */
ALL(b):[b in a => b in c] & ALL(b):[b in c => f(b)=b]] => ALL(b):[b in a => f(b)=b]]

The tree tool of Wolfgang Schwartz can prove it, you
can also prove it with DC Proof, can't you:

/* Encoding E__ stands for _ in _ */
∀x(Exa → Exc), ∀x(Exc → f(x)=x) entails ∀x(Exa → f(x)=x).
https://www.umsu.de/trees/#~6x%28Exa~5Exb%29,~6x%28Exb~5f%28x%29=x%29|=~6x%28Exa~5f%28x%29=x%29

Do you know what it says?
Do you know why it prevents EXISTUNIQUE?

[Mostowski Collapse]

It prevents EXISTUNIQUE because you can find (f1,c1)
and (f2, c2), where f1≠f2 and:

ALL(b):[b in c1 => f1(b)=b]]

ALL(b):[b in c2 => f2(b)=b]]

Now if you futher have constructed (f1,c1) and (f2,c2)
such that a ⊆ c1 and a ⊆ c2, then you can downgrade
the two and you get:

ALL(b):[b in a => f1(b)=b]]

ALL(b):[b in a => f2(b)=b]]

And this breaks EXISTUNIQUE. Its really extremly trivial.
Just use the Downgrade Lemma.

[Dan Christensen]

On Tuesday, December 14, 2021 at 2:49:03 PM UTC-5, Mostowski Collapse wrote:
> The deeper problem is not Terrence Tao versus Dan-O-Matik
> versus metamath. In all 3 approaches this here then fails:
>

You fail, Jan Burse, because you do not understand that f(x) is meaningless (undefined) if x is outside the domain of function f. You have painted yourself into this corner and now look like an idiot claiming the functions so defined are not unique. They are unique. Deal with it.

[Mostowski Collapse]

By all means, you cannot prove this here:

/* Not Provable */
ALL(a):[Set(a) => EXISTUNIQUE(f):ALL(b):[b in a => f(b)=b]]

This is impossible. Prove me wrong. Prove it.

[Mostowski Collapse]

BTW: This is also not provable:

/* Not Provable */
ALL(a):[Set(a) => EXISTUNIQUE(f):[Set'(f) & ALL(b):[b in a => (b,b) in f]]]

Maybe easier to first digest that this is not provable.

[Dan Christensen]

On Tuesday, December 14, 2021 at 3:53:36 PM UTC-5, Mostowski Collapse wrote:
> By all means, you cannot prove this here:
> /* Not Provable */
> ALL(a):[Set(a) => EXISTUNIQUE(f):ALL(b):[b in a => f(b)=b]]
> This is impossible. Prove me wrong. Prove it.

Still in denial, I see. Really kind of pathetic.

I have posted the proof twice in this thread. Deal with it, Jan Burse.

Dan

[Mostowski Collapse]

No denial, you didn't post a proof, where you use EXISTUNIQUE(f).
You used your custom EXISTUNIQUE(f,a), which has a domain
parameter a. But the question is whether this here is provable or not:

On Tuesday, December 14, 2021 at 3:53:36 PM UTC-5, Mostowski Collapse wrote:
> By all means, you cannot prove this here:
> /* Not Provable */
> ALL(a):[Set(a) => EXISTUNIQUE(f):ALL(b):[b in a => f(b)=b]]
> This is impossible. Prove me wrong. Prove it.

Maybe you are a little confused?

[Mostowski Collapse]

You used this EXISTUNIQUE(f,a):

EXISTUNIQUE(f,a):A(f) <=> EXIST(h):[A(h) & ALL(f):ALL(g):[A(f) & A(g) => f|a = g|a]]

But my claim is you cannot prove it with EXISTUNIQUE(f):

EXISTUNIQUE(f):A(f) <=> EXIST(h):[A(h) & ALL(f):ALL(g):[A(f) & A(g) => f = g]]

I made this pretty clear here, when I wrote:
"You didn't prove f=g"

Whats wrong with you?

