Here is an updated version of my proof constructing the natural numbers.
..
THEOREM
..
From any Dedekind infinite set x can be selected a subset n on which the Peano Axioms hold. I let f be a function on set x that is injective and not surjective, i.e. x is Dedekind infinite. I let x0 be an element of x having no pre-image under f. I prove:
..
ALL(x):ALL(f):[Set(x)
..
& ALL(a):[a in x => f(a) in x]
..
& ALL(a):ALL(b):[a in x & b in x => [f(a)=f(b) => a=b]] (f is injective)
..
& EXIST(a):[a in x & ALL(b):[b in x => ~f(b)=a]] (f is not surjective)
..
=> EXIST(n):EXIST(x0):[Set(n) & ALL(a):[a in n => a in x]
..
& x0 in n (PA1)
..
& ALL(a):[a in n => f(a) in n] (PA2)
..
& ALL(a):ALL(b):[a in n & b in n => [f(a)=f(b) => a=b]] (PA3)
..
& ALL(a):[a in n => ~f(a)=x0] (PA4)
..
& ALL(a):[Set(a) & ALL(b):[b in a => b in n] (PA5)
=> [x0 in a & ALL(b):[b in a => f(b) in a]
=> ALL(b):[b in n => b in a]]]]]
..
..
FORMAL PROOF (only 104 lines)
..
https://dcproof.com/ConstructN.htm
..
..
Dan
..
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

Whats the update, what was the revision?

Dan Christensen schrieb:
> Here is an updated version of my proof constructing the natural numbers.
> .
> THEOREM
> .
> From any Dedekind infinite set x can be selected a subset n on which the Peano Axioms hold. I let f be a function on set x that is injective and not surjective, i.e. x is Dedekind infinite. I let x0 be an element of x having no pre-image under f. I prove:
> .
> ALL(x):ALL(f):[Set(x)
> .
> & ALL(a):[a in x => f(a) in x]
> .
> & ALL(a):ALL(b):[a in x & b in x => [f(a)=f(b) => a=b]] (f is injective)
> .
> & EXIST(a):[a in x & ALL(b):[b in x => ~f(b)=a]] (f is not surjective)
> .
> => EXIST(n):EXIST(x0):[Set(n) & ALL(a):[a in n => a in x]
> .
> & x0 in n (PA1)
> .
> & ALL(a):[a in n => f(a) in n] (PA2)
> .
> & ALL(a):ALL(b):[a in n & b in n => [f(a)=f(b) => a=b]] (PA3)
> .
> & ALL(a):[a in n => ~f(a)=x0] (PA4)
> .
> & ALL(a):[Set(a) & ALL(b):[b in a => b in n] (PA5)
> => [x0 in a & ALL(b):[b in a => f(b) in a]
> => ALL(b):[b in n => b in a]]]]]
> .
> .
> FORMAL PROOF (only 104 lines)
> .
> https://dcproof.com/ConstructN.htm
> .
> .
> Dan
> .
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>

BTW: Its still not a construction. You only proved:

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

But can you also prove in your universe of discourse that:

EXIST(N):Peano(N) ?

Or do you claim this fallacy:

/* Dan-O-Matiks aka Slow-O-Matiks Fallacy */
ALL(f):[Dedekind(f) => EXIST(N):Peano(N)] => EXIST(N):Peano(N) ?

You can check yourself that it would be a fallacy:

∀f(Df → ∃nPn) → ∃nPn is invalid.
Countermodel:
Domain: { 0 }
D: { }
P: { }
https://www.umsu.de/trees

