You are still subject to the same fallacy all over,
that Ben already pointed out. Which is getting
annoying slowly.

Mostowski Collapse schrieb am Montag, 30. Mai 2022 um 00:11:34 UTC+2:
> You cannot force logic to consider only one model,
> if your axioms don't says so. Otherwise if the only model where:
> Even={0, 2, 4, 6, ... }
> Odd={1, 3, 5, 7, ... }
> You could prove:
>
> ALL(a):[a e n => [Even(a) => Even(a+2)]]
> ALL(a):[a e n => [Odd(a) => Even(a+1)]]
>
> But you cannot prove it. Because your axioms
> have multiple models.
> Dan Christensen schrieb am Sonntag, 29. Mai 2022 um 21:31:56 UTC+2:
> > On Sunday, May 29, 2022 at 3:03:58 PM UTC-4, Mostowski Collapse wrote:
> > > Because your nonsense has among its models, in a model where a*b=0:
> > > Even={0}
> > > Odd={1,2,3,..}
> > >
> > Pay attention, Jan Burse! We have:
> >
> > Even={0, 2, 4, 6, ... }
> > Odd={1, 3, 5, 7, ... }
> > > You can only prove:
> > > ALL(a):[a e n => [Even(a) => Odd(a+1)]]
> > Also:
> > ALL(a):[a e n => Even(2*a) & Odd(2*a+1)]
> > > https://dcproof.com/EvenNextOdd.htm
> > >
> > > But you cannot prove, the above model is counter model:
> > Again, the only "model" we are concerned here is:
> >
> > n={0, 1, 2, 3, ...}
> > Even={0, 2, 4, 6, ... }
> > Odd={1, 3, 5, 7, ... }
> >
> > Deal with it, Jan Burse. Quit embarrassing yourself like this.
> > 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:

> You are still subject to the same fallacy all over,
> that Ben already pointed out. Which is getting annoying slowly.
>
> Mostowski Collapse schrieb am Montag, 30. Mai 2022 um 00:11:34 UTC+2:

Basically, using UNDEFINED to indicate an
undecided fact, we can only say that your model is:

Even(-1) = UNDEFINED
Even(0) = UNDEFINED
Even(1) = UNDEFINED
Even(2) = UNDEFINED
Etc..

because from your axioms, we even don't know
whether there is some b such that 2*b = 0 etc..

For example nothing indicates in your axioms of
EvenNextOdd that 2*0 = 0 would hold.

https://dcproof.com/EvenNextOdd.htm

LMAO!

Mostowski Collapse schrieb am Montag, 30. Mai 2022 um 00:16:23 UTC+2:
> You are still subject to the same fallacy all over,
> that Ben already pointed out. Which is getting
> annoying slowly.
> Mostowski Collapse schrieb am Montag, 30. Mai 2022 um 00:11:34 UTC+2:
> > You cannot force logic to consider only one model,
> > if your axioms don't says so. Otherwise if the only model where:
> > Even={0, 2, 4, 6, ... }
> > Odd={1, 3, 5, 7, ... }
> > You could prove:
> >
> > ALL(a):[a e n => [Even(a) => Even(a+2)]]
> > ALL(a):[a e n => [Odd(a) => Even(a+1)]]
> >
> > But you cannot prove it. Because your axioms
> > have multiple models.
> > Dan Christensen schrieb am Sonntag, 29. Mai 2022 um 21:31:56 UTC+2:
> > > On Sunday, May 29, 2022 at 3:03:58 PM UTC-4, Mostowski Collapse wrote:
> > > > Because your nonsense has among its models, in a model where a*b=0:
> > > > Even={0}
> > > > Odd={1,2,3,..}
> > > >
> > > Pay attention, Jan Burse! We have:
> > >
> > > Even={0, 2, 4, 6, ... }
> > > Odd={1, 3, 5, 7, ... }
> > > > You can only prove:
> > > > ALL(a):[a e n => [Even(a) => Odd(a+1)]]
> > > Also:
> > > ALL(a):[a e n => Even(2*a) & Odd(2*a+1)]
> > > > https://dcproof.com/EvenNextOdd.htm
> > > >
> > > > But you cannot prove, the above model is counter model:
> > > Again, the only "model" we are concerned here is:
> > >
> > > n={0, 1, 2, 3, ...}
> > > Even={0, 2, 4, 6, ... }
> > > Odd={1, 3, 5, 7, ... }
> > >
> > > Deal with it, Jan Burse. Quit embarrassing yourself like this.
> > > Dan
> > >
> > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > Visit my Math Blog at http://www.dcproof.wordpress.com

On Sunday, May 29, 2022 at 6:23:44 PM UTC-4, Mostowski Collapse wrote:
> Basically, using UNDEFINED to indicate an
> undecided fact, we can only say that your model is:
>
> Even(-1) = UNDEFINED
> Even(0) = UNDEFINED
> Even(1) = UNDEFINED
> Even(2) = UNDEFINED
> Etc..
>

Wrong again, Jan Burse.

