On Wednesday, July 20, 2022 at 4:21:07 PM UTC-4, Mostowski Collapse wrote:
> The inconsistency was derived from the function equality axiom,

Wrong again, Jan Burse.

[snip]

> 1 ALL(dom):ALL(cod):ALL(f1):ALL(f2):[Set(dom) & Set(cod)
> & ALL(a):[a ε dom => f1(a) ε cod]
> & ALL(a):[a ε dom => f2(a) ε cod]
> => [f1=f2 <=> ALL(a):[a ε dom => f1(a)=f2(a)]]]
> Fn Equality

> 2 ~0=1
> Axiom
>

Following is where your problem started. If, instead of simply defining these functions willy-nilly as here, you had used the Function Axiom to prove their existence, you would not be able to arbitrarily change your domains as you do later in your "proof." If you want to change the domain, codomain or graph set, you would have assign another name to the required function, as you would expect. It would then, of course, be a DIFFERENT FUNCTION.

> 3 f1(0)=0
> Axiom
>
> 4 f2(0)=1
> Axiom
>
[snip]

Dan

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

It was here:

Mostowski Collapse schrieb am Sonntag, 17. Juli 2022 um 08:13:23 UTC+2:
> Here is a proof of f1(0)=0. I guess you can also prove f2(0)=1.
> Its extremly extremly long, because DC Proof is sooooooooo stupid:
>
> 158 f1(0)=0
> Detach, 150, 157
https://groups.google.com/g/sci.logic/c/iRJvlsKToKc/m/xEjrYVzdAAAJ

So to summarize we have proofs of:
a) Existence of f1 and f2
b) f1=f2 & ~(f2=f2)

Whats your point? It doesn't make your Function Equality axiom
less inconsistent. Your conclusion is still that your Function
Equality axiom leads to an inconsistent theory.

And its not an inconsistency by some handwaving when
WM claims that set theory is inconsistent, its an inconsistency
which is not only a conjecture but which is demonstrable,

since it is demonatrable for some A, that A & ~A is
provable. The A here is f1=f2. So your Function Equality
axiom leads to an inconsistent system.

Also the presensence or absence of existence proofs
is no guarantee for consistency. Especially inconsistent
systems can prove anything, through ex falso quodlibet,

and therefore you can prove all kind of existence in
inconsistent systems. There is no such method as show
existence of an object, and then the system is not inconsistent,

thats just utter nonsense as well.

Dan Christensen schrieb am Donnerstag, 21. Juli 2022 um 01:58:12 UTC+2:
> On Wednesday, July 20, 2022 at 4:21:07 PM UTC-4, Mostowski Collapse wrote:
> > The inconsistency was derived from the function equality axiom,
> Wrong again, Jan Burse.
>
> [snip]
> > 1 ALL(dom):ALL(cod):ALL(f1):ALL(f2):[Set(dom) & Set(cod)
> > & ALL(a):[a ε dom => f1(a) ε cod]
> > & ALL(a):[a ε dom => f2(a) ε cod]
> > => [f1=f2 <=> ALL(a):[a ε dom => f1(a)=f2(a)]]]
> > Fn Equality
>
> > 2 ~0=1
> > Axiom
> >
> Following is where your problem started. If, instead of simply defining these functions willy-nilly as here, you had used the Function Axiom to prove their existence, you would not be able to arbitrarily change your domains as you do later in your "proof." If you want to change the domain, codomain or graph set, you would have assign another name to the required function, as you would expect. It would then, of course, be a DIFFERENT FUNCTION.
> > 3 f1(0)=0
> > Axiom
> >
> > 4 f2(0)=1
> > Axiom
> >
> [snip]
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

You are like the guy, who didn't clean his room, and
then mother steps in and asks why you didn't clean
the room, and you come up with more and more excuses.