Mostowski Collapse schrieb am Mittwoch, 17. November 2021 um 21:40:02 UTC+1:
> Whats the update, what was the revision?
>
> Dan Christensen schrieb:
> > Here is an updated version of my proof constructing the natural numbers..
> > .
> > THEOREM
> > .
> > From any Dedekind infinite set x can be selected a subset n on which the Peano Axioms hold. I let f be a function on set x that is injective and not surjective, i.e. x is Dedekind infinite. I let x0 be an element of x having no pre-image under f. I prove:
> > .
> > ALL(x):ALL(f):[Set(x)
> > .
> > & ALL(a):[a in x => f(a) in x]
> > .
> > & ALL(a):ALL(b):[a in x & b in x => [f(a)=f(b) => a=b]] (f is injective)
> > .
> > & EXIST(a):[a in x & ALL(b):[b in x => ~f(b)=a]] (f is not surjective)
> > .
> > => EXIST(n):EXIST(x0):[Set(n) & ALL(a):[a in n => a in x]
> > .
> > & x0 in n (PA1)
> > .
> > & ALL(a):[a in n => f(a) in n] (PA2)
> > .
> > & ALL(a):ALL(b):[a in n & b in n => [f(a)=f(b) => a=b]] (PA3)
> > .
> > & ALL(a):[a in n => ~f(a)=x0] (PA4)
> > .
> > & ALL(a):[Set(a) & ALL(b):[b in a => b in n] (PA5)
> > => [x0 in a & ALL(b):[b in a => f(b) in a]
> > => ALL(b):[b in n => b in a]]]]]
> > .
> > .
> > FORMAL PROOF (only 104 lines)
> > .
> > https://dcproof.com/ConstructN.htm
> > .
> > .
> > Dan
> > .
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com
> >

See my reply to your identical posting at sci.logic just now.

Dan

On Wednesday, November 17, 2021 at 4:40:41 PM UTC-5, Mostowski Collapse wrote:
> BTW: Its still not a construction. You only proved:
>
> ALL(f):[Dedekind(f) => EXIST(N):Peano(N)]
>
> But can you also prove in your universe of discourse that:
>

[snip]

But in ZFC this existence is cast into axioms.
What would be your axioms?
What is your foundation?

Dan Christensen schrieb am Donnerstag, 18. November 2021 um 02:51:29 UTC+1:
> See my reply to your identical posting at sci.logic just now.
>
> Dan
> On Wednesday, November 17, 2021 at 4:40:41 PM UTC-5, Mostowski Collapse wrote:
> > BTW: Its still not a construction. You only proved:
> >
> > ALL(f):[Dedekind(f) => EXIST(N):Peano(N)]
> >
> > But can you also prove in your universe of discourse that:
> >
> [snip]

This here can be also used for ZFC set theory:
https://www.umsu.de/trees

ZFC set theory is just some FOL axioms.

But when using a binary relation such as ∈ ,
it might not find counter models. Monadic logic
does have a complete model finder, but otherwise

model finding wouldn't necessarely terminate, and
it would also not recursively enumerate the axiom
schemas for you. So because the ZFC FOL axioms

are infinitely many and because a binary relation is
in use, the application of the tool would be limited.
Not sure whether these two limitations can be

removed. This paper shows a simplification ZFC0
which uses only a simple comprehension axiom, but
I guess it cannot lift the limitation:

Automated ZFC Theorem Proving with E
John Hester
https://arxiv.org/abs/1902.00818

Mostowski Collapse schrieb am Donnerstag, 18. November 2021 um 11:09:45 UTC+1:
> But in ZFC this existence is cast into axioms.
> What would be your axioms?
> What is your foundation?
> Dan Christensen schrieb am Donnerstag, 18. November 2021 um 02:51:29 UTC+1:
> > See my reply to your identical posting at sci.logic just now.
> >
> > Dan
> > On Wednesday, November 17, 2021 at 4:40:41 PM UTC-5, Mostowski Collapse wrote:
> > > BTW: Its still not a construction. You only proved:
> > >
> > > ALL(f):[Dedekind(f) => EXIST(N):Peano(N)]
> > >
> > > But can you also prove in your universe of discourse that:
> > >
> > [snip]

On Thursday, November 18, 2021 at 5:09:45 AM UTC-5, Mostowski Collapse wrote:
> But in ZFC this existence is cast into axioms.
> What would be your axioms?
> What is your foundation?
..
See the Logic and Sets menu on the DC Proof main screen. Each option is documented in the user manual.
..
Dan
..
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

Dan Christensen wrote:

> having no pre-image under f. I prove:
> . ALL(x):ALL(f):[Set(x) . & ALL(a):[a in x => f(a) in x]

How is it that more than 190 governments from all over the world ended up
dealing with the COVID-19 pandemic in almost exactly the same manner,
with lockdowns, mask mandates, and vaccination cards now being commonplace
everywhere? The answer may lie in the Young Global Leaders school, which
was established and managed by Klaus Schwab of the World Economic Forum,
and that many of today’s prominent political and business leaders passed
through on their way to the top.

