First learn how my historic geometric identity of January 2020 proves your mainstream formulation of calculus is an elaborate fraud:

https://drive.google.com/file/d/1RDulODvgncItTe7qNI1d8KTN5bl0aTXj

Seeing is believing, so download the following applet and watch how my historic geometric theorem works for any given function:

https://drive.google.com/file/d/1ON1GQ7b6UNpZSEEsbG14eAFCPv8p03pv

There are no viruses or spy software in my applets and to my knowledge most browsers will warn you before you download any malicious code in applets. Don't waste your time on the fraudulent mainstream formulation of calculus which cannot be understood because it is anti-mathematical nonsense. Download my applet now and gain understanding and knowledge you couldn't find at school or university!

Second, learn how it fixes the mainstream bogus definition of definite integral:

https://drive.google.com/file/d/1uIBgJ1ObroIbkt0V2YFQEpPdd8l-xK6y

Applet which explains the definite integral without any ill-formed concepts such as infinity, infinitesimals or the circular rot of limit theory:

https://drive.google.com/file/d/1JYRxjGb3MxlYWp_2KqVXwXNr5XUvUNz7

A video explaining the new applet:

https://youtu.be/TJqvbshIGtg

Third, study my free eBook because not only your mathematical future depends on it but also your sanity!

https://drive.google.com/file/d/1CIul68phzuOe6JZwsCuBuXUR8X-AkgEO

Lastly, ignore all the trolls and cranks such as Dan Christensen, Jean Pierre Messager (aka Python aka YBM aka JPM), Jan Burse, Michael MORONey, QB, Earle, Jens Stuckelberger, Jan Bilawski, Zelos Malum, Markus Klyver, Alan McKenzie, Ludwig Poehlman (certified crank aka Archimedes Plutonium) etc. These fools have nothing to say about my work. All they know is how to throw shade because they are ignorant, arrogant, incompetent and incorrigibly stupid. Unfortunately, sci.math has become the mainstream's "Truth Network" - replete with paid trolls, some of whose names are mentioned in this paragraph. These fools have lost the privilege of being educated by me. Gracing them with any kind of response is like placing a prized pearl on a pile of dung.

I shall not respond to any of them again.

At times it's good to laugh at the stupidity and dishonesty of mainstream academics. However, the situation is dire when religion trumps common sense:

https://drive.google.com/file/d/1520NjhgiakcrssQxtbxRCDXus_aHXpI9

This comment will be periodically reposted for the sake of new visitors.

My lecture on the true Foundations of Mathematics was recorded.

You can download it here:

https://youtu.be/_WZJY1xgJTk

The Gift applet (almost in power point format!):

https://drive.google.com/file/d/1SYT-MbYtXUAYgwPeTgZ8QC8gxNpj-fap

Remember, you cannot understand mathematics without me because no one understands mathematics as well as I do. The lecture is a bit on the slow side (because of my health issues), BUT you will learn more about the foundations of mathematics than you ever knew your entire life!

The Elements is not about geometry. It's not about algebra. It's not about constructions. It is a template for the realisation of all SCIENCE.

I quote a Greek scholar (http://www.physics.ntua.gr/mourmouras/euclid/common/anti_prologou.html):

Τα Στοιχεία Ευκλείδου δέν είναι Γεωμετρία (ούτε επιπεδομετρία, ούτε γεωμετρία χώρου), απλώς μπορούν να παραχθούν από αύτα «Γεωμετρίες» (είτε «Ευκλείδειες,» είτε «μή Ευκλείδειες», κλπ), όπως επίσης μπορούν να παραχθούν πολλές άλλες θεωρίες «Μαθηματικές» καί όχι μόνο, π.χ. Θεωρία Αριθμών, Aναλογίες, Aσύμμετρα Μεγέθη, Θεωρία της Σχετικότητας, κ.α.

Τα Στοιχεία τού Ευκλείδου δεν είναι ούτε αντικείμενα του Πραγματικού κόσμου, καίούτε είναι αντικείμενα τών Μαθηματικών, αλλά χωρίς αυτά, όλα τά παραπάνω δέν μπορούν να περιγραφούν, να κατανοηθούν, να ερμηνευτούν.

Πολλοί συγχέουν τον πραγματικό χώρο τών «Φυσικών» επιστημών με τά Στοιχεία Ευκλείδου. Aπό τα Στοιχεία Ευκλείδου προέρχονται οί πλείστες μαθηματικές καί φυσικές επιστήμες, τόσο οί παραδοσιακές όσο καί οί μοντέρνες. Γιά τις τεχνολογίες αυτό είναι προφανές.

