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!

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
>
....

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...

Dan Christensen schrieb am Montag, 22. November 2021 um 14:20:34 UTC+1:
> 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
> >
> ...

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.

Mostowski Collapse schrieb am Montag, 22. November 2021 um 17:31:14 UTC+1:
> 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...
> Dan Christensen schrieb am Montag, 22. November 2021 um 14:20:34 UTC+1:
> > 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 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/

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

Mostowski Collapse wrote:

> Hey Luigi, how is mario cart doing?

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

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/

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

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.

-

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%.

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:
> 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%.

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.

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.

Michael Moroney wrote:

> 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.

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.

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:
> 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.

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

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

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/

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 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)..
>
> 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 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
..
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

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.

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 schrieb am Dienstag, 23. November 2021 um 08:36:12 UTC+1:
> 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.

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 schrieb am Dienstag, 23. November 2021 um 08:45:39 UTC+1:
> 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 schrieb am Dienstag, 23. November 2021 um 08:36:12 UTC+1:
> > 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.

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 schrieb am Dienstag, 23. November 2021 um 08:50:56 UTC+1:
> 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 schrieb am Dienstag, 23. November 2021 um 08:45:39 UTC+1:
> > 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 schrieb am Dienstag, 23. November 2021 um 08:36:12 UTC+1:
> > > 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.

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.

Mostowski Collapse schrieb am Dienstag, 23. November 2021 um 09:00:18 UTC+1:
> 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 schrieb am Dienstag, 23. November 2021 um 08:50:56 UTC+1:
> > 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 schrieb am Dienstag, 23. November 2021 um 08:45:39 UTC+1:
> > > 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 schrieb am Dienstag, 23. November 2021 um 08:36:12 UTC+1:
> > > > 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.

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

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