[Mostowski Collapse]

I dont think DC proof correctly associates functions and function graphs.
You can also try Terence Tao decrement example:

Example 3.3.2:
Unfortunately this does not define a function, because
when x = 0 there is no natural number y whose increment
is equal to x (Axiom 2.3). On the other hand, we can legitimately
define a decrement function h : N\{0} → N associated to
the property P(x,y) defined by y++ = x.
https://lms.umb.sk/pluginfile.php/111477/mod_page/content/5/TerenceTao_Analysis.I.Third.Edition.pdf

I think its is provable:

~EXIST(y):P(0,y)

Or do you claim otherwise?
Do you also have dark booleans for decrement?

LMAO!

[Mostowski Collapse]

Or asked differently how many decrement functions are
there in Peano arithmetic? Can you prove:

EXISTUNIQUE(h):"h is the decrement function"

[Dan Christensen]

On Tuesday, December 14, 2021 at 4:39:00 PM UTC-5, Mostowski Collapse wrote:
> No denial, you didn't post a proof, where you use EXISTUNIQUE(f).
> You used your custom EXISTUNIQUE(f,a), which has a domain
> parameter a. But the question is whether this here is provable or not:

Still in denial, I see. I have twice posted proof of the uniqueness of the identity function on any given set, i.e. the same input will result in the same output. Deal with it, Jan Burse. You are looking like a complete idiot here, in the same league as AP, JG and WM. If you are simply going to keep repeating your absurd claims as here, do not expect a reply from me on this matter.

Dan

[Mostowski Collapse]

The harder I look the less your nonsense makes any sense. Those
people calling other people cranks, are often cranks themselves.

Can you show us the decrement function in DC Proof. If
I take your approach and only define it as:

ALL(a):[a in N \ {0} => h(a)++ = a]

I can still prove in DC Proof:

EXIST(b):h(0)=b

But Terrence Tao says "decrement function h : N\{0} → N

associated to the property P(x,y) defined by y++ = x".

[Mostowski Collapse]

Now take Terence Taos definition of f=g. He says
f=g for f, g: X->Y: f=g iff forall x(x e X => f(x)=g(x)).

Now take f, g: N -> N with:

f(x)=if x=\=0 then x-1 else 3
g(x)=if x=\=0 then x-1 else 4

Both satisfy a Dan-O-Matik definition:

ALL(a):[a in N \ {0} => f(a)++ = a]
ALL(a):[a in N \ {0} => g(a)++ = a]

But according to Terence Tao we have f=\=g.
So Dan-O-Matik has a hell of a lot of decrement functions.

LoL

[Mostowski Collapse]

What is easier to use, DC proof or ZFC? DC proof
claims that certain things are given:

> ALL(a):[a in D => f(a) in C] where D and C are the given domain and
> codomain sets. In this case, "f(x)" is undefined and quite meaningless for x not in D.

Is this true when we talk about functions? The domains
and co-domains cannot always be given so easily. Thats the
Dan-O-Matik paradox. For example there is this theorem:

"Every injective function f has an inverse function g on its range"

If f: A -> B, whats the domain of g? What does need less
steps to prove the above, DC proof or ZFC?

[Mostowski Collapse]

With ZFC I only need to do the following:

1) let g = { (y, x) | (x, y) in f}, show it is a set

2) show g is a function when f is injective

3) conclude dom(g) = ran(f)

I am pretty sure DC proof needs much more work. Since to
start with ALL(a):[a in A => f(a) in B] and end with existence
of g, D and C such that ALL(a):[a in D => g(a) in C],

it will anyway first construct g in the form of a set and then
use its function axiom. So its kind of set theory with extra
steps. Utter unnecessary nonsense to be precise.

[Dan Christensen]

On Friday, December 24, 2021 at 4:24:14 AM UTC-5, Mostowski Collapse wrote:
> What is easier to use, DC proof or ZFC? DC proof
> claims that certain things are given:
>
> > ALL(a):[a in D => f(a) in C] where D and C are the given domain and
> > codomain sets. In this case, "f(x)" is undefined and quite meaningless for x not in D.
>
> Is this true when we talk about functions? The domains

> and co-domains cannot always be given so easily. ...

When you want to introduce an arbitrary function in a proof, you should, at the very least, attach a name to the function, domain and codomain.

Example in DC Proof notation

1. Set(x) & Set(y) & ALL(a):[a in x => f(a) in y]
(Premise)

Where f is an arbitrary function on arbitrary sets x and y.