Τα Στοιχεία Ευκλείδου αποτελούν ένα ενιαίον όλο, ένα «Σύστημα», τό οποίο είναι «κλειστό καί ἀνοικτό», τόσο όσο καί ή σκέψη, ό λόγος καί ή γραμματική τής ανθρώπινης υπόστασης. H πληρότητα καί ἡ συνέπεια είναι δομημένες μέ έναν εκπληκτικό τρόπο, πού τό καθιστούν μοναδικό.

My καθαρεύουσα (high Greek) Greek is very poor, but here is my translation:

Euclidean Elements are not Geometry (neither plane geometry nor space geometry), from these can be produced "Geometries" (either "Euclidean" or "non-Euclidean", etc.), as well as many other "Mathematical" theories can be produced and not only, e.g. Number Theory, Proportions, Asymmetric Sizes, Theory of Relativity, etc.

The Elements of Euclid are neither objects of the real world, nor are they objects of Mathematics, but without them, all the above cannot be described, understood, interpreted. Many confuse the real world of the "Natural" sciences with the Euclidean Elements. Most of the mathematical and natural sciences, both traditional and modern, come from the Euclidean Elements. For technologies this is obvious.

The Euclidean Elements constitute a single whole, a "System", which is "closed and open", as much as the thought, reason and grammar of the human condition. Completeness and consistency are structured in an amazing way, which make it unique.

----------------------------------------------------------

This professor (of Physics and mathematics) also reads all the Ancient texts and gives lectures. He is quite an interesting academic and very learned. He is currently reading all of Aristotle's texts:

https://www.youtube.com/c/DimitriosMourmouras

What you probably do not know is that there is no mention of any of the following words in the Elements:

axiom
geometry
straight-edge or ruler
compass

The diagrams are merely used as a means of communicating the relationships between points (location) and distances (lines).

The Elements is a product of pure thought that was realised first by the brilliant Ancient Greeks.

Become a subscriber to my New Calculus YT channel to learn more mathematics than you learned in all your school and university years:

https://www.youtube.com/c/JohnGabriel/videos

Mainstream mathematics has turned into a universe of alternate "facts". The majority of mainstream academics are stupid and ignorant beyond belief so that mainstream math is today more of a cult than the science it was meant to be.

Unfortunately mainstream academia does not teach one how to think, only to memorise the prescribed doctrines that form their official statement of beliefs. Free yourself from the utter bullshit you are forced to memorise and if you're honest with yourself, will admit that you never actually understand.

Mainstream academia are shitting themselves because their days are numbered.. I am their prime target because the more I publish facts (truth), the more I am hated for it. Even the cranks (Archimedes Plutonium aka Ludwig Poehlman) join in the mocking spectacle.

Don't believe a word I say, but prove that what I say is indeed the case. How, you ask? Use your brain!

There's more....

Learn about the BIG LIE "Calculus was made rigorous...":

https://www.academia.edu/45408445/Big_Lie_Calculus_was_made_rigorous

"Why can’t you understand the difference between assuming that f'(x)=3x^2, as a “fact” upon which to build further proofs, and hypothesizing that f'(x) might equal 3x^2, as a guess to be treated with extreme suspicion and checked using the definition before I’m allowed to write f'(x)=3x^2?" - Anders Kaesorg.

The 5 Step method is outlined below and it summarises the mindset of most morons from MIT:

1. Assumption of fact
2. Hypothesis
3. Probability
4. Suspicion
5. Verification

;-)

https://youtu.be/gX5Bt8BEdNM

Subscribe to my YT Channel - the best math channel on the web:

https://www.youtube.com/c/JohnGabriel/videos

Who gave you the 0.999... and 0.333...:

Euler wrote a very good book on algebra but unfortunately it was soiled with his delusional notions.

Daher ist uns Bruch 1/(1+a) gleich dieser unendlichen Reihe 1 - a + aa -aaa + ...&c.

Therefore is our fraction 1/(1+a) equal to this infinite series 1 - a + aa -aaa + ...&c.

1/(1+a) is the <<limit>> of the series <<1 - a + aa -aaa + ...&c.>>

Ergo, Lim S = S or S = Lim S.

There is much more in the following link:

https://drive.google.com/file/d/12oUJAfIMFMcXFb8DvgsYxuPfdaB99XYH

Free your mind from the religious rot of mainstream mathematics academia which is a full-blown cult, not too different from Trump's supporters.

STUDENTS BEWARE: Don't be a victim of JG's fake math

On Wednesday, April 6, 2022 at 8:47:54 PM UTC-4, I am Super Rectum (aka John Gabriel (JG), Troll Boy) wrote:

> First learn how ...

When will YOU learn, Troll Boy?

