Count down is starting 9, 8, 7, 6, 5, 4, ... there could be
soon an Aha Moment, unlike jcsantos@fc.up.pt who
can go fuck himself. Probably the most annoying idiot

on this planet. But Dan Christensen has really the
chance to become a better human than homo stack
exchangis, in the clade of homo trollensis:

Watch this table, what magic is going on?

Function Declared Computed
Declaration Codomain Range

abs : R -> R, x |-> |x| R [0,oo)
sqr : R -> R, x |-> x^2 R [0,oo)

Everything clear now? This is for non-Bourbaki.
For Bourbaki we could replace Declared Domain,

by cod(f). There would be a function cod(f).
But the table wouldn't look any different

essentially, also for Bourbaki its possible
cod(f)=\=rng(f).

Dan Christensen schrieb am Freitag, 22. April 2022 um 00:04:10 UTC+2:
> My Axiom of Functional Equality, as it stands, does not work. Thanks all for your persistence on this issue. I will either replace or withdraw it altogether within the next few days.
>
> Regards,
> Dan

But I already answered the question even for non-ZFC,
if you would use something more powerful. Like NBG,
only the most crazy idiot by the name José Carlos Santos

interfered on MSE. I already wrote in my answer that img(_)
respectively rng(_) can be extended. But it would not
change anything for this table, since this table

is a geometric fact:

Function Declared Computed
Declaration Codomain Range

abs : R -> R, x |-> |x| R [0,oo)
sqr : R -> R, x |-> x^2 R [0,oo)

The function declaration abs : R -> R defines the
drawing board, R x R, the whole Euclidean plan.
Then rng(f) is the projection, the shadow of f on

the y axis. Whats the shadow of abs(_) on the
y axis? Its always [0,oo) n C where C is the codomain.
You can use whathever formalization, ZFC or non-ZFC,

Bourbaki or non-Bourbaki, its always [0,oo). And the
intention of abs : R -> R is also always the same, to
tell us what is our bigger surround world we are

considering, i.e. R x R. Of course if you only consider
a half Euclidean plane R x [0,oo), and say abs : R -> [0,oo),
then suddently abs becomes surjective, because the rng(f)

shadow now covers all of the y axis, which is now only a half y axis.

Mostowski Collapse schrieb am Freitag, 29. April 2022 um 20:03:04 UTC+2:
> Count down is starting 9, 8, 7, 6, 5, 4, ... there could be
> soon an Aha Moment, unlike jcsa...@fc.up.pt who
> can go fuck himself. Probably the most annoying idiot
>
> on this planet. But Dan Christensen has really the
> chance to become a better human than homo stack
> exchangis, in the clade of homo trollensis:
>
> Watch this table, what magic is going on?
>
> Function Declared Computed
> Declaration Codomain Range
>
> abs : R -> R, x |-> |x| R [0,oo)
> sqr : R -> R, x |-> x^2 R [0,oo)
>
> Everything clear now? This is for non-Bourbaki.
> For Bourbaki we could replace Declared Domain,
>
> by cod(f). There would be a function cod(f).
> But the table wouldn't look any different
>
> essentially, also for Bourbaki its possible
> cod(f)=\=rng(f).
> Dan Christensen schrieb am Freitag, 22. April 2022 um 00:04:10 UTC+2:
> > My Axiom of Functional Equality, as it stands, does not work. Thanks all for your persistence on this issue. I will either replace or withdraw it altogether within the next few days.
> >
> > Regards,
> > Dan

The core problem is that people are not anymore
grounded in geometry. Easy notations like f : X -> Y
and rng(f) that can be easily explained geometrically,

are not anymore understood by folks such as Dan
Christensen und by utter morons like José Carlos Santos
from the group of homo stack exchangis, in the clade

of homo trollensis. They have totally lost their
grounding of mathematics in geometry and are only
good for politics and polemics.

Example:
- Dan Christensen: The set theory most commonly
used in math textbooks. (Hint: not ZFC)
- José Carlos Santos: “There is no distinction made
between the codomain and what is commonly called
the range of a function”. And, in your answer, you
seem to be unaware of that.

Blistering nonsense to confound codomain and range.
Doesn't make any sense geometrically. If domain codomain
span some space, that houses some functions,

this does not imply that codomain is the range of a
function. This is everywhere like that. If you model
the problem with or without set theory.

Thats just geometry.