Then f(z) will be undefined (i.e. meaningless) if z is not in the domain set x. This is the usual practice in math textbooks. Quit pretending otherwise, Jan Burse.

[Mostowski Collapse]

You are changing topic. I dont mind if you need Set(_)
as well in DC poop. The question is rather can you
prove this textbook theorem, i. e. EXIST(g) :

"Every injective function f has an inverse function g on its range"

But I guess you already gave up. A few weeks ago you
boasted ZFC or set theory is not found in everyday math.
What about the above theorem, what is needed in DC poop?

[Dan Christensen]

On Friday, December 24, 2021 at 5:49:24 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Freitag, 24. Dezember 2021 um 16:44:49 UTC+1:
> > On Friday, December 24, 2021 at 4:24:14 AM UTC-5, Mostowski Collapse wrote:
> > > What is easier to use, DC proof or ZFC? DC proof
> > > claims that certain things are given:
> > >
> > > > ALL(a):[a in D => f(a) in C] where D and C are the given domain and
> > > > codomain sets. In this case, "f(x)" is undefined and quite meaningless for x not in D.
> > >
> > > Is this true when we talk about functions? The domains
> > > and co-domains cannot always be given so easily. ...
> >
> > When you want to introduce an arbitrary function in a proof, you should, at the very least, attach a name to the function, domain and codomain.
> >
> > Example in DC Proof notation
> >
> > 1. Set(x) & Set(y) & ALL(a):[a in x => f(a) in y]
> > (Premise)
> >
> > Where f is an arbitrary function on arbitrary sets x and y.
> >
> > Then f(z) will be undefined (i.e. meaningless) if z is not in the domain set x. This is the usual practice in math textbooks. Quit pretending otherwise, Jan Burse.

> You are changing topic.

The topic was your bizarre notion of functions outside of their domain of definition, Jan Burse. Just admit you were wrong.

> I dont mind if you need Set(_)
> as well in DC poop. The question is rather can you
> prove this textbook theorem, i. e. EXIST(g) :
> "Every injective function f has an inverse function g on its range"

Trying to change the subject, Jan Burse? In any case, you might have a look at my formal proof of the Cantor-Burnstein-Schroeder theorem at http://dcproof.com/CBS.htm

[Mostowski Collapse]

So you cannot show existence of g?

Hiding behind some absurd claims the encoding
of functions as set of pairs is bizzar?

How do you show existence of g = f^-1 in DC proof?

LoL

[Dan Christensen]

On Sunday, December 26, 2021 at 5:04:17 AM UTC-5, Mostowski Collapse wrote:
> So you cannot show existence of g?
>

See the proof of "Existence of Inverse Function Lemma" at the above link. It proves the existence of the inverse of a bijection.

[Mostowski Collapse]

It only proves:

"If a function is surjective and injection, then it has an inverse."
http://dcproof.com/ExistenceOfInverse.htm

The theorem that was demanded was:

"Every injective function f has an inverse function g on its range"

[Mostowski Collapse]

Can you prove your ExistenceOfInverse without
using bizzar sets, like this here:

Define: f' (as subset of yx)
17 Set'(f')
Split, 16
18 ALL(a):ALL(b):[(a,b) e f' <=> (a,b) e yx & f(b)=a]

Or do you admit that you were loosing your head,
accusing other people of bizzar ideas, whereas
your DC poop cannot do without just these bizzar

ideas. I guess everybody will agree that

ALL(a) : ALL(b) : [~(a, b) e yx => ~(a, b) e f']

Maybe you stole the proof somewhere? And you are
cheater? How can you not know that you yourself
use bizzar concepts in DC poop?

[Dan Christensen]

On Sunday, December 26, 2021 at 10:23:12 AM UTC-5, Dan Christensen wrote:
> On Sunday, December 26, 2021 at 5:04:17 AM UTC-5, Mostowski Collapse wrote:
> > So you cannot show existence of g?
> >
> See the proof of "Existence of Inverse Function Lemma" at the above link. It proves the existence of the inverse of a bijection.
> Dan
>

Just now completed a proof (138 lines) of the following:

ALL(x):ALL(y):ALL(f):[Set(x)
& Set(y)
& ALL(c):[c in x => f(c) in y]
& ALL(c):ALL(d):[c in x & d in x => [f(c)=f(d) => c=d]]

=> EXIST(rf):EXIST(g):[Set(rf) & ALL(b):[b in rf <=> b in y & EXIST(c):[c in x & f(c)=b]]
& ALL(a):[a in rf => g(a) in x]
& ALL(a):[a in rf => f(g(a))=a]]]

[Mostowski Collapse]

Why don't you show the proof?
Does it use bizzarro set theory?