JG here claims to have a discovered a shortcut to mastering calculus without using limits. Unfortunately for him, this means he has no workable a definition of the derivative of a function. It blows up for functions as simple f(x)=|x|. Or even f(x)=0. As a result, he has had to ban 0, negative numbers and instantaneous rates of change rendering his goofy little system quite useless.

Forget calculus. JG has also banned all axioms because he cannot even derive the most elementary results of basic arithmetic, e.g. 2+2=4. Such results require the use of axioms, so he must figure he's now off the hook.

Even at his advanced age (60+?), John Gabriel is STILL struggling with basic, elementary-school arithmetic. As he has repeatedly posted here:

"There are no points on a line."
--April 12, 2021

"Pi is NOT a number of ANY kind!"
--July 10, 2020

"1/2 not equal to 2/4"
--October 22, 2017

“1/3 does NOT mean 1 divided by 3 and never has meant that”
-- February 8, 2015

"3 =< 4 is nonsense.”
--October 28, 2017

"Zero is not a number."
-- Dec. 2, 2019

"0 is not required at all in mathematics, just like negative numbers."
-- Jan. 4, 2017

“There is no such thing as an empty set.”
--Oct. 4, 2019

“3 <=> 2 + 1 or 3 <=> 8 - 5, etc, are all propositions” (actually all are meaningless gibberish)
--Oct. 22, 2019

No math genius our JG, though he actually lists his job title as “mathematician” at Linkedin.com. Apparently, they do not verify your credentials.

Though really quite disturbing, interested readers should see: “About the spamming troll John Gabriel in his own words...” (lasted updated March 10, 2020) at https://groups.google.com/forum/#!msg/sci.math/PcpAzX5pDeY/1PDiSlK_BwAJ

Dan

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

STUDENTS BEWARE: Dan-O-Matik, the only Ultra Moron on
this planet who doesn't understand how ALL(_) quantifier
works. If you state an axiom such as:

1 ALL(dom):ALL(cod):ALL(f):ALL(g):[Set(dom) & Set(cod) &
ALL(a):[a e dom => f(a) e cod] & ALL(a):[a e dom => g(a) e cod]
=> [f=g <=> ALL(a):[a e dom => f(a)=g(a)]]]
Axiom
https://dcproof.com/EmptyFunctionsUniqueV3.htm

It MUST be satisfied by ALL possible combinations
of f, g, dom and cod, not only those that Dan-O-Matik has
in mind. It must be the full cross product.

If you want something else than a full cross product,
you musts stated this explicitly, like for example:

ALL(d):ALL(c):ALL(f):ALL(g):[Set(d) & Set(c) &
d = dom(f) & c = cod(f) & d = dom(g) & c = cod(g) &
ALL(a):[a e d=> f(a) e c] & ALL(a):[a e d=> g(a) e c]
=> [f=g <=> ALL(a):[a e d=> f(a)=g(a)]]]

If you do not have functions lilke dom(_) and cod(_)
that retrieve the dom and cod of a function and
if you don't state the FUNCTIONAL DEPENDENCIES

such as d = dom(f) & c = cod(f) & d = dom(g) & c = cod(g),
then your axiom becomes pure nonsense.

Dan Christensen schrieb am Donnerstag, 7. April 2022 um 07:19:20 UTC+2:
> STUDENTS BEWARE: Don't be a victim of JG's fake math

On Thursday, April 7, 2022 at 4:48:58 AM UTC-4, Mostowski Collapse wrote:
> STUDENTS BEWARE: Dan-O-Matik, the only Ultra Moron on
> this planet who doesn't understand how ALL(_) quantifier
> works. If you state an axiom such as:
>

See my reply in the thread, "Equality of Functions: Big Controversy"

Dan

If there were a controversy, this was like 100 years ago.

LMAO!

Dan Christensen schrieb am Donnerstag, 7. April 2022 um 17:47:22 UTC+2:
> On Thursday, April 7, 2022 at 4:48:58 AM UTC-4, Mostowski Collapse wrote:
> > STUDENTS BEWARE: Dan-O-Matik, the only Ultra Moron on
> > this planet who doesn't understand how ALL(_) quantifier
> > works. If you state an axiom such as:
> >
> See my reply in the thread, "Equality of Functions: Big Controversy"
>
> Dan

On Thursday, April 7, 2022 at 12:12:16 PM UTC-4, Mostowski Collapse (aka Jan Burse) wrote:
> If there were a controversy, this was like 100 years ago.
>

There is a controversy in YOUR mind only, Jan Burse. Deal with it. Just admit you were wrong.