Mostowski Collapse schrieb am Donnerstag, 21. Juli 2022 um 16:43:08 UTC+2:
> It was here:
> Mostowski Collapse schrieb am Sonntag, 17. Juli 2022 um 08:13:23 UTC+2:
> > Here is a proof of f1(0)=0. I guess you can also prove f2(0)=1.
> > Its extremly extremly long, because DC Proof is sooooooooo stupid:
> >
> > 158 f1(0)=0
> > Detach, 150, 157
> https://groups.google.com/g/sci.logic/c/iRJvlsKToKc/m/xEjrYVzdAAAJ
>
> So to summarize we have proofs of:
> a) Existence of f1 and f2
> b) f1=f2 & ~(f2=f2)
>
> Whats your point? It doesn't make your Function Equality axiom
> less inconsistent. Your conclusion is still that your Function
> Equality axiom leads to an inconsistent theory.
>
> And its not an inconsistency by some handwaving when
> WM claims that set theory is inconsistent, its an inconsistency
> which is not only a conjecture but which is demonstrable,
>
> since it is demonatrable for some A, that A & ~A is
> provable. The A here is f1=f2. So your Function Equality
> axiom leads to an inconsistent system.
>
> Also the presensence or absence of existence proofs
> is no guarantee for consistency. Especially inconsistent
> systems can prove anything, through ex falso quodlibet,
>
> and therefore you can prove all kind of existence in
> inconsistent systems. There is no such method as show
> existence of an object, and then the system is not inconsistent,
>
> thats just utter nonsense as well.
> Dan Christensen schrieb am Donnerstag, 21. Juli 2022 um 01:58:12 UTC+2:
> > On Wednesday, July 20, 2022 at 4:21:07 PM UTC-4, Mostowski Collapse wrote:
> > > The inconsistency was derived from the function equality axiom,
> > Wrong again, Jan Burse.
> >
> > [snip]
> > > 1 ALL(dom):ALL(cod):ALL(f1):ALL(f2):[Set(dom) & Set(cod)
> > > & ALL(a):[a ε dom => f1(a) ε cod]
> > > & ALL(a):[a ε dom => f2(a) ε cod]
> > > => [f1=f2 <=> ALL(a):[a ε dom => f1(a)=f2(a)]]]
> > > Fn Equality
> >
> > > 2 ~0=1
> > > Axiom
> > >
> > Following is where your problem started. If, instead of simply defining these functions willy-nilly as here, you had used the Function Axiom to prove their existence, you would not be able to arbitrarily change your domains as you do later in your "proof." If you want to change the domain, codomain or graph set, you would have assign another name to the required function, as you would expect. It would then, of course, be a DIFFERENT FUNCTION.
> > > 3 f1(0)=0
> > > Axiom
> > >
> > > 4 f2(0)=1
> > > Axiom
> > >
> > [snip]
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

On Thursday, July 21, 2022 at 10:43:08 AM UTC-4, Mostowski Collapse wrote:
> It was here:
> Mostowski Collapse schrieb am Sonntag, 17. Juli 2022 um 08:13:23 UTC+2:
> > Here is a proof of f1(0)=0. I guess you can also prove f2(0)=1.
> > Its extremly extremly long, because DC Proof is sooooooooo stupid:
> >
> > 158 f1(0)=0
> > Detach, 150, 157
> https://groups.google.com/g/sci.logic/c/iRJvlsKToKc/m/xEjrYVzdAAAJ
>
> So to summarize we have proofs of:
> a) Existence of f1 and f2
> b) f1=f2 & ~(f2=f2)
>

Should be: f1=f2 & ~(f1=f2)?

Where you derived that nonsense, you failed actually prove the existence of functions f1 and f2 using the Function Axiom. It is a cautionary tale for students: If you are are simply go to define functions without proving their existence, you must be VERY careful. As I recall, you quite carelessly mixed up the different domain sets and made a complete mess of things. Did your little buddy and fellow troll JG here "help" you with that proof? (HA, HA, HA!!!)

Dan

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

Please tell this your tool DC Proof, it shows me:

130 EXIST(fun):ALL(a1):ALL(b):[a1 ε dom1 & b ε cod1
=> [fun(a1)=b <=> (a1,b) ε gra1]]
Detach, 15, 129
https://groups.google.com/g/sci.logic/c/iRJvlsKToKc/m/xEjrYVzdAAAJ

Do you say your tool has an error?

Dan Christensen schrieb am Donnerstag, 21. Juli 2022 um 19:44:04 UTC+2:
> Should be: f1=f2 & ~(f1=f2)?
> Where you derived that nonsense, you failed actually prove
the existence of functions f1 and f2 using the Function Axiom.

Well if you disable function equality, then
indirectly you also disable function spaces.

If I open dcproof16.exe I see this nonsense:
- Function Axiom
- Function Equality
- Function Space

Whats a function space S good for, if you
have no definition for f = g, for some f, g e S?

Not much use...

Mostowski Collapse schrieb am Donnerstag, 21. Juli 2022 um 21:45:46 UTC+2:
> Please tell this your tool DC Proof, it shows me:
>
> 130 EXIST(fun):ALL(a1):ALL(b):[a1 ε dom1 & b ε cod1
> => [fun(a1)=b <=> (a1,b) ε gra1]]
> Detach, 15, 129
> https://groups.google.com/g/sci.logic/c/iRJvlsKToKc/m/xEjrYVzdAAAJ
>
> Do you say your tool has an error?
> Dan Christensen schrieb am Donnerstag, 21. Juli 2022 um 19:44:04 UTC+2:
> > Should be: f1=f2 & ~(f1=f2)?
> > Where you derived that nonsense, you failed actually prove
> the existence of functions f1 and f2 using the Function Axiom.

Also we can play the same f1=f2 & ~(f1=f2) game
most likely with your function space axiom.
Its of course with high probability also botched.