Even(0) = True
Even (1) = False
Even(2) = True
Even(3) = False
etc.

Well you got one of them right. More than I expected.

> because from your axioms, we even don't know
> whether there is some b such that 2*b = 0 etc..
>

The theorems on lines 1-14 are all derivable from Peano's Axioms. They are just basic results from arithmetic on N. Deal with it, Jan Burse.

https://dcproof.com/EvenNextOdd.htm

Just admit you were wrong and quit embarrassing yourself like this, 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

Thats not what your axioms prove:

You cannot prove:

Even(0)
~Even(1)
Even(2)
~Even(3)
Etc..

From this nonsense here:
https://dcproof.com/EvenNextOdd.htm

Dan Christensen schrieb am Montag, 30. Mai 2022 um 00:41:37 UTC+2:
> On Sunday, May 29, 2022 at 6:23:44 PM UTC-4, Mostowski Collapse wrote:
> > Basically, using UNDEFINED to indicate an
> > undecided fact, we can only say that your model is:
> >
> > Even(-1) = UNDEFINED
> > Even(0) = UNDEFINED
> > Even(1) = UNDEFINED
> > Even(2) = UNDEFINED
> > Etc..
> >
> Wrong again, Jan Burse.
>
> Even(0) = True
> Even (1) = False
> Even(2) = True
> Even(3) = False
> etc.
>
> Well you got one of them right. More than I expected.
> > because from your axioms, we even don't know
> > whether there is some b such that 2*b = 0 etc..
> >
> The theorems on lines 1-14 are all derivable from Peano's Axioms. They are just basic results from arithmetic on N. Deal with it, Jan Burse.
>
> https://dcproof.com/EvenNextOdd.htm
>
> Just admit you were wrong and quit embarrassing yourself like this, 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 Sunday, May 29, 2022 at 6:46:24 PM UTC-4, Mostowski Collapse wrote:
> Thats not what your axioms prove:
>
> You cannot prove:
>
> Even(0)
> ~Even(1)
> Even(2)
> ~Even(3)
> Etc..
>

They may well require theorems in addition to those listed on lines 1-14, e.g. that 2*0=0. So what? Get a life, Jan Burse.

https://dcproof.com/EvenNextOdd.htm

Dan

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

Well it only shows the misery of your nonsense.
What do we know about your Even? Nothing.

The minimal requirement would be that for
a Peano numeral n* = s^n(0), we should be able

to have the following provable:

Even(n*) for n is even
~Even(n*) for n is not even

Dan Christensen schrieb am Montag, 30. Mai 2022 um 00:55:41 UTC+2:
> On Sunday, May 29, 2022 at 6:46:24 PM UTC-4, Mostowski Collapse wrote:
> > Thats not what your axioms prove:
> >
> > You cannot prove:
> >
> > Even(0)
> > ~Even(1)
> > Even(2)
> > ~Even(3)
> > Etc..
> >
> They may well require theorems in addition to those listed on lines 1-14, e.g. that 2*0=0. So what? Get a life, Jan Burse.
>
> https://dcproof.com/EvenNextOdd.htm
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

On Sunday, May 29, 2022 at 7:30:09 PM UTC-4, Mostowski Collapse wrote:
> Well it only shows the misery of your nonsense.
> What do we know about your Even?

See for yourself. From various theorems list there (lines 1-14), each being derivable from Peano's Axiom, and the definitions of Even and Odd (lines 15-16), we can obtain:

(1) ALL(a):[a e n => [Even(a) => Odd(a+1)]]

(2) ALL(a):[a e n => Even(2*a) & Odd(2*a+1)]

where n = the usual set of natural numbers.

https://dcproof.com/EvenNextOdd.htm

Did with it, Jan Burse. And do GET A LIFE!!!

Dan

On Sunday, May 29, 2022 at 7:30:09 PM UTC-4, Mostowski Collapse wrote:

> Dan Christensen schrieb am Montag, 30. Mai 2022 um 00:55:41 UTC+2:
> > On Sunday, May 29, 2022 at 6:46:24 PM UTC-4, Mostowski Collapse wrote:
> > > Thats not what your axioms prove:
> > >
> > > You cannot prove:
> > >
> > > Even(0)
> > > ~Even(1)
> > > Even(2)
> > > ~Even(3)
> > > Etc..
> > >
> > They may well require theorems in addition to those listed on lines 1-14, e.g. that 2*0=0. So what? Get a life, Jan Burse.
> >
> > https://dcproof.com/EvenNextOdd.htm

> What do we know about your Even?
>

See for yourself. From various theorems listed there (lines 1-14), each being derivable from Peano's Axiom, we can obtain:

(1) ALL(a):[a e n => [Even(a) => Odd(a+1)]]

(2) ALL(a):[a e n => Even(2*a) & Odd(2*a+1)]

where n = the usual set of natural numbers.

https://dcproof.com/EvenNextOdd.htm

Deal with it, Jan Burse. Do you really want to end up like JG, WM or AP?