Dan

STUDENTS Beware: Dan Christensen doesn't understand
this simple mathematical fact:

This here:
ALL(a):[a e X => fun(a) e Y]
And this here:
fun : X -> Y
Are not logically equivalent.
Counter Example:
fun : R -> R, fun(x) = abs(x)

It satisfy:
ALL(a):[x e R+ => fun(x) e R+]
It does not satisfy:
fun : R+ -> R+

Dan Christensen schrieb am Donnerstag, 7. April 2022 um 18:28:52 UTC+2:
> On Thursday, April 7, 2022 at 12:12:16 PM UTC-4, Mostowski Collapse (aka Jan Burse) wrote:
> > If there were a controversy, this was like 100 years ago.
> >
> There is a controversy in YOUR mind only, Jan Burse. Deal with it. Just admit you were wrong.
>
> Dan

On Tuesday, May 3, 2022 at 10:00:13 AM UTC+2, Mostowski Collapse wrote:

Here's another simple example.

Consider the identity function id_{0, 1} defined on {0, 1} with id_{0, 1}(x) = x for all x in {0, 1}: the domain of this function is just {0, 1}.

Now:

ALL(a):[a e {0} => id_{0, 1}(a) e {0}]

and

ALL(a):[a e {1} => id_{0, 1}(a) e {1}]

Hence DC would claim that id_{0, 1} is a function with domain {1} and codomain {1} as well as a function with domain {2} and codoman {2}. So in DAN-O-Matic a certain function may have more than one domain and one codomain. *strange*

> STUDENTS Beware: Dan Christensen doesn't understand
> this simple mathematical fact:
>
> This here:
> ALL(a):[a e X => fun(a) e Y]
> And this here:
> fun : X -> Y
>
> Are not logically equivalent.
> Counter Example:
>
> fun : R -> R, fun(x) = abs(x)
>
> It satisfy:
> ALL(a):[x e R+ => fun(x) e R+]
> It does not satisfy:
> fun : R+ -> R+

On Tuesday, May 3, 2022 at 4:41:51 AM UTC-4, Fritz Feldhase wrote:
> On Tuesday, May 3, 2022 at 10:00:13 AM UTC+2, Mostowski Collapse wrote:
>
> Here's another simple example.
>
> Consider the identity function id_{0, 1} defined on {0, 1} with id_{0, 1}(x) = x for all x in {0, 1}: the domain of this function is just {0, 1}.
>
> Now:
>
> ALL(a):[a e {0} => id_{0, 1}(a) e {0}]
>
> and
>
> ALL(a):[a e {1} => id_{0, 1}(a) e {1}]
>
> Hence DC would claim that id_{0, 1} is a function with domain {1} and codomain {1} as well as a function with domain {2} and codoman {2}.

Fritz the Crank here believes this is an important distinction, though he hasn't explained why. It's all he has got and he tries to make the most of it. Really kind of pathetic.

(So cute the way these two cranks here are running interference for their little buddy and fellow crank JG.)

Dan

Its easy to show that it matters. When we know:
fun : {0} -> {0}

Then we can prove:
ALL(x):ALL(y):[fun(x,y) => x=0 & y=0]

When we only know:
ALL(a):[a e {0} => fun(a) e {0}]

Then we cannot prove:
ALL(x):ALL(y):[fun(x,y) => x=0 & y=0]

Dan Christensen schrieb am Dienstag, 3. Mai 2022 um 15:42:20 UTC+2:
> On Tuesday, May 3, 2022 at 4:41:51 AM UTC-4, Fritz Feldhase wrote:
> > On Tuesday, May 3, 2022 at 10:00:13 AM UTC+2, Mostowski Collapse wrote:
> >
> > Here's another simple example.
> >
> > Consider the identity function id_{0, 1} defined on {0, 1} with id_{0, 1}(x) = x for all x in {0, 1}: the domain of this function is just {0, 1}.
> >
> > Now:
> >
> > ALL(a):[a e {0} => id_{0, 1}(a) e {0}]
> >
> > and
> >
> > ALL(a):[a e {1} => id_{0, 1}(a) e {1}]
> >
> > Hence DC would claim that id_{0, 1} is a function with domain {1} and codomain {1} as well as a function with domain {2} and codoman {2}.
> Fritz the Crank here believes this is an important distinction, though he hasn't explained why. It's all he has got and he tries to make the most of it. Really kind of pathetic.
>
> (So cute the way these two cranks here are running interference for their little buddy and fellow crank JG.)
>
> Dan

On Tuesday, May 3, 2022 at 10:10:17 AM UTC-4, Mostowski Collapse wrote:
> Its easy to show that it matters. When we know:
> fun : {0} -> {0}
>
> Then we can prove:
> ALL(x):ALL(y):[fun(x,y) => x=0 & y=0]
>

Makes no sense. Not going to try to correct your homework here. Start over.