The German economist, journalist, and author Ernst Wolff has revealed some
facts about Schwab’s “Young Global Leaders” school that are relevant for
understanding world events during the pandemic in a video from the German
Corona Committee podcast. While Wolff is mainly known as a critic of the
globalist financial system, recently he has focused on bringing to light
what he sees as the hidden agenda behind the anti-Covid measures being
enacted around the world.

KLAUS SCHWAB'S SCHOOL FOR COVID DICTATORS TRAINED TO SERVE CABAL'S ORDERS
(Pt 2) https://www.bitchute.com/video/Cgkut0hFZjGx/

about 1300 "leaders". Including military. This fascist world is fucked. So
it's not ONLY the big-pharma and bill gaytes, but mainly the deadly
globalist fascist WEF, the "world economic forum". In this aspect is a war
between the the people, unorganized, say in communist, socialist etc
parties, unions etc, and the global fascist force, making the whole thing
the /* biggest_war_ever_existent_on_this_capitalist_planet */.

But DC proof doesn't implement ZFC. It has Set(), which
is not needed in ZFC. Why do you mention theses menus?

BTW: Any progress transforming your possibly vacously
true proof of the form:

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

Into a proof hat also concludes:

EXIST(f):Dedekind(f) ?

Dan Christensen schrieb am Donnerstag, 18. November 2021 um 15:16:21 UTC+1:
> On Thursday, November 18, 2021 at 5:09:45 AM UTC-5, Mostowski Collapse wrote:
> > But in ZFC this existence is cast into axioms.
> > What would be your axioms?
> > What is your foundation?
> .
> See the Logic and Sets menu on the DC Proof main screen. Each option is documented in the user manual.
> .
> Dan
> .
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

I even dont know how to prove EXIST(f):Dedekind(f) in
ZFC, you possibly want to prove it in DC Proof.
In ZFC the axiom of infinity allone doesn't give

EXIST(f):Dedekind(f), since you require that f is
injective, and injectivity of f(x) = x u {x} doesn't
follow from the axiom of infinity. So how do you

show EXIST(f):Dedekind(f) ?

Mostowski Collapse schrieb am Sonntag, 21. November 2021 um 19:14:52 UTC+1:
> But DC proof doesn't implement ZFC. It has Set(), which
> is not needed in ZFC. Why do you mention theses menus?
>
> BTW: Any progress transforming your possibly vacously
> true proof of the form:
>
> ALL(f):[Dedekind(f) => EXIST(omega):Peano(omega)]
>
> Into a proof hat also concludes:
>
> EXIST(f):Dedekind(f) ?
> Dan Christensen schrieb am Donnerstag, 18. November 2021 um 15:16:21 UTC+1:
> > On Thursday, November 18, 2021 at 5:09:45 AM UTC-5, Mostowski Collapse wrote:
> > > But in ZFC this existence is cast into axioms.
> > > What would be your axioms?
> > > What is your foundation?
> > .
> > See the Logic and Sets menu on the DC Proof main screen. Each option is documented in the user manual.
> > .
> > 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, November 21, 2021 at 1:14:52 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Donnerstag, 18. November 2021 um 15:16:21 UTC+1:
> > On Thursday, November 18, 2021 at 5:09:45 AM UTC-5, Mostowski Collapse wrote:
> > > But in ZFC this existence is cast into axioms.
> > > What would be your axioms?
> > > What is your foundation?
> > .
> > See the Logic and Sets menu on the DC Proof main screen. Each option is documented in the user manual.
> > .

> But DC proof doesn't implement ZFC.
..
That is good thing if you want to learn the basic methods of proof as used in most math textbooks.
..
> It has Set(), which
> is not needed in ZFC. Why do you mention theses menus?
..
It lists the name of all the axioms of set theory in DC Proof. Each is documented in the user manual.
..
>
> BTW: Any progress transforming your possibly vacously
> true proof of the form:
>
> ALL(f):[Dedekind(f) => EXIST(omega):Peano(omega)]
>
..
See my original posting here.
..
> Into a proof hat also concludes:
>
> EXIST(f):Dedekind(f) ?
..
Impossible since DC Proof does not assume a non-empty universe.
..
Dan
..
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

Well you could add some more axioms. ZFC is also
FOL plus a couple of axioms and axiom schemas.
These axioms would express some ontological
commitment. Does DC Proof not be able to

reason with EXIST? That the universe can be empty
doesn't change much. For example I pretty sure
you can prove in DC Proof the following:

ALL(x):[A(x) & B(x)] => ALL(x):A(x)

But then you can also prove:

EXIST(x):C(x) => EXIST(x):[C(X) v D(x)]

So basically you can also reason with ontological
commitment, and FOL provides some inference
rules about ontological commitment.