[Mostowski Collapse]

It seems there are ways to do things without set theory,
You find here an example:

Example 2.2. The sentence
states that every function has a range and is provable in NK2.
https://www.dmg.tuwien.ac.at/hetzl/teaching/hol.20190125.pdf

But you need lambda expressions. Actually lambda
expressions about predications. The PM called these

fictitious object:
https://en.wikipedia.org/wiki/Principia_Mathematica#Introduction_to_the_notation_of_the_theory_of_classes_and_relations

So what is the DC Proof, plain set theory nevertheless?

[Mostowski Collapse]

Ha Ha, Dan-O-Matik is quite desperate, he now
denies valid proofs in mathematics:

Dan Christensen schrieb am Freitag, 31. Dezember 2021 um 16:41:43 UTC+1:
> Some universal function-like f just pops into existence,
a consequence of ANY theorem??? Maybe in philosophy class.
Not in math. Sorry. Likewise, (4) and (5).
https://groups.google.com/g/sci.math/c/Wr4n-Hb4b98/m/AjdUTKPeAgAJ

But there is nothing invalid in this proof:

1. f(a)=f(a) ⊢ f(a)=f(a) (ax)
2. f(a)=f(a) ⊢ ∃yf(a)=y (R∃)
3. ∀zz=z ⊢ ∃yf(a)=y (L∀)
4. ∀zz=z ⊢ ∀x∃yf(x)=y (R∀)
5. ⊢ ∀zz=z ⇒ ∀x∃yf(x)=y (R⇒)

http://www.xlog.ch/izytab/moblet/en/docs/18_live/40_bin2021/paste07/package.html

Used inference rules, besides reflexivity of equality:

https://en.wikipedia.org/wiki/Sequent_calculus#Inference_rules

[Mostowski Collapse]

Can this be proved in DC Proof?

ALL(x):[P(x)=>ALL(y):[P(y) & P(f(y))]]

Here is a LK calculus proof:

?- prove0((![X='x']:p(X) => ![Y='y']:(p(Y) & p(f(Y)))), 1).

1. p(a) ⊢ p(a) (ax)
2. ∀xp(x) ⊢ p(a) (L∀)
3. p(f(a)) ⊢ p(f(a)) (ax)
4. ∀xp(x) ⊢ p(f(a)) (L∀)
5. ∀xp(x) ⊢ p(a) ∧ p(f(a)) (R∧, 2)
6. ∀xp(x) ⊢ ∀y(p(y) ∧ p(f(y))) (R∀)
7. ⊢ ∀xp(x) ⇒ ∀y(p(y) ∧ p(f(y))) (R⇒)

http://www.xlog.ch/izytab/moblet/en/docs/18_live/40_bin2021/paste07/package.html

Adapted from an example for another proof method:
https://en.wikipedia.org/wiki/Method_of_analytic_tableaux#First-order_tableau_with_unification

[Mostowski Collapse]

Corr.: Typo

ALL(x):P(x)=>ALL(y):[P(y) & P(f(y))]

[Scot Dino]

Mostowski Collapse wrote:

> Can this be proved in DC Proof?
>
> ALL(x):[P(x)=>ALL(y):[P(y) & P(f(y))]]

APPOLLO 11 THE REAL MOON LANDING
https://seed125.bitchute.com/x4t2DPNQVTsV/S120GRdJlfKH.mp4

[Mostowski Collapse]

The next challenge is to become cyborgs:

"I Tried To Warn You" | Elon Musk's Last Warning (2022)
https://www.youtube.com/watch?v=AQNBO6WbQo8

Scot Dino schrieb:

[mitchr...@gmail.com]

Glorified calculators can't program the human brain.

