When will this nonsense be fixed?

Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 16:16:29 UTC+2:
> Also dcproof17.exe has the same old bug, its unusable.
> This bug was reported on sci.logic or sci.math in the past.
>
> Try entering an axiom:
>
> ALL(x):ALL(y):[P((x,y)) <=> Q]
>
> It will show you:
>
> 1 ALL(x):ALL(y):[P(x,y) <=> Q]
> Axiom

This here is also broken, you cannot enter an expression:

z = (x,y)

No equality for pairs? Maybe do equality for pairs first,
before you do equality of functions? Here is the screenshot

https://www.rubycap.ch/dcproof17_nopaireq.png

where it is seen that the nonsense tool even doesn't
support equality over pairs. How should a sane person

work with this tool and do the usual math?

LMAO!

Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 18:06:47 UTC+2:
> When will this nonsense be fixed?
> Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 16:16:29 UTC+2:
> > Also dcproof17.exe has the same old bug, its unusable.
> > This bug was reported on sci.logic or sci.math in the past.
> >
> > Try entering an axiom:
> >
> > ALL(x):ALL(y):[P((x,y)) <=> Q]
> >
> > It will show you:
> >
> > 1 ALL(x):ALL(y):[P(x,y) <=> Q]
> > Axiom

On Tuesday, August 23, 2022 at 12:06:47 PM UTC-4, Mostowski Collapse wrote:
> When will this nonsense be fixed?
> Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 16:16:29 UTC+2:
> > Also dcproof17.exe has the same old bug, its unusable.
> > This bug was reported on sci.logic or sci.math in the past.
> >
> > Try entering an axiom:
> >
> > ALL(x):ALL(y):[P((x,y)) <=> Q]
> >
> > It will show you:
> >
> > 1 ALL(x):ALL(y):[P(x,y) <=> Q]
> > Axiom

Not a bug. Redundant brackets are automatically removed by the parser. What are you trying to do?

Dan

On Tuesday, August 23, 2022 at 12:14:38 PM UTC-4, Mostowski Collapse wrote:
> This here is also broken, you cannot enter an expression:
>
> z = (x,y)
>

Not a bug. A conscious design choice. It would introduce too many complications. You might try to introduce a function, e.g. f(x,y)=z. Do you have a particular application in mind?
> No equality for pairs?

Correct. Using the same function f, you could have, for example, f(x,y)=f(a,b).

Dan

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

Well they are not redundant you moron.
What logic does DC Proof implement?

A logic for squirells?

Dan Christensen schrieb am Dienstag, 23. August 2022 um 18:33:29 UTC+2:
> On Tuesday, August 23, 2022 at 12:06:47 PM UTC-4, Mostowski Collapse wrote:
> > When will this nonsense be fixed?
> > Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 16:16:29 UTC+2:
> > > Also dcproof17.exe has the same old bug, its unusable.
> > > This bug was reported on sci.logic or sci.math in the past.
> > >
> > > Try entering an axiom:
> > >
> > > ALL(x):ALL(y):[P((x,y)) <=> Q]
> > >
> > > It will show you:
> > >
> > > 1 ALL(x):ALL(y):[P(x,y) <=> Q]
> > > Axiom
> Not a bug. Redundant brackets are automatically removed by the parser. What are you trying to do?
>
> Dan

On Tuesday, August 23, 2022 at 12:49:19 PM UTC-4, Mostowski Collapse wrote:

> A logic for squirells?
> Dan Christensen schrieb am Dienstag, 23. August 2022 um 18:33:29 UTC+2:
> > On Tuesday, August 23, 2022 at 12:06:47 PM UTC-4, Mostowski Collapse wrote:
> > > When will this nonsense be fixed?
> > > Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 16:16:29 UTC+2:
> > > > Also dcproof17.exe has the same old bug, its unusable.
> > > > This bug was reported on sci.logic or sci.math in the past.
> > > >
> > > > Try entering an axiom:
> > > >
> > > > ALL(x):ALL(y):[P((x,y)) <=> Q]
> > > >
> > > > It will show you:
> > > >
> > > > 1 ALL(x):ALL(y):[P(x,y) <=> Q]
> > > > Axiom
> > Not a bug. Redundant brackets are automatically removed by the parser. What are you trying to do?
> >

> Well they are not redundant

[snip abuse]

If you can think of an important application where defining a function won't work, let me know.

Dan

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

How do you prove this valid theorem in your nonsense tool:

ALL(y):ALL(z):[EXIST(x):[x = (y,z) & P(x)] <=> P((y,z))]

You can even not enter it.