Dan Christensen schrieb am Montag, 22. November 2021 um 01:01:04 UTC+1:
> On Sunday, November 21, 2021 at 1:14:52 PM UTC-5, Mostowski Collapse wrote:
>
> > Dan Christensen schrieb am Donnerstag, 18. November 2021 um 15:16:21 UTC+1:
> > > On Thursday, November 18, 2021 at 5:09:45 AM UTC-5, Mostowski Collapse wrote:
> > > > But in ZFC this existence is cast into axioms.
> > > > What would be your axioms?
> > > > What is your foundation?
> > > .
> > > See the Logic and Sets menu on the DC Proof main screen. Each option is documented in the user manual.
> > > .
> > But DC proof doesn't implement ZFC.
> .
> That is good thing if you want to learn the basic methods of proof as used in most math textbooks.
> .
> > It has Set(), which
> > is not needed in ZFC. Why do you mention theses menus?
> .
> It lists the name of all the axioms of set theory in DC Proof. Each is documented in the user manual.
> .
> >
> > BTW: Any progress transforming your possibly vacously
> > true proof of the form:
> >
> > ALL(f):[Dedekind(f) => EXIST(omega):Peano(omega)]
> >
> .
> See my original posting here.
> .
> > Into a proof hat also concludes:
> >
> > EXIST(f):Dedekind(f) ?
> .
> Impossible since DC Proof does not assume a non-empty universe.
> .
> Dan
> .
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Of course you can doubt this inference rules.
But they are there, also in DC Proof, unless
you not only allow the domain of discourse to

be empty, which has no effect on ontological
commitment reasoning. You can only tweak
ontological commitment when you for example

abandon deMorgan of quantifiers. But you provide
the same as well, you have a QUANTIFIER SWITCHING
rule. For example the quantifier switching of DC

Proof is not generally valid in intuitionistic logic
for all directions. So allowing an empty domain
of discourse has nothing todo with ontological

commitment, its only facet of first order logic.
What counts is whether you have classical logic
or not. Classical logic allows quite some

juggling with ontological commitment.

Mostowski Collapse schrieb am Montag, 22. November 2021 um 01:55:01 UTC+1:
> Well you could add some more axioms. ZFC is also
> FOL plus a couple of axioms and axiom schemas.
> These axioms would express some ontological
> commitment. Does DC Proof not be able to
>
> reason with EXIST? That the universe can be empty
> doesn't change much. For example I pretty sure
> you can prove in DC Proof the following:
>
> ALL(x):[A(x) & B(x)] => ALL(x):A(x)
>
> But then you can also prove:
>
> EXIST(x):C(x) => EXIST(x):[C(X) v D(x)]
>
> So basically you can also reason with ontological
> commitment, and FOL provides some inference
> rules about ontological commitment.
> Dan Christensen schrieb am Montag, 22. November 2021 um 01:01:04 UTC+1:
> > On Sunday, November 21, 2021 at 1:14:52 PM UTC-5, Mostowski Collapse wrote:
> >
> > > Dan Christensen schrieb am Donnerstag, 18. November 2021 um 15:16:21 UTC+1:
> > > > On Thursday, November 18, 2021 at 5:09:45 AM UTC-5, Mostowski Collapse wrote:
> > > > > But in ZFC this existence is cast into axioms.
> > > > > What would be your axioms?
> > > > > What is your foundation?
> > > > .
> > > > See the Logic and Sets menu on the DC Proof main screen. Each option is documented in the user manual.
> > > > .
> > > But DC proof doesn't implement ZFC.
> > .
> > That is good thing if you want to learn the basic methods of proof as used in most math textbooks.
> > .
> > > It has Set(), which
> > > is not needed in ZFC. Why do you mention theses menus?
> > .
> > It lists the name of all the axioms of set theory in DC Proof. Each is documented in the user manual.
> > .
> > >
> > > BTW: Any progress transforming your possibly vacously
> > > true proof of the form:
> > >
> > > ALL(f):[Dedekind(f) => EXIST(omega):Peano(omega)]
> > >
> > .
> > See my original posting here.
> > .
> > > Into a proof hat also concludes:
> > >
> > > EXIST(f):Dedekind(f) ?
> > .
> > Impossible since DC Proof does not assume a non-empty universe.
> > .
> > 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, November 21, 2021 at 7:55:01 PM UTC-5, Mostowski Collapse wrote:
> Well you could add some more axioms. ZFC is also
> FOL plus a couple of axioms and axiom schemas.
> These axioms would express some ontological
> commitment. Does DC Proof not be able to
> reason with EXIST? That the universe can be empty
> doesn't change much. For example I pretty sure
> you can prove in DC Proof the following:
>
> ALL(x):[A(x) & B(x)] => ALL(x):A(x)
>
..
In DC Proof, you would need to make domain of discussion explicit (e.g. a set d) to obtain:
..
ALL(x):[x in d => A(x) & B(x)] => ALL(x):[x in d => A(x)]
..
> But then you can also prove:
>
> EXIST(x):C(x) => EXIST(x):[C(X) v D(x)]
>
..
Yes.
..
> So basically you can also reason with ontological
> commitment, and FOL provides some inference
> rules about ontological commitment.
..
There is no need in either case here to assume a non-empty universe/domain of discussion.
..
Dan
..
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