[Mostowski Collapse]

These calculators are quite good in controlling
robots. I guess cars are all now manifactured

by robots. So where are the jobs in the future?

[Dan Christensen]

On Sunday, January 2, 2022 at 2:20:35 PM UTC-5, Mostowski Collapse wrote:

> > Can this be proved in DC Proof?
> >

> > > ALL(x):P(x)=>ALL(y):[P(y) & P(f(y))] (Corr.)
> >

Yes, but you have to make the domain of discourse explicit. In mathematics, it can't just be lurking unseen in the shadows.

Define: f on the domain of discourse u

1. ALL(a):[a in u => f(a) in u]
(Axiom)

Suppose...

2. ALL(a):[a in u => P(a)]
(Premise)

Suppose...

3. x in u
(Premise)

Apply initial premise

4. x in u => P(x)
(U Spec, 2)

5. P(x)
(Detach, 4, 3)

Apply definition of f

6. x in u => f(x) in u
(U Spec, 1)

7. f(x) in u
(Detach, 6, 3)

Apply initial premise

8. f(x) in u => P(f(x))
(U Spec, 2, 7)

9. P(f(x))
(Detach, 8, 7)

10. P(x) & P(f(x))
(Join, 5, 9)

As Required:

11. ALL(a):[a in u => P(a) & P(f(a))]
(Conclusion, 3)

As Required:

12. ALL(a):[a in u => P(a)] => ALL(a):[a in u => P(a) & P(f(a))]
(Conclusion, 2)

Dan

[Mostowski Collapse]

Dan-O-Matik fights the evils of darkness:

> In mathematics, it can't just be lurking unseen in the shadows.

Well you can replace P(a) by a in u, and you get:

ALL(a):[a in u => a in u] => ALL(a):[a in u => a in u & f(a) in u]

Since ALL(a):[a in u => a in u] is trivial you get:

ALL(a):[a in u => a in u & f(a) in u]

I thought in DC Puff f must always have a domain?

LMAO!

[Mostowski Collapse]

From ALL(a):[a in u => a in u & f(a) in u] it also follows:

ALL(a):[a in u => f(a) in u]

But what if you had elsewhere ALL(a):[a in dom => f(a) in cod],
even maybe with dom ⊆ u and cod ⊆ u, you will nevertheless
obtain the above? Which is nothing else than confirmation

that FOL function symbols are total, even when you have
the Dan-O-Matik ALL(a):[a in dom => f(a) in cod], which makes
Dan-O-Matik cite Terrence Tao, we nevertheless fall back

to essentially ∀x∃yf(x)=y. Isn't this amazing that dcproof4.exe
with its adhoc fixes, does not solve the problem of Burses Paradox,
it rather aggravates it, since it leads us digging deeper into

the evils of darkness, and when we lift the curtains of DC Puff
we find more totality of FOL function symbols lurking in the
shadows. To adapt a famous proverb by Dan-O-Matik:

"Deal with it, Dan-O-Matik. You are looking like a complete
idiot here in the same league as AP, JG and WM."

Disclaimer: Maybe there are more and more fixes of
dcproof4.exe, a whole tower of patch work, and the above
exact same derivation doesn't work.

But what do we know about an adhoc logic such as DC Puff
which even doesn't have a name and is nowhere found in
any mathematics textbooks?

[Dan Christensen]

> Dan-O-Matik fights the evils of darkness:

>

> I thought in DC Proof f must always have a domain?
>

Do pay attention, Jan Burse. Both the domain and codomain are u. See line 1.

Dan

[Mostowski Collapse]

Thats FOL semantics, yes.

[Mostowski Collapse]

My expectation was rather that:

ALL(x):P(x)=>ALL(y):[P(y) & P(f(y))]

Isn't provable in DC Proof.