Mostowski Collapse schrieb:
> But I already answered the question even for non-ZFC,
> if you would use something more powerful. Like NBG,
> only the most crazy idiot by the name José Carlos Santos
>
> interfered on MSE. I already wrote in my answer that img(_)
> respectively rng(_) can be extended. But it would not
> change anything for this table, since this table
>
> is a geometric fact:
>
> Function Declared Computed
> Declaration Codomain Range
>
> abs : R -> R, x |-> |x| R [0,oo)
> sqr : R -> R, x |-> x^2 R [0,oo)
>
> The function declaration abs : R -> R defines the
> drawing board, R x R, the whole Euclidean plan.
> Then rng(f) is the projection, the shadow of f on
>
> the y axis. Whats the shadow of abs(_) on the
> y axis? Its always [0,oo) n C where C is the codomain.
> You can use whathever formalization, ZFC or non-ZFC,
>
> Bourbaki or non-Bourbaki, its always [0,oo). And the
> intention of abs : R -> R is also always the same, to
> tell us what is our bigger surround world we are
>
> considering, i.e. R x R. Of course if you only consider
> a half Euclidean plane R x [0,oo), and say abs : R -> [0,oo),
> then suddently abs becomes surjective, because the rng(f)
>
> shadow now covers all of the y axis, which is now only a half y axis.
>
> Mostowski Collapse schrieb am Freitag, 29. April 2022 um 20:03:04 UTC+2:
>> Count down is starting 9, 8, 7, 6, 5, 4, ... there could be
>> soon an Aha Moment, unlike jcsa...@fc.up.pt who
>> can go fuck himself. Probably the most annoying idiot
>>
>> on this planet. But Dan Christensen has really the
>> chance to become a better human than homo stack
>> exchangis, in the clade of homo trollensis:
>>
>> Watch this table, what magic is going on?
>>
>> Function Declared Computed
>> Declaration Codomain Range
>>
>> abs : R -> R, x |-> |x| R [0,oo)
>> sqr : R -> R, x |-> x^2 R [0,oo)
>>
>> Everything clear now? This is for non-Bourbaki.
>> For Bourbaki we could replace Declared Domain,
>>
>> by cod(f). There would be a function cod(f).
>> But the table wouldn't look any different
>>
>> essentially, also for Bourbaki its possible
>> cod(f)=\=rng(f).
>> Dan Christensen schrieb am Freitag, 22. April 2022 um 00:04:10 UTC+2:
>>> My Axiom of Functional Equality, as it stands, does not work. Thanks all for your persistence on this issue. I will either replace or withdraw it altogether within the next few days.
>>>
>>> Regards,
>>> Dan

José Carlos Santos addressed me, about my unawareness,
while I was in process of explaining when and why
codmain and range are not the same.

Thats possibly the most bold trolling somebody ever
did, or José Carlos Santos had the some unfortunate
brain stroke the moment he used his keyboard,

and we are back to:

Our conjecture is that the cycle will prove most
reliable. A doctor who gets an unlucky string of
misleading results will do the least damage there.
https://plato.stanford.edu/entries/formal-epistemology/#TheZolEff

But this is a much too benevolent interpretation,
possibly José Carlos Santos is just a pervert, that
saw somebody with 101 points and started a fight.