ALL(x):[A(x) & B(x)] => ALL(x):A(x)

isn't provable in DC proof?

Dan Christensen schrieb am Montag, 22. November 2021 um 04:58:47 UTC+1:
> On Sunday, November 21, 2021 at 7:55:01 PM UTC-5, Mostowski Collapse wrote:
> > Well you could add some more axioms. ZFC is also
> > FOL plus a couple of axioms and axiom schemas.
> > These axioms would express some ontological
> > commitment. Does DC Proof not be able to
> > reason with EXIST? That the universe can be empty
> > doesn't change much. For example I pretty sure
> > you can prove in DC Proof the following:
> >
> > ALL(x):[A(x) & B(x)] => ALL(x):A(x)
> >
> .
> In DC Proof, you would need to make domain of discussion explicit (e.g. a set d) to obtain:
> .
> ALL(x):[x in d => A(x) & B(x)] => ALL(x):[x in d => A(x)]
> .
> > But then you can also prove:
> >
> > EXIST(x):C(x) => EXIST(x):[C(X) v D(x)]
> >
> .
> Yes.
> .
> > So basically you can also reason with ontological
> > commitment, and FOL provides some inference
> > rules about ontological commitment.
> .
> There is no need in either case here to assume a non-empty universe/domain of discussion.
> .
> Dan
> .
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

∀x(Ax ∧ Bx) → ∀xAx is valid.
https://www.umsu.de/trees

Its also true for the empty universe, allowing the
empty universe doesn't change its general validity.

Mostowski Collapse schrieb am Montag, 22. November 2021 um 10:44:51 UTC+1:
> ALL(x):[A(x) & B(x)] => ALL(x):A(x)
> isn't provable in DC proof?
> Dan Christensen schrieb am Montag, 22. November 2021 um 04:58:47 UTC+1:
> > On Sunday, November 21, 2021 at 7:55:01 PM UTC-5, Mostowski Collapse wrote:
> > > Well you could add some more axioms. ZFC is also
> > > FOL plus a couple of axioms and axiom schemas.
> > > These axioms would express some ontological
> > > commitment. Does DC Proof not be able to
> > > reason with EXIST? That the universe can be empty
> > > doesn't change much. For example I pretty sure
> > > you can prove in DC Proof the following:
> > >
> > > ALL(x):[A(x) & B(x)] => ALL(x):A(x)
> > >
> > .
> > In DC Proof, you would need to make domain of discussion explicit (e.g. a set d) to obtain:
> > .
> > ALL(x):[x in d => A(x) & B(x)] => ALL(x):[x in d => A(x)]
> > .
> > > But then you can also prove:
> > >
> > > EXIST(x):C(x) => EXIST(x):[C(X) v D(x)]
> > >
> > .
> > Yes.
> > .
> > > So basically you can also reason with ontological
> > > commitment, and FOL provides some inference
> > > rules about ontological commitment.
> > .
> > There is no need in either case here to assume a non-empty universe/domain of discussion.
> > .
> > 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 4:51:47 AM UTC-5, Mostowski Collapse wrote:
> ∀x(Ax ∧ Bx) → ∀xAx is valid.
> https://www.umsu.de/trees
>
..
This Tree Proof Generator assumes a non-empty universe/domain of discourse limiting its usefulness to mathematicians. Lazy logicians, love such shortcuts.
..
> Its also true for the empty universe, allowing the
> empty universe doesn't change its general validity.
..
True, but you could not get at this useful fact from standard FOL since it automatically assumes a non-empty domain. Its users seem not to even want to consider the possibility. Too much work.
..
Dan
..
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

And where based on what axioms do you show
existence of a Dedekind infinite set?