Here is a proof in Wolfgang Schwartz tree tool, I use
p(_,_) for pair (_,_), because he currently doesn't have pairs
syntax, but this doesn't matter, the theorem is valid:

∀y∀z(∃x(x=p(y,z) ∧ Px) ↔ Pp(y,z)) is valid.
https://www.umsu.de/trees/#~6y~6z%28~7x%28x=p%28y,z%29~1P%28x%29%29~4P%28p%28y,z%29%29%29

Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 18:49:19 UTC+2:
> Well they are not redundant you moron.
> What logic does DC Proof implement?
>
> A logic for squirells?
> Dan Christensen schrieb am Dienstag, 23. August 2022 um 18:33:29 UTC+2:
> > On Tuesday, August 23, 2022 at 12:06:47 PM UTC-4, Mostowski Collapse wrote:
> > > When will this nonsense be fixed?
> > > Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 16:16:29 UTC+2:
> > > > Also dcproof17.exe has the same old bug, its unusable.
> > > > This bug was reported on sci.logic or sci.math in the past.
> > > >
> > > > Try entering an axiom:
> > > >
> > > > ALL(x):ALL(y):[P((x,y)) <=> Q]
> > > >
> > > > It will show you:
> > > >
> > > > 1 ALL(x):ALL(y):[P(x,y) <=> Q]
> > > > Axiom
> > Not a bug. Redundant brackets are automatically removed by the parser. What are you trying to do?
> >
> > Dan

Two bugs in one row, Bug 1 and Bug 2, so one cannot do Bourbaki:

f = (g, a, b)

Even little Bourbaki does not work:

f = (g, c)

g = function graph set of pairs, has domain intrinsic
c = co-domain, further component, because extrinsic

Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 19:02:22 UTC+2:
> How do you prove this valid theorem in your nonsense tool:
>
> ALL(y):ALL(z):[EXIST(x):[x = (y,z) & P(x)] <=> P((y,z))]
>
> You can even not enter it.
>
> Here is a proof in Wolfgang Schwartz tree tool, I use
> p(_,_) for pair (_,_), because he currently doesn't have pairs
> syntax, but this doesn't matter, the theorem is valid:
>
> ∀y∀z(∃x(x=p(y,z) ∧ Px) ↔ Pp(y,z)) is valid.
> https://www.umsu.de/trees/#~6y~6z%28~7x%28x=p%28y,z%29~1P%28x%29%29~4P%28p%28y,z%29%29%29
> Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 18:49:19 UTC+2:
> > Well they are not redundant you moron.
> > What logic does DC Proof implement?
> >
> > A logic for squirells?
> > Dan Christensen schrieb am Dienstag, 23. August 2022 um 18:33:29 UTC+2:
> > > On Tuesday, August 23, 2022 at 12:06:47 PM UTC-4, Mostowski Collapse wrote:
> > > > When will this nonsense be fixed?
> > > > Mostowski Collapse schrieb am Dienstag, 23. August 2022 um 16:16:29 UTC+2:
> > > > > Also dcproof17.exe has the same old bug, its unusable.
> > > > > This bug was reported on sci.logic or sci.math in the past.
> > > > >
> > > > > Try entering an axiom:
> > > > >
> > > > > ALL(x):ALL(y):[P((x,y)) <=> Q]
> > > > >
> > > > > It will show you:
> > > > >
> > > > > 1 ALL(x):ALL(y):[P(x,y) <=> Q]
> > > > > Axiom
> > > Not a bug. Redundant brackets are automatically removed by the parser.. What are you trying to do?
> > >
> > > Dan

On Tuesday, August 23, 2022 at 1:02:22 PM UTC-4, Mostowski Collapse wrote:
> How do you prove this valid theorem in your nonsense tool:
>
> ALL(y):ALL(z):[EXIST(x):[x = (y,z) & P(x)] <=> P((y,z))]
>

Another quirk of FOL???

You will never see P((y,z)) in math text books. It would only lead to confusion. Why should I incorporate in DC Proof?

Dan

On Tuesday, August 23, 2022 at 8:26:06 PM UTC-4, Mostowski Collapse wrote:
> Two bugs in one row, Bug 1 and Bug 2, so one cannot do Bourbaki:
>
> f = (g, a, b)

Lacking somewhat in detail!

>

Here is my function axiom based on Terence Tao's definition ("Analysis I", p.49)

1. ALL(dom):ALL(cod):ALL(gra):[Set(dom) & Set(cod) & Set'(gra)

=> [ALL(a1):ALL(b):[(a1,b) in gra => a1 in dom & b in cod]

& ALL(a1):[a1 in dom => EXIST(b):[b in cod & (a1,b) in gra]]

& ALL(a1):ALL(b1):ALL(b2):[a1 in dom & b1 in cod & b2 in cod
=> [(a1,b1) in gra & (a1,b2) in gra => b1=b2]]

=> EXIST(fun):[Function(fun,dom,cod)

& ALL(a1):ALL(b):[a1 in dom & b in cod => [fun(a1)=b <=> (a1,b) in gra]]]]]

Function

Where
dom = domain set
cod = codomain set
gra = graph set
fun = function name

Dan

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

On Wednesday, August 24, 2022 at 4:54:28 AM UTC+2, Dan Christensen wrote:

> You will never see P((y,z)) in math text books. It would only lead to confusion.