> When we only know:
> ALL(a):[a e {0} => fun(a) e {0}]
>
> Then we cannot prove:
> ALL(x):ALL(y):[fun(x,y) => x=0 & y=0]

Dan

On Tuesday, May 3, 2022 at 5:42:14 PM UTC+2, Dan Christensen wrote:
> On Tuesday, May 3, 2022 at 10:10:17 AM UTC-4, Mostowski Collapse wrote:
> >
> > Its easy to show that it matters.

Look, you silly idiot.

If we have f: {0} -> {0} then
the function f is defined for the argument 0.
Moreover f(0) e {0}, hence f(0) = 0.

If, on the ither hand, we have f: {1} -> {1} then
the function f is NOT "defined" for the argument 0.
This means that f(0) might be "everything" (i. e. we can't just assume that f(0) e {1}).

It certainly may be the case that these state of affairs doesn't matter in the context of Dan-O-Matik, but it certainly matters in the context of mathematics.

Hell, are you on drugs, or what?

On Tuesday, May 3, 2022 at 3:42:20 PM UTC+2, Dan Christensen wrote:
> On Tuesday, May 3, 2022 at 4:41:51 AM UTC-4, Fritz Feldhase wrote:
> > On Tuesday, May 3, 2022 at 10:00:13 AM UTC+2, Mostowski Collapse wrote:
> >
> > Here's another simple example.
> >
> > Consider the identity function id_{0, 1} defined on {0, 1} with id_{0, 1}(x) = x for all x in {0, 1}: the domain of this function is just {0, 1}.
> >
> > Now:
> >
> > ALL(a):[a e {0} => id_{0, 1}(a) e {0}]
> >
> > and
> >
> > ALL(a):[a e {1} => id_{0, 1}(a) e {1}]
> >
> > Hence DC would claim that id_{0, 1} is a function with domain {1} and codomain {1} as well as a function with domain {2} and codoman {2}.
> >
> Fritz [...] believes this is an important distinction, though he hasn't explained why.

Holy shit!!!

Look, you silly idiot.

If we have a function f with domain {0} and codomain {0}, then
the function f is "defined" for the argument 0.
Moreover f(0) e {0}, hence f(0) = 0.

If, on the other hand, we have a function f with domain {1} and codomain {1}, then
the function f is NOT "defined" (as we say) for the argument 0.
This means that f(0) might be "everything" (i. e. we can't just assume that f(0) e {1} and hence f(0) = 1).

It certainly may be the case that these state of affairs doesn't matter in the context of Dan-O-Matik, but it certainly matters in the context of mathematics.

Hell, are you on drugs, or what?

> (So cute

Shut up you demented idiot!

if f : {1} -> {1} then f(0) = {} with Fritz application.
Not everything.

Feldhase schrieb am Dienstag, 3. Mai 2022 um 18:14:49 UTC+2:
> On Tuesday, May 3, 2022 at 3:42:20 PM UTC+2, Dan Christensen wrote:
> > On Tuesday, May 3, 2022 at 4:41:51 AM UTC-4, Fritz Feldhase wrote:
> > > On Tuesday, May 3, 2022 at 10:00:13 AM UTC+2, Mostowski Collapse wrote:
> > >
> > > Here's another simple example.
> > >
> > > Consider the identity function id_{0, 1} defined on {0, 1} with id_{0, 1}(x) = x for all x in {0, 1}: the domain of this function is just {0, 1}.
> > >
> > > Now:
> > >
> > > ALL(a):[a e {0} => id_{0, 1}(a) e {0}]
> > >
> > > and
> > >
> > > ALL(a):[a e {1} => id_{0, 1}(a) e {1}]
> > >
> > > Hence DC would claim that id_{0, 1} is a function with domain {1} and codomain {1} as well as a function with domain {2} and codoman {2}.
> > >
> > Fritz [...] believes this is an important distinction, though he hasn't explained why.
>
> Holy shit!!!
>
> Look, you silly idiot.
>
> If we have a function f with domain {0} and codomain {0}, then
> the function f is "defined" for the argument 0.
> Moreover f(0) e {0}, hence f(0) = 0.
> If, on the other hand, we have a function f with domain {1} and codomain {1}, then
> the function f is NOT "defined" (as we say) for the argument 0.
> This means that f(0) might be "everything" (i. e. we can't just assume that f(0) e {1} and hence f(0) = 1).
> It certainly may be the case that these state of affairs doesn't matter in the context of Dan-O-Matik, but it certainly matters in the context of mathematics.
>
> Hell, are you on drugs, or what?
> > (So cute
>
> Shut up you demented idiot!