[Mostowski Collapse]

That it wouldn't be provable can be simulated in FOL
by thinking in terms of (Function over U) and not in terms
of (Function from U to U). This would somehow cover

emptyness from free logic. Since a (Function over U)
might send an argument to outside of U. So we then
have, if we think only in terms of (Function over U):

Normally with FOL:
∀xPx → ∀x(Px ∧ Pf(x)) is valid.
https://www.umsu.de/trees/#~6xPx~5~6x%28Px~1Pf%28x%29%29

Abnormally with "free logic" touch:
∀x(Ux → Px) → ∀x(Ux → (Px ∧ Pf(x))) is invalid.
Countermodel:
Domain: { 0, 1 }
f: { (0,1), (1,0) }
U: { 0 }
P: { 0 }
https://www.umsu.de/trees/#~6x%28Ux~5Px%29~5~6x%28Ux~5%28Px~1Pf%28x%29%29%29

No need to postulate ∀x(Ux → Uf(x)).

[Dan Christensen]

On Monday, January 3, 2022 at 8:25:36 AM UTC-5, Mostowski Collapse wrote:
> My expectation was rather that:
> ALL(x):P(x)=>ALL(y):[P(y) & P(f(y))]
> Isn't provable in DC Proof.

You will have to settle for:

ALL(a):[a in u => P(a)] => ALL(a):[a in u => P(a) & P(f(a))]

Too much detail for you? Too bad.

[Mostowski Collapse]

You are changing topic, are you hunted by some shadows, Dr. Frankenstein?
This here shouldn't be provable as well in DC Proof, its already not provable in FOL:

∀x(Exu → Px) → ∀x(Exu → (Px ∧ Pf(x))) is invalid.
Countermodel:
Domain: { 0, 1 }
u: 0
f: { (0,1), (1,0) }
E: { (0,0) }
P: { 0 }
https://www.umsu.de/trees/#~6x%28Exu~5Px%29~5~6x%28Exu~5%28Px~1Pf%28x%29%29%29

[Dan Christensen]

On Monday, January 3, 2022 at 11:19:04 AM UTC-5, Mostowski Collapse wrote:
> You are changing topic, are you hunted by some shadows, Dr. Frankenstein?
> This here shouldn't be provable as well in DC Proof, its already not provable in FOL:
>
> ∀x(Exu → Px) → ∀x(Exu → (Px ∧ Pf(x))) is invalid.
> Countermodel:
> Domain: { 0, 1 }
> u: 0
> f: { (0,1), (1,0) }
> E: { (0,0) }
> P: { 0 }
> https://www.umsu.de/trees/#~6x%28Exu~5Px%29~5~6x%28Exu~5%28Px~1Pf%28x%29%29%29

You really MUST pay attention, Jan Burse! Once again, you forgot the initial axiom on line 1.

1. ALL(a):[a in u => f(a) in u]
(Axiom)

From TPG: "(∀x(Exu → Ef(x)u) ∧ ∀x(Exu → Px)) → ∀x(Exu → (Px ∧ Pf(x))) is valid."

https://www.umsu.de/trees/#~6x(Exu~5Ef(x)u)~1~6x(Exu~5Px)~5~6x(Exu~5(Px~1Pf(x)))

[Mostowski Collapse]

I do not assume something like this, when I ask whether:

ALL(x):P(x)=>ALL(y):[P(y) & P(f(y))]

is provable in DC Proof. I didn't say the above is a FOL formula.
If it were a FOL formula I wouldn't use the idiosyncratic ALL(x)
or EXIST(z). Most of the time when I post problems towards

DC Proof, I do not assume they are translated to FOL. I
am genuinely interested what is provable and what is not provable
in DC Proof. So this here is a DC Proof problem, there is no idea

that it means a FOL formula, it doesn't mean FOL, it means
what ever from day to day changing semantics of ALL(x)
is in DC Proof:

ALL(x):P(x)=>ALL(y):[P(y) & P(f(y))]

On the other hand this here is a FOL problem:

∀xPx → ∀x(Px ∧ Pf(x))

Got it? Or are you too stupid?