LoL

Dan Christensen schrieb am Mittwoch, 17. November 2021 um 21:29:37 UTC+1:
> Here is an updated version of my proof constructing the natural numbers.
> .
> THEOREM
> .
> From any Dedekind infinite set x can be selected a subset n on which the Peano Axioms hold. I let f be a function on set x that is injective and not surjective, i.e. x is Dedekind infinite. I let x0 be an element of x having no pre-image under f. I prove:
> .
> ALL(x):ALL(f):[Set(x)
> .
> & ALL(a):[a in x => f(a) in x]
> .
> & ALL(a):ALL(b):[a in x & b in x => [f(a)=f(b) => a=b]] (f is injective)
> .
> & EXIST(a):[a in x & ALL(b):[b in x => ~f(b)=a]] (f is not surjective)
> .
> => EXIST(n):EXIST(x0):[Set(n) & ALL(a):[a in n => a in x]
> .
> & x0 in n (PA1)
> .
> & ALL(a):[a in n => f(a) in n] (PA2)
> .
> & ALL(a):ALL(b):[a in n & b in n => [f(a)=f(b) => a=b]] (PA3)
> .
> & ALL(a):[a in n => ~f(a)=x0] (PA4)
> .
> & ALL(a):[Set(a) & ALL(b):[b in a => b in n] (PA5)
> => [x0 in a & ALL(b):[b in a => f(b) in a]
> => ALL(b):[b in n => b in a]]]]]
> .
> .
> FORMAL PROOF (only 104 lines)
> .
> https://dcproof.com/ConstructN.htm
> .
> .
> Dan
> .
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

See my reply to your identical posting just now at sci.logic

Dan

On Monday, June 27, 2022 at 5:19:19 AM UTC-4, Mostowski Collapse wrote:
> And where based on what axioms do you show
> existence of a Dedekind infinite set?
>
> LoL
> Dan Christensen schrieb am Mittwoch, 17. November 2021 um 21:29:37 UTC+1:
> > Here is an updated version of my proof constructing the natural numbers..
> > .
> > THEOREM
> > .
> > From any Dedekind infinite set x can be selected a subset n on which the Peano Axioms hold. I let f be a function on set x that is injective and not surjective, i.e. x is Dedekind infinite. I let x0 be an element of x having no pre-image under f. I prove:
> > .
> > ALL(x):ALL(f):[Set(x)
> > .
> > & ALL(a):[a in x => f(a) in x]
> > .
> > & ALL(a):ALL(b):[a in x & b in x => [f(a)=f(b) => a=b]] (f is injective)
> > .
> > & EXIST(a):[a in x & ALL(b):[b in x => ~f(b)=a]] (f is not surjective)
> > .
> > => EXIST(n):EXIST(x0):[Set(n) & ALL(a):[a in n => a in x]
> > .
> > & x0 in n (PA1)
> > .
> > & ALL(a):[a in n => f(a) in n] (PA2)
> > .
> > & ALL(a):ALL(b):[a in n & b in n => [f(a)=f(b) => a=b]] (PA3)
> > .
> > & ALL(a):[a in n => ~f(a)=x0] (PA4)
> > .
> > & ALL(a):[Set(a) & ALL(b):[b in a => b in n] (PA5)
> > => [x0 in a & ALL(b):[b in a => f(b) in a]
> > => ALL(b):[b in n => b in a]]]]]
> > .
> > .
> > FORMAL PROOF (only 104 lines)
> > .
> > https://dcproof.com/ConstructN.htm
> > .
> > .
> > Dan
> > .
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

Dan Christensen wrote:

> See my reply to your identical posting just now at sci.logic Dan
>
> On Monday, June 27, 2022 at 5:19:19 AM UTC-4, Mostowski Collapse wrote:
>> And where based on what axioms do you show existence of a Dedekind

irrelevant. Your latino prime-minister says he was successful in
"vaccinating" with injections and metal rigid needles 90% of you the
stupid population

and now he wants to vaccinate the rest of 10%. Because in your fake_money
country, populated by morons, abortion is healthcare and mortal serum in
"vaccines" are also healthcare.

you nazi idiot.