We should give it a try whether we can prove.
Who is up to this challenge, who can find such
a little proof quickly with DC Poop?

f e S & ~(f e S)

For some function space. Not sure, maybe.
Didn't we do this already here in sci.logic
or sci.math? Dunno, I do not track all the nonsense

of DC Proof. If you would make use of GitHub,
we could open issues for all of your blunders. And we
would now have an overview of all the broken axioms.

P.S.: What I remember your function spaces
could not yet prove, at least on your side such
a proof was not yet presented:

|2^1| =\= |2^0|

This has a certain smell that this was not yet
readily proved. But the idea to hunt for f e S &
~(f e S), I got this idea only right now. Thx.

Mostowski Collapse schrieb am Samstag, 23. Juli 2022 um 01:07:37 UTC+2:
> Well if you disable function equality, then
> indirectly you also disable function spaces.
>
> If I open dcproof16.exe I see this nonsense:
> - Function Axiom
> - Function Equality
> - Function Space
>
> Whats a function space S good for, if you
> have no definition for f = g, for some f, g e S?
>
> Not much use...
> Mostowski Collapse schrieb am Donnerstag, 21. Juli 2022 um 21:45:46 UTC+2:
> > Please tell this your tool DC Proof, it shows me:
> >
> > 130 EXIST(fun):ALL(a1):ALL(b):[a1 ε dom1 & b ε cod1
> > => [fun(a1)=b <=> (a1,b) ε gra1]]
> > Detach, 15, 129
> > https://groups.google.com/g/sci.logic/c/iRJvlsKToKc/m/xEjrYVzdAAAJ
> >
> > Do you say your tool has an error?
> > Dan Christensen schrieb am Donnerstag, 21. Juli 2022 um 19:44:04 UTC+2:
> > > Should be: f1=f2 & ~(f1=f2)?
> > > Where you derived that nonsense, you failed actually prove
> > the existence of functions f1 and f2 using the Function Axiom.

Using GitHub would give you a kind of Co-Pilot, since
it seems you can even not Pilot your own DC Proof
tool. Presenting and withdrawing axioms ad infinitum,

well, this looks rather like the real developer of this
tool, which is probably not Dan Christensen, has long
ago pushed the button of his ejection seat.

Mostowski Collapse schrieb:
> Also we can play the same f1=f2 & ~(f1=f2) game
> most likely with your function space axiom.
> Its of course with high probability also botched.
>
> We should give it a try whether we can prove.
> Who is up to this challenge, who can find such
> a little proof quickly with DC Poop?
>
> f e S & ~(f e S)
>
> For some function space. Not sure, maybe.
> Didn't we do this already here in sci.logic
> or sci.math? Dunno, I do not track all the nonsense
>
> of DC Proof. If you would make use of GitHub,
> we could open issues for all of your blunders. And we
> would now have an overview of all the broken axioms.
>
> P.S.: What I remember your function spaces
> could not yet prove, at least on your side such
> a proof was not yet presented:
>
> |2^1| =\= |2^0|
>
> This has a certain smell that this was not yet
> readily proved. But the idea to hunt for f e S &
> ~(f e S), I got this idea only right now. Thx.
>
> Mostowski Collapse schrieb am Samstag, 23. Juli 2022 um 01:07:37 UTC+2:
>> Well if you disable function equality, then
>> indirectly you also disable function spaces.
>>
>> If I open dcproof16.exe I see this nonsense:
>> - Function Axiom
>> - Function Equality
>> - Function Space
>>
>> Whats a function space S good for, if you
>> have no definition for f = g, for some f, g e S?
>>
>> Not much use...
>> Mostowski Collapse schrieb am Donnerstag, 21. Juli 2022 um 21:45:46 UTC+2:
>>> Please tell this your tool DC Proof, it shows me:
>>>
>>> 130 EXIST(fun):ALL(a1):ALL(b):[a1 ε dom1 & b ε cod1
>>> => [fun(a1)=b <=> (a1,b) ε gra1]]
>>> Detach, 15, 129
>>> https://groups.google.com/g/sci.logic/c/iRJvlsKToKc/m/xEjrYVzdAAAJ
>>>
>>> Do you say your tool has an error?
>>> Dan Christensen schrieb am Donnerstag, 21. Juli 2022 um 19:44:04 UTC+2:
>>>> Should be: f1=f2 & ~(f1=f2)?
>>>> Where you derived that nonsense, you failed actually prove
>>> the existence of functions f1 and f2 using the Function Axiom.

Mostowski Collapse wrote:

> Using GitHub would give you a kind of Co-Pilot, since it seems you can
> even not Pilot your own DC Proof tool. Presenting and withdrawing axioms
> ad infinitum,

this good american is uKraine residential, and apparently the cocaine
zelenske is not there for you anymore. But the russian wants him alive, to
collect the weapons from the nazi west. This is complicated.

2022 07 23 Zelensky Is About To Be Assassinated By The Americans
https://www.youtube.com/watch?v=mactoSForFY