Dan

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

Your theorms don't make any sense. Multiplication is undefined.

The minimal requirement would be that for
a Peano numeral n* = s^n(0), we should be able

to have the following provable:

Even(n*) for n is even
~Even(n*) for n is not even

Dan Christensen schrieb am Montag, 30. Mai 2022 um 05:57:49 UTC+2:
> On Sunday, May 29, 2022 at 7:30:09 PM UTC-4, Mostowski Collapse wrote:
>
> > Dan Christensen schrieb am Montag, 30. Mai 2022 um 00:55:41 UTC+2:
> > > On Sunday, May 29, 2022 at 6:46:24 PM UTC-4, Mostowski Collapse wrote:
> > > > Thats not what your axioms prove:
> > > >
> > > > You cannot prove:
> > > >
> > > > Even(0)
> > > > ~Even(1)
> > > > Even(2)
> > > > ~Even(3)
> > > > Etc..
> > > >
> > > They may well require theorems in addition to those listed on lines 1-14, e.g. that 2*0=0. So what? Get a life, Jan Burse.
> > >
> > > https://dcproof.com/EvenNextOdd.htm
> > What do we know about your Even?
> >
> See for yourself. From various theorems listed there (lines 1-14), each being derivable from Peano's Axiom, we can obtain:
>
> (1) ALL(a):[a e n => [Even(a) => Odd(a+1)]]
>
> (2) ALL(a):[a e n => Even(2*a) & Odd(2*a+1)]
>
> where n = the usual set of natural numbers.
>
> https://dcproof.com/EvenNextOdd.htm
>
> Deal with it, Jan Burse. Do you really want to end up like JG, WM or AP?
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Your proof would be legit, if you had:

1) A proof of:
7 ALL(a):ALL(b):[a e n & b e n => a+b=b+a]
Axiom

2) A definition of multiplication

3) A proof of:
13 ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n & ~a=0 => [a*b<a*c => b<c]]
Axiom

4) A proof of:
14 ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n => [c<b => a*(b-c)=a*b-a*c]]
Axiom

5) A proof of:
9 ~EXIST(a):[a e n & 2*a=1]
Axiom

LMAO!

Mostowski Collapse schrieb am Montag, 30. Mai 2022 um 10:02:17 UTC+2:
> Your theorms don't make any sense. Multiplication is undefined.
> The minimal requirement would be that for
> a Peano numeral n* = s^n(0), we should be able
>
> to have the following provable:
>
> Even(n*) for n is even
> ~Even(n*) for n is not even
> Dan Christensen schrieb am Montag, 30. Mai 2022 um 05:57:49 UTC+2:
> > On Sunday, May 29, 2022 at 7:30:09 PM UTC-4, Mostowski Collapse wrote:
> >
> > > Dan Christensen schrieb am Montag, 30. Mai 2022 um 00:55:41 UTC+2:
> > > > On Sunday, May 29, 2022 at 6:46:24 PM UTC-4, Mostowski Collapse wrote:
> > > > > Thats not what your axioms prove:
> > > > >
> > > > > You cannot prove:
> > > > >
> > > > > Even(0)
> > > > > ~Even(1)
> > > > > Even(2)
> > > > > ~Even(3)
> > > > > Etc..
> > > > >
> > > > They may well require theorems in addition to those listed on lines 1-14, e.g. that 2*0=0. So what? Get a life, Jan Burse.
> > > >
> > > > https://dcproof.com/EvenNextOdd.htm
> > > What do we know about your Even?
> > >
> > See for yourself. From various theorems listed there (lines 1-14), each being derivable from Peano's Axiom, we can obtain:
> >
> > (1) ALL(a):[a e n => [Even(a) => Odd(a+1)]]
> >
> > (2) ALL(a):[a e n => Even(2*a) & Odd(2*a+1)]
> >
> > where n = the usual set of natural numbers.
> >
> > https://dcproof.com/EvenNextOdd.htm
> >
> > Deal with it, Jan Burse. Do you really want to end up like JG, WM or AP?
> > 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, May 30, 2022 at 5:31:20 AM UTC-4, Mostowski Collapse wrote:
> Your proof would be legit, if you had:
>
> 1) A proof of:
> 7 ALL(a):ALL(b):[a e n & b e n => a+b=b+a]
> Axiom
>
> 2) A definition of multiplication
>
> 3) A proof of:
> 13 ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n & ~a=0 => [a*b<a*c => b<c]]
> Axiom
>
> 4) A proof of:
> 14 ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n => [c<b => a*(b-c)=a*b-a*c]]
> Axiom
>
> 5) A proof of:
> 9 ~EXIST(a):[a e n & 2*a=1]
> Axiom
>

For readability, I usually do not list axioms/definitions/theorems that are not cited in a given proof.