Mostowski Collapse schrieb:
> The core problem is that people are not anymore
> grounded in geometry. Easy notations like f : X -> Y
> and rng(f) that can be easily explained geometrically,
>
> are not anymore understood by folks such as Dan
> Christensen und by utter morons like José Carlos Santos
> from the group of homo stack exchangis, in the clade
>
> of homo trollensis. They have totally lost their
> grounding of mathematics in geometry and are only
> good for politics and polemics.
>
> Example:
> - Dan Christensen: The set theory most commonly
>   used in math textbooks. (Hint: not ZFC)
> - José Carlos Santos: “There is no distinction made
>   between the codomain and what is commonly called
>   the range of a function”. And, in your answer, you
>   seem to be unaware of that.
>
> Blistering nonsense to confound codomain and range.
> Doesn't make any sense geometrically. If domain codomain
> span some space, that houses some functions,
>
> this does not imply that codomain is the range of a
> function. This is everywhere like that. If you model
> the problem with or without set theory.
>
> Thats just geometry.
>
> Mostowski Collapse schrieb:
>> But I already answered the question even for non-ZFC,
>> if you would use something more powerful. Like NBG,
>> only the most crazy idiot by the name José Carlos Santos
>>
>> interfered on MSE. I already wrote in my answer that img(_)
>> respectively rng(_) can be extended. But it would not
>> change anything for this table, since this table
>>
>> is a geometric fact:
>>
>> Function                  Declared        Computed
>> Declaration               Codomain        Range
>>
>> abs :  R -> R, x |-> |x|  R               [0,oo)
>> sqr : R -> R, x |-> x^2   R               [0,oo)
>>
>> The function declaration abs : R -> R defines the
>> drawing board, R x R, the whole Euclidean plan.
>> Then rng(f) is the projection, the shadow of f on
>>
>> the y axis. Whats the shadow of abs(_) on the
>> y axis? Its always [0,oo) n C where C is the codomain.
>> You can use whathever formalization, ZFC or non-ZFC,
>>
>> Bourbaki or non-Bourbaki, its always [0,oo). And the
>> intention of abs : R -> R is also always the same, to
>> tell us what is our bigger surround world we are
>>
>> considering, i.e. R x R. Of course if you only consider
>> a half Euclidean plane R x [0,oo), and say abs : R -> [0,oo),
>> then suddently abs becomes surjective, because the rng(f)
>>
>> shadow now covers all of the y axis, which is now only a half y axis.
>>
>> Mostowski Collapse schrieb am Freitag, 29. April 2022 um 20:03:04 UTC+2:
>>> Count down is starting 9, 8, 7, 6, 5, 4, ... there could be
>>> soon an Aha Moment, unlike jcsa...@fc.up.pt who
>>> can go fuck himself. Probably the most annoying idiot
>>>
>>> on this planet. But Dan Christensen has really the
>>> chance to become a better human than homo stack
>>> exchangis, in the clade of homo trollensis:
>>>
>>> Watch this table, what magic is going on?
>>>
>>> Function Declared Computed
>>> Declaration Codomain Range
>>>
>>> abs : R -> R, x |-> |x| R [0,oo)
>>> sqr : R -> R, x |-> x^2 R [0,oo)
>>>
>>> Everything clear now? This is for non-Bourbaki.
>>> For Bourbaki we could replace Declared Domain,
>>>
>>> by cod(f). There would be a function cod(f).
>>> But the table wouldn't look any different
>>>
>>> essentially, also for Bourbaki its possible
>>> cod(f)=\=rng(f).
>>> Dan Christensen schrieb am Freitag, 22. April 2022 um 00:04:10 UTC+2:
>>>> My Axiom of Functional Equality, as it stands, does not work. Thanks
>>>> all for your persistence on this issue. I will either replace or
>>>> withdraw it altogether within the next few days.
>>>>
>>>> Regards,
>>>> Dan
>

Mostowski Collapse wrote:

> Count down is starting 9, 8, 7, 6, 5, 4, ... there could be soon an Aha
> Moment, unlike jcsantos@fc.up.pt who can go fuck himself. Probably the
> most annoying idiot on this planet. But Dan Christensen has really the
> chance to become a better human than homo stack exchangis, in the clade
> of homo trollensis:

We smell you two's stinking nazi dicks along large distances.

https://www.veteranstoday.com/wp-content/uploads/2022/04/Ukie-Nazis-semi-undercover_002.jpg

Maybe José Carlos Santos had really a stroke of some
sort, maybe he is a chain smoker? Since he wrote:

"Beacause the OP wrote that, in ZFC,"

Whats the typo in because? Also ZFC has a specific language
like set-like function, otherwise a function is a class. So there is
a THIRD CATEGORY MISTAKE in the question:

If I understand correctly, in ZFC set theory, a function is just a set
of ordered pairs with certain properties.
https://math.stackexchange.com/q/4438429/1002973

The correct phrasing would be:

If I understand correctly, in ZFC set theory, a set-like function is
just a set of ordered pairs with certain properties.
https://math.stackexchange.com/q/4438429/1002973

You can check yourself:

6.8 Definition A class S is said to be a (binary) relation if
evey member x of S is an ordered pair.
6.12 Definition A relation F is said to be a function if ...
Basic Set Theory - Azriel Levy
https://www.amazon.de/Basic-Theory-Dover-Books-Mathematics/dp/0486420795

3 category mistakes in one MSE question.
Congratulations Dan Christensen, thats
something for the Guiness Book of Records.