On Monday, June 27, 2022 at 4:12:21 PM UTC-4, Josh Kano wrote:
> Dan Christensen wrote:
>
> > See my reply to your identical posting just now at sci.logic Dan
> >
> > On Monday, June 27, 2022 at 5:19:19 AM UTC-4, Mostowski Collapse wrote:
> >> And where based on what axioms do you show existence of a Dedekind
> irrelevant. Your latino prime-minister says he was successful in
> "vaccinating" with injections and metal rigid needles 90% of you the
> stupid population
>
> and now he wants to vaccinate the rest of 10%. Because in your fake_money
> country, populated by morons, abortion is healthcare and mortal serum in
> "vaccines" are also healthcare.
>

And what is Reichsfuhrer Putin doing about it in your RuSSia, Nazi Boy? Not allowed to discuss that? Oh, well...

Escape while you can, Nazi Boy. The Finnish border is already closed.

Dan Christensen wrote:

>> and now he wants to vaccinate the rest of 10%. Because in your
>> fake_money country, populated by morons, abortion is healthcare and
>> mortal serum in "vaccines" are also healthcare.
>
> And what is Reichsfuhrer Putin doing about it in your RuSSia, Nazi Boy?

your latino prime_minister of cacanada, not know woman is to blame for
listening to the devil. His nazis government is saying, abortion and
mortal serum in "vaccines" are "healthcare". Which is absurd.

he wants you penetrated with mortal serum said "vaccines" 100%, otherwise
you will suffer the consequences staying alive.

On Monday, June 27, 2022 at 7:14:43 PM UTC-4, Brain Nakanoi wrote:
> Dan Christensen wrote:
>
> >> and now he wants to vaccinate the rest of 10%. Because in your
> >> fake_money country, populated by morons, abortion is healthcare and
> >> mortal serum in "vaccines" are also healthcare.
> >
> > And what is Reichsfuhrer Putin doing about it in your RuSSia, Nazi Boy?

Can't say??? Can't blame you, Nazi Boy! But really, you must escape now while you still can.

Dan Christensen wrote:

> See my reply to your identical posting just now at sci.logic Dan

are you insane??

Revelations from US Vice President Kamala Harris for CNN(Clinton News
Network): "I am a woman's daughter and a woman's granddaughter"
Officialy the second most powerful person in the USA..
https://www.bitchute.com/video/6fQ8qlyJZIzV/

why the nazis hides in schools and kindergartens
https://www.bitchute.com/video/qfhOQA049Jk0/
Kornet ATGM with a 9M133F-2 thermobaric warhead. They hit them with
literally the worst round possible for this situation. If they had used a
fragmentation or HE round, there would have been survivors. With a
thermobaric one, not even a big wood trunk or a trench will save you. I'd
bet money they all died. Those that didn't die immediately died a very
slow death within few hours after. Internal bleeding is so much worse than
shrapnel wounds since you can't treat it and there's no tourniquet for
internal bleeding.

On Monday, June 27, 2022 at 8:00:13 PM UTC-4, Hyman Sugawara wrote:
> Dan Christensen wrote:
>
> > See my reply to your identical posting just now at sci.logic Dan
> are you insane??
>
> Revelations from US Vice President Kamala Harris for CNN(Clinton News
> Network): "I am a woman's daughter and a woman's granddaughter"
> Officialy the second most powerful person in the USA..
> https://www.bitchute.com/video/6fQ8qlyJZIzV/
>
> why the nazis hides in schools and kindergartens
> https://www.bitchute.com/video/qfhOQA049Jk0/
> Kornet ATGM with a 9M133F-2 thermobaric warhead. They hit them with
> literally the worst round possible for this situation. If they had used a
> fragmentation or HE round, there would have been survivors. With a
> thermobaric one, not even a big wood trunk or a trench will save you. I'd
> bet money they all died. Those that didn't die immediately died a very
> slow death within few hours after. Internal bleeding is so much worse than
> shrapnel wounds since you can't treat it and there's no tourniquet for
> internal bleeding.

Do you think this shit will help your case at your war crimes trial, Nazi Boy? Don't count on it. You will be strung up the next morning.