Again, the purpose of the proof in question was simply to demonstrate how to use definitions of predicates: Even and and Odd being the simplest, least controversial examples. https://dcproof.com/EvenNextOdd.htm

Dan

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

I think its quite controversial having Even with:

Even(-1) = UNDEFINED
Even(0) = UNDEFINED
Even(1) = UNDEFINED
Even(2) = UNDEFINED
Etc..

Thats not a correct definition of Even.

Dan Christensen schrieb am Montag, 30. Mai 2022 um 16:00:01 UTC+2:
> On Monday, May 30, 2022 at 5:31:20 AM UTC-4, Mostowski Collapse wrote:
> > Your proof would be legit, if you had:
> >
> > 1) A proof of:
> > 7 ALL(a):ALL(b):[a e n & b e n => a+b=b+a]
> > Axiom
> >
> > 2) A definition of multiplication
> >
> > 3) A proof of:
> > 13 ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n & ~a=0 => [a*b<a*c => b<c]]
> > Axiom
> >
> > 4) A proof of:
> > 14 ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n => [c<b => a*(b-c)=a*b-a*c]]
> > Axiom
> >
> > 5) A proof of:
> > 9 ~EXIST(a):[a e n & 2*a=1]
> > Axiom
> >
> For readability, I usually do not list axioms/definitions/theorems that are not cited in a given proof.
>
> Again, the purpose of the proof in question was simply to demonstrate how to use definitions of predicates: Even and and Odd being the simplest, least controversial examples. https://dcproof.com/EvenNextOdd.htm
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Anyway I guess Dan Christensen has perfected the art of
dark elements. Whereby one would usually expect the following
for even natural numbers, including zero:

Even(-2) = FALSE
Even(-1) = FALSE
Even(0) = TRUE
Even(1) = FALSE
Even(2) = TRUE
Even(3) = FALSE
etc...

We seen in Dan Christensens EvenNextOdd an abundance of
dark elements. He is a real master of dark elements:

Even(-2) = UNDEFINED
Even(-1) = UNDEFINED
Even(0) = UNDEFINED
Even(1) = UNDEFINED
Even(2) = UNDEFINED
Even(3) = UNDEFINED
etc...

Mostowski Collapse schrieb am Montag, 30. Mai 2022 um 16:09:42 UTC+2:
> I think its quite controversial having Even with:
> Even(-1) = UNDEFINED
> Even(0) = UNDEFINED
> Even(1) = UNDEFINED
> Even(2) = UNDEFINED
> Etc..
> Thats not a correct definition of Even.
> Dan Christensen schrieb am Montag, 30. Mai 2022 um 16:00:01 UTC+2:
> > On Monday, May 30, 2022 at 5:31:20 AM UTC-4, Mostowski Collapse wrote:
> > > Your proof would be legit, if you had:
> > >
> > > 1) A proof of:
> > > 7 ALL(a):ALL(b):[a e n & b e n => a+b=b+a]
> > > Axiom
> > >
> > > 2) A definition of multiplication
> > >
> > > 3) A proof of:
> > > 13 ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n & ~a=0 => [a*b<a*c => b<c]]
> > > Axiom
> > >
> > > 4) A proof of:
> > > 14 ALL(a):ALL(b):ALL(c):[a e n & b e n & c e n => [c<b => a*(b-c)=a*b-a*c]]
> > > Axiom
> > >
> > > 5) A proof of:
> > > 9 ~EXIST(a):[a e n & 2*a=1]
> > > Axiom
> > >
> > For readability, I usually do not list axioms/definitions/theorems that are not cited in a given proof.
> >
> > Again, the purpose of the proof in question was simply to demonstrate how to use definitions of predicates: Even and and Odd being the simplest, least controversial examples. https://dcproof.com/EvenNextOdd.htm
> > 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, May 30, 2022 at 10:09:42 AM UTC-4, Mostowski Collapse wrote:
> I think its quite controversial having Even with:
> Even(-1) = UNDEFINED
> Even(0) = UNDEFINED
> Even(1) = UNDEFINED
> Even(2) = UNDEFINED
> Etc..

Liar. Again, using nothing more than basic arithmetic (e.g . 2*0=0 and 2*1=2) we can obtain from my proof:

Even(0) = True
Even(1) = False
Even(2) = True
etc.

Aren't you getting tired of looking like an complete idiot, Jan Burse? I am getting bored with your silly games.

Dan