Lets also include the burping administrators
of stack exchange in the Guiness Book of
Records entry. Giving the cloud of comments

the two remaining answers have on MSE,
maybe MSE would have been wise to first
sift the question terminological wise, it

seems that the confusion remains...

LMAO!

Mostowski Collapse schrieb:
> Maybe José Carlos Santos had really a stroke of some
> sort, maybe he is a chain smoker? Since he wrote:
>
> "Beacause the OP wrote that, in ZFC,"
>
> Whats the typo in because? Also ZFC has a specific language
> like set-like function, otherwise a function is a class. So there is
> a THIRD CATEGORY MISTAKE in the question:
>
> If I understand correctly, in ZFC set theory, a function is just a set
> of ordered pairs with certain properties.
> https://math.stackexchange.com/q/4438429/1002973
>
> The correct phrasing would be:
>
> If I understand correctly, in ZFC set theory, a set-like function is
> just a set of ordered pairs with certain properties.
> https://math.stackexchange.com/q/4438429/1002973
>
> You can check yourself:
>
> 6.8 Definition A class S is said to be a (binary) relation if
> evey member x of S is an ordered pair.
> 6.12 Definition A relation F is said to be a function if ...
> Basic Set Theory - Azriel Levy
> https://www.amazon.de/Basic-Theory-Dover-Books-Mathematics/dp/0486420795

Mostowski Collapse wrote:

> 3 category mistakes in one MSE question.
> Congratulations Dan Christensen, thats
> something for the Guiness Book of Records.

*_brainwashed_english_pig_* with kids at home, go to war fighting for the
nazis. Now fractured hands etc. And demanding mercy from the russians. How
incredible stupid this kind of wankers can be. Ohh my butt. You go to war
fighting for the nazis.

[download] Destination: Another British mercenary captured by Russians -
Andrew Hill is from Plymouth https://www.bitchute.com/video/Du3RUvd8S08d/

BTW: I was quite impressed by stalin inspired
anti-capitalist sketch of some mathematics by
Arturo Magidin. I am not sure whether they

would fit stack exchange, maybe they would
more match private closet web site, I could
just imagine some farmer who would sporadically

have some mythical subjective experience, and
then go on make some crop circles just for the
few lot of them. I agree they dont need an english

wikipedia, they possibly grunt in proto-russian,
they also dont need it because this mathematics
would have its fullfilment in autistic giggling.

privat closet web site = where you FTP
your manuscripts and dont show it anybody,
drop box with a private setting:

for no-one else, its all mine.

Gollum: My Precious
https://youtu.be/Iz-8CSa9xj8

LoL

Mostowski Collapse schrieb am Freitag, 29. April 2022 um 23:29:04 UTC+2:
> BTW: I was quite impressed by stalin inspired
> anti-capitalist sketch of some mathematics by
> Arturo Magidin. I am not sure whether they
>
> would fit stack exchange, maybe they would
> more match private closet web site, I could
> just imagine some farmer who would sporadically
>
> have some mythical subjective experience, and
> then go on make some crop circles just for the
> few lot of them. I agree they dont need an english
>
> wikipedia, they possibly grunt in proto-russian,
> they also dont need it because this mathematics
> would have its fullfilment in autistic giggling.

oops, my bad, english is capitalist, here are some
translations for comrad Arturo Magidin:

"MY PRECIOUS" in Different Languages
https://youtu.be/yxpY--HfdQI

Mostowski Collapse schrieb am Freitag, 29. April 2022 um 23:38:12 UTC+2:
> privat closet web site = where you FTP
> your manuscripts and dont show it anybody,
> drop box with a private setting:
>
> for no-one else, its all mine.
>
> Gollum: My Precious
> https://youtu.be/Iz-8CSa9xj8
>
> LoL
> Mostowski Collapse schrieb am Freitag, 29. April 2022 um 23:29:04 UTC+2:
> > BTW: I was quite impressed by stalin inspired
> > anti-capitalist sketch of some mathematics by
> > Arturo Magidin. I am not sure whether they
> >
> > would fit stack exchange, maybe they would
> > more match private closet web site, I could
> > just imagine some farmer who would sporadically
> >
> > have some mythical subjective experience, and
> > then go on make some crop circles just for the
> > few lot of them. I agree they dont need an english
> >
> > wikipedia, they possibly grunt in proto-russian,
> > they also dont need it because this mathematics
> > would have its fullfilment in autistic giggling.