it follows from f ⊆ {0} x {0}

see here:

a function f: X→Y is a binary relation between X and Y
https://en.m.wikipedia.org/wiki/Function_(mathematics)

Dan Christensen schrieb am Dienstag, 3. Mai 2022 um 17:42:14 UTC+2:
> On Tuesday, May 3, 2022 at 10:10:17 AM UTC-4, Mostowski Collapse wrote:
> > Its easy to show that it matters. When we know:
> > fun : {0} -> {0}
> >
> > Then we can prove:
> > ALL(x):ALL(y):[fun(x,y) => x=0 & y=0]
> >
> Makes no sense. Not going to try to correct your homework here. Start over.
> > When we only know:
> > ALL(a):[a e {0} => fun(a) e {0}]
> >
> > Then we cannot prove:
> > ALL(x):ALL(y):[fun(x,y) => x=0 & y=0]
>
> Dan

And because of seriality, you will be able to prove:

ALL(x):ALL(y):[fun(x,y) => x=0 & y=0]

See also:

A binary relation is serial (also called left-total) if
\forall x\in X,\exists y\in Y,\quad (x,y)\in R.
https://en.m.wikipedia.org/wiki/Function_(mathematics)

fun(x,y) is alternative notation for (x,y) e fun.

LoL

Mostowski Collapse schrieb am Dienstag, 3. Mai 2022 um 18:56:57 UTC+2:
> it follows from f ⊆ {0} x {0}
>
> see here:
>
> a function f: X→Y is a binary relation between X and Y
> https://en.m.wikipedia.org/wiki/Function_(mathematics)
> Dan Christensen schrieb am Dienstag, 3. Mai 2022 um 17:42:14 UTC+2:
> > On Tuesday, May 3, 2022 at 10:10:17 AM UTC-4, Mostowski Collapse wrote:
> > > Its easy to show that it matters. When we know:
> > > fun : {0} -> {0}
> > >
> > > Then we can prove:
> > > ALL(x):ALL(y):[fun(x,y) => x=0 & y=0]
> > >
> > Makes no sense. Not going to try to correct your homework here. Start over.
> > > When we only know:
> > > ALL(a):[a e {0} => fun(a) e {0}]
> > >
> > > Then we cannot prove:
> > > ALL(x):ALL(y):[fun(x,y) => x=0 & y=0]
> >
> > Dan

On Tuesday, 3 May 2022 at 18:56:57 UTC+2, Mostowski Collapse wrote:
> it follows from f ⊆ {0} x {0}
>
> see here:
>
> a function f: X→Y is a binary relation between X and Y
> https://en.m.wikipedia.org/wiki/Function_(mathematics)

.... that associates to each element of X exactly one element of Y, or it is not a function.

A (binary) relation in turn is a subset of the cartesian product of domain and codomain.

Overall: functions is a subclass of relations, and relations is a subclass of cartesian products.

Corollary: there is no such thing (not even) in set theory as a function, and before that a relation, that doesn't come with an explicit domain and codomain attached. Or it is not a function, resp. a relation, it's just a set.

Anyway, my 2c on the overall diatribe.

Julio

Every relation has an intrinsic property:

dom(rel) = { x | EXIST(y) (x,y) e rel }

So do functions as well, which are a special case of relations.

Same with range, rng(rel).

ju...@diegidio.name schrieb am Dienstag, 3. Mai 2022 um 19:10:31 UTC+2:
> On Tuesday, 3 May 2022 at 18:56:57 UTC+2, Mostowski Collapse wrote:
> > it follows from f ⊆ {0} x {0}
> >
> > see here:
> >
> > a function f: X→Y is a binary relation between X and Y
> > https://en.m.wikipedia.org/wiki/Function_(mathematics)
> ... that associates to each element of X exactly one element of Y, or it is not a function.
>
> A (binary) relation in turn is a subset of the cartesian product of domain and codomain.
>
> Overall: functions is a subclass of relations, and relations is a subclass of cartesian products.
>
> Corollary: there is no such thing (not even) in set theory as a function, and before that a relation, that doesn't come with an explicit domain and codomain attached. Or it is not a function, resp. a relation, it's just a set.
>
> Anyway, my 2c on the overall diatribe.
>
> Julio