[Mostowski Collapse]

You can check yourself, I nowhere stated that the formula
must be interpreted by FOL. You violated Occams Razor
by adding stuff that was never asked:

Mostowski Collapse schrieb am Sonntag, 2. Januar 2022 um 20:18:53 UTC+1:

> Can this be proved in DC Proof?
>

> ALL(x):[P(x)=>ALL(y):[P(y) & P(f(y))]]
>
> Here is a LK calculus proof:
>
> ?- prove0((![X='x']:p(X) => ![Y='y']:(p(Y) & p(f(Y)))), 1).
>
> 1. p(a) ⊢ p(a) (ax)
> 2. ∀xp(x) ⊢ p(a) (L∀)
> 3. p(f(a)) ⊢ p(f(a)) (ax)
> 4. ∀xp(x) ⊢ p(f(a)) (L∀)
> 5. ∀xp(x) ⊢ p(a) ∧ p(f(a)) (R∧, 2)
> 6. ∀xp(x) ⊢ ∀y(p(y) ∧ p(f(y))) (R∀)
> 7. ⊢ ∀xp(x) ⇒ ∀y(p(y) ∧ p(f(y))) (R⇒)
>
> http://www.xlog.ch/izytab/moblet/en/docs/18_live/40_bin2021/paste07/package.html
>
> Adapted from an example for another proof method:
> https://en.wikipedia.org/wiki/Method_of_analytic_tableaux#First-order_tableau_with_unification

https://groups.google.com/g/sci.math/c/SOIECV0E-Dc/m/30qnX6aHAwAJ

Whats wrong with you?

[Dan Christensen]

On Monday, January 3, 2022 at 12:17:01 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Montag, 3. Januar 2022 um 18:07:19 UTC+1:
> > On Monday, January 3, 2022 at 11:19:04 AM UTC-5, Mostowski Collapse wrote:
> > > You are changing topic, are you hunted by some shadows, Dr. Frankenstein?
> > > This here shouldn't be provable as well in DC Proof, its already not provable in FOL:
> > >
> > > ∀x(Exu → Px) → ∀x(Exu → (Px ∧ Pf(x))) is invalid.
> > > Countermodel:
> > > Domain: { 0, 1 }
> > > u: 0
> > > f: { (0,1), (1,0) }
> > > E: { (0,0) }
> > > P: { 0 }
> > > https://www.umsu.de/trees/#~6x%28Exu~5Px%29~5~6x%28Exu~5%28Px~1Pf%28x%29%29%29
> > You really MUST pay attention, Jan Burse! Once again, you forgot the initial axiom on line 1.
> >
> > 1. ALL(a):[a in u => f(a) in u]
> > (Axiom)
> >
> > From TPG: "(∀x(Exu → Ef(x)u) ∧ ∀x(Exu → Px)) → ∀x(Exu → (Px ∧ Pf(x))) is valid."
> >
> > https://www.umsu.de/trees/#~6x(Exu~5Ef(x)u)~1~6x(Exu~5Px)~5~6x(Exu~5(Px~1Pf(x)))

> I do not assume something like this,

That is why you are getting an invalid result.

> when I ask whether:
> ALL(x):P(x)=>ALL(y):[P(y) & P(f(y))]
> is provable in DC Proof.

Too many missing details. No wonder it is not a required subject even in pure math programs (e.g. at MIT).

> I didn't say the above is a FOL formula.

Messy details, eh, Jan Burse? Unfortunately, that's what properly formal proofs are all about.

> If it were a FOL formula I wouldn't use the idiosyncratic ALL(x)
> or EXIST(z). Most of the time when I post problems towards
>
> DC Proof, I do not assume they are translated to FOL.

I'm afraid, it is your version of FOL that is widely viewed as idiosyncratic with its mysterious, unspecified, but non-empty domains of discourse, etc. It seems mathematicians had to develop their own system of logic, more like that used in DC Proof.

[Mostowski Collapse]

See other thread "Re: Minor updates to DC Proof 2.0".
The question is still unanswered, and your rant doesn't help either.

[Mostowski Collapse]

Also there is no such thing as my version of FOL.
See here, I am not the author of www.umsu.de/trees:

Normally with FOL:
∀xPx → ∀x(Px ∧ Pf(x)) is valid.
https://www.umsu.de/trees/#~6xPx~5~6x%28Px~1Pf%28x%29%29

Whats wrong with you?

[Mostowski Collapse]

Dan-O-Matik went full berserk:

> I'm afraid, it is your version of FOL

Also the prover on my website doesn't originate
from me. Its the prover written by Jens Otten,
which goes by then name leanseq_v5.pl:

Build Your Own First-Order Prover
Invited Tutorial at CADE in Natal/Brazil
(24 August 2019)
http://jens-otten.de/tutorial_cade19/

Whats wrong with you?