What clsss/set bssed relations/functions do not have
attached, what is extrinsic, is the codomain, but the
range is also an intrinsic property:

rng(rel) = { y | EXIST(x) (x,y) e rel }

Mostowski Collapse schrieb am Dienstag, 3. Mai 2022 um 19:21:03 UTC+2:
> Every relation has an intrinsic property:
>
> dom(rel) = { x | EXIST(y) (x,y) e rel }
>
> So do functions as well, which are a special case of relations.
>
> Same with range, rng(rel).
> ju...@diegidio.name schrieb am Dienstag, 3. Mai 2022 um 19:10:31 UTC+2:
> > On Tuesday, 3 May 2022 at 18:56:57 UTC+2, Mostowski Collapse wrote:
> > > it follows from f ⊆ {0} x {0}
> > >
> > > see here:
> > >
> > > a function f: X→Y is a binary relation between X and Y
> > > https://en.m.wikipedia.org/wiki/Function_(mathematics)
> > ... that associates to each element of X exactly one element of Y, or it is not a function.
> >
> > A (binary) relation in turn is a subset of the cartesian product of domain and codomain.
> >
> > Overall: functions is a subclass of relations, and relations is a subclass of cartesian products.
> >
> > Corollary: there is no such thing (not even) in set theory as a function, and before that a relation, that doesn't come with an explicit domain and codomain attached. Or it is not a function, resp. a relation, it's just a set.
> >
> > Anyway, my 2c on the overall diatribe.
> >
> > Julio

On Tuesday, May 3, 2022 at 6:54:09 PM UTC+2, Mostowski Collapse wrote:

> if f : {1} -> {1} then f(0) = {} with Fritz application.
>
> Not everything.

Indeed. But I didn't "assume" Fritz application (which I've borrowed from Kelly and/or Gödel) here. Who knows, maybe someone defines

f(x) := the y such that (x, y) e f, if there is exactly one such y; and x otherwise.

:-P

But I know what you mean. Take my claim with a grain of salt.

Do you have an idea what's the matter with Dan?

Drugs? Dementia or just plain crankhhood?

> Feldhase schrieb am Dienstag, 3. Mai 2022 um 18:14:49 UTC+2:
> > On Tuesday, May 3, 2022 at 3:42:20 PM UTC+2, Dan Christensen wrote:
> > > On Tuesday, May 3, 2022 at 4:41:51 AM UTC-4, Fritz Feldhase wrote:
> > > > On Tuesday, May 3, 2022 at 10:00:13 AM UTC+2, Mostowski Collapse wrote:
> > > >
> > > > Here's another simple example.
> > > >
> > > > Consider the identity function id_{0, 1} defined on {0, 1} with id_{0, 1}(x) = x for all x in {0, 1}: the domain of this function is just {0, 1}.
> > > >
> > > > Now:
> > > >
> > > > ALL(a):[a e {0} => id_{0, 1}(a) e {0}]
> > > >
> > > > and
> > > >
> > > > ALL(a):[a e {1} => id_{0, 1}(a) e {1}]
> > > >
> > > > Hence DC would claim that id_{0, 1} is a function with domain {1} and codomain {1} as well as a function with domain {2} and codoman {2}.
> > > >
> > > Fritz [...] believes this is an important distinction, though he hasn't explained why.
> >
> > Holy shit!!!
> >
> > Look, you silly idiot.
> >
> > If we have a function f with domain {0} and codomain {0}, then
> > the function f is "defined" for the argument 0.
> > Moreover f(0) e {0}, hence f(0) = 0.
> > If, on the other hand, we have a function f with domain {1} and codomain {1}, then
> > the function f is NOT "defined" (as we say) for the argument 0.
> > This means that f(0) might be "everything" (i. e. we can't just assume that f(0) e {1} and hence f(0) = 1).
> > It certainly may be the case that these state of affairs doesn't matter in the context of Dan-O-Matik, but it certainly matters in the context of mathematics.
> >
> > Hell, are you on drugs, or what?
> > > (So cute
> >
> > Shut up you demented idiot!

On Tuesday, May 3, 2022 at 7:10:31 PM UTC+2, ju...@diegidio.name wrote:
> On Tuesday, 3 May 2022 at 18:56:57 UTC+2, Mostowski Collapse wrote:
> > it follows from f ⊆ {0} x {0}
> >
> > see here:
> >
> > a function f: X→Y is a binary relation between X and Y
> > https://en.m.wikipedia.org/wiki/Function_(mathematics)
> >
> ... that associates to each element of X exactly one element of Y, or it is not a function.

Indeed. Somtimes this property is referred to as "functional".

> A (binary) relation in turn is a subset of the cartesian product of [two sets].o
>
> Overall: [Set theoretic] functions is a [proper] subclass of relations [....]

Right.

> Corollary: there is no such thing (not even) in set theory as a function, and before that a relation, that doesn't come with an explicit domain and codomain attached.

This claim is "problematic".

Look, The set f := {(0, 0), (1, 1}} is a [set theoretic] function. Since it is a "functional" set of ordered pairs.

Now certainly f c IN x IN, and f c IR x IR etc.

But neither IN nor IR is _the_ domain (and/or _the_ codomain) of f.

Actually, dom(f) = {0, 1} and _the_ codomain of f is simply not "determined".

And no, {0, 1} is not _the_ "codomain" of f but its image (sometimes referred to as "range").

On Tuesday, 3 May 2022 at 20:18:39 UTC+2, Fritz Feldhase wrote:
> On Tuesday, May 3, 2022 at 7:10:31 PM UTC+2, ju...@diegidio.name wrote:
> > On Tuesday, 3 May 2022 at 18:56:57 UTC+2, Mostowski Collapse wrote:
> > > it follows from f ⊆ {0} x {0}
> > >
> > > see here:
> > >
> > > a function f: X→Y is a binary relation between X and Y
> > > https://en.m.wikipedia.org/wiki/Function_(mathematics)
> > >
> > ... that associates to each element of X exactly one element of Y, or it is not a function.
> Indeed. Somtimes this property is referred to as "functional".
>
> > A (binary) relation in turn is a subset of the cartesian product of [two sets].o
> >
> > Overall: [Set theoretic] functions is a [proper] subclass of relations [...]
>
> Right.
>
> > Corollary: there is no such thing (not even) in set theory as a function, and before that a relation, that doesn't come with an explicit domain and codomain attached.
>
> This claim is "problematic".
>
> Look, The set f := {(0, 0), (1, 1}} is a [set theoretic] function.

Not unless you tell from which domain cross codomain,
that was my very point, if you want to call it a function:
indeed, your line of reasoning here (which might have
been Dan's) is just not valid...

> Since it is a "functional" set of ordered pairs.
>
> Now certainly f c IN x IN, and f c IR x IR etc.
>
> But neither IN nor IR is _the_ domain (and/or _the_ codomain) of f.
>
> Actually, dom(f) = {0, 1} and _the_ codomain of f is simply not "determined".
>
> And no, {0, 1} is not _the_ "codomain" of f but its image (sometimes referred to as "range").

Julio

On Tuesday, 3 May 2022 at 20:35:43 UTC+2, Julio Di Egidio wrote:
> On Tuesday, 3 May 2022 at 20:18:39 UTC+2, Fritz Feldhase wrote:
> > On Tuesday, May 3, 2022 at 7:10:31 PM UTC+2, ju...@diegidio.name wrote:

> > Look, The set f := {(0, 0), (1, 1}} is a [set theoretic] function.
>
> Not unless you tell from which domain cross codomain,
> that was my very point, if you want to call it a function:

But you have snipped the most substantial parts, indeed all
the "connections", while rather injecting what was obvious
in the original. So you should rather look at my initial post...

Julio

On Tuesday, May 3, 2022 at 8:35:43 PM UTC+2, ju...@diegidio.name wrote:

> > Look, The set f := {(0, 0), (1, 1}} is a [set theoretic] function.
> >
> Not unless you tell from which domain cross codomain,

Actually, I can only tell you how things are treated in (some, many, most?) standard textbooks.

There a [set theoretic] function is (usually) defined as a "functional" set of ordered pairs.

Just an example:

"A /relation/ is a set of ordered pairs (x, y).

Example: The set {(1,a), (1, b), (2,b), (3,c), (3, a), (4,a)} is a relation.

A /function/ is a relation (so, it is the set of ordered pairs) that does not contain two pairs with the same
first component."

Source: https://faculty.fiu.edu/~wlodarcz/chapter3.pdf

EOD

On Tuesday, 3 May 2022 at 20:55:24 UTC+2, Fritz Feldhase wrote:
> On Tuesday, May 3, 2022 at 8:35:43 PM UTC+2, ju...@diegidio.name wrote:
>
> > > Look, The set f := {(0, 0), (1, 1}} is a [set theoretic] function.
> > >
> > Not unless you tell from which domain cross codomain,ù
>
> Actually, I can only tell you how things are treated in (some, many, most?) standard textbooks.
>
> There a [set theoretic] function is (usually) defined as a "functional" set of ordered pairs.

"A function is a binary relation that is functional and serial", from the link originally given by Jan Burse... I haven't in fact dug into "serial", just to say the textbooks you use and/or how you read them is altogether a different problem: and just yet another fallacy from your side. Indeed, you as Dan, in terms of logic, aren't at all atypical.

Here is the practical case: if I give you an arbitrary set of pairs that is functional, it's still not any function until we fix a domain and codomain.. Terminology quite matters.

HTH and EOD.

Julio