No, it wouldn't, you silly idiot!

> Why should I incorporate in DC Proof?

Because it is "necessary" from a formal point of view.

Hint: We may define

ordered_pair(z) :<-> ExEy(z = (x,y)).

Now if, say,

z0 := (1, 2) (*)

Then it should be able to prove

ordered_pair(z0)

Moreover "identity theory" should allow to derive/conclude

ordered_pair((1, 2)). (After all z0 = (1, 2) from (*).)

Which FROM A FORMAL POINT OF VIEW clearly is NOT the same as

ordered_pair(1, 2).

Hint: ordered_pair(.) is an UNARY predicate, in some systems even written as ordered_pair^1(.), while ordered_pair(., .) would be a BINARY predicate, in some systems even written as ordered_pair^2(., .). Now neither "ordered_pair^1(x, y)" would be a wff, nor "ordered_pair^2(z)". Hence in a formal system it should not be allowed just to write ordered_pair(1, 2) instead of ordered_pair((1, 2)) (and vice versa).

Moreover, from a semantic point of view, how would you express that, say, the set z0 = (1, 2) is an ordered pair without an UNARY predicate "ordered_pair(.)"?

On Tuesday, August 23, 2022 at 11:41:33 PM UTC-4, Fritz Feldhase wrote:
> On Wednesday, August 24, 2022 at 4:54:28 AM UTC+2, Dan Christensen wrote:
>
> > You will never see P((y,z)) in math text books. It would only lead to confusion.

> No, it wouldn't [snip abuse]

> > Why should I incorporate in DC Proof?

> Because it is "necessary" from a formal point of view.
>

Debatable. I can see maybe P( { (y,z) } ).

Dan

> Hint: We may define
>
> ordered_pair(z) :<-> ExEy(z = (x,y)).
>
> Now if, say,
>
> z0 := (1, 2) (*)
>
> Then it should be able to prove
>
> ordered_pair(z0)
>
> Moreover "identity theory" should allow to derive/conclude
>
> ordered_pair((1, 2)). (After all z0 = (1, 2) from (*).)
>
> Which FROM A FORMAL POINT OF VIEW clearly is NOT the same as
>
> ordered_pair(1, 2).
>
> Hint: ordered_pair(.) is an UNARY predicate, in some systems even written as ordered_pair^1(.), while ordered_pair(., .) would be a BINARY predicate, in some systems even written as ordered_pair^2(., .). Now neither "ordered_pair^1(x, y)" would be a wff, nor "ordered_pair^2(z)". Hence in a formal system it should not be allowed just to write ordered_pair(1, 2) instead of ordered_pair((1, 2)) (and vice versa).
>
> Moreover, from a semantic point of view, how would you express that, say, the set z0 = (1, 2) is an ordered pair without an UNARY predicate "ordered_pair(.)"?

On Wednesday, August 24, 2022 at 5:03:46 AM UTC+2, Dan Christensen wrote:
> On Tuesday, August 23, 2022 at 8:26:06 PM UTC-4, Mostowski Collapse wrote:
> > Two bugs in one row, Bug 1 and Bug 2, so one cannot do Bourbaki:
> >
> > f = (g, a, b)
> >
> Lacking somewhat in detail!

So what? We already told you the details.

Hint: If f is a function then f = (graph(f), dom(f), cod(f)).

> Here is my function axiom based on Terence Tao's definition ("Analysis I", p.49)
>
> 1. ALL(dom):ALL(cod):ALL(gra):[Set(dom) & Set(cod) & Set'(gra)
>
> => [ALL(a1):ALL(b):[(a1,b) in gra => a1 in dom & b in cod]
>
> & ALL(a1):[a1 in dom => EXIST(b):[b in cod & (a1,b) in gra]]
>
> & ALL(a1):ALL(b1):ALL(b2):[a1 in dom & b1 in cod & b2 in cod
> => [(a1,b1) in gra & (a1,b2) in gra => b1=b2]]
>
> => EXIST(fun):[Function(fun,dom,cod)
>
> & ALL(a1):ALL(b):[a1 in dom & b in cod => [fun(a1)=b <=> (a1,b) in gra]]]]]

Funny thing, you allow for gra c dom x cod (i. e. a set of pairs), you even use the notation of an ordered pair, namely (., .), but you won't define a function as a certain triple, namely as (gra, dom, cod). :-)

Moreover you have to use two undefined primitives, namely the ternary predicate "Function(., ., .)" and the binary "operation" (the binary FOPL function) ".(.)". (No problem from a logical point of view, of course.)

This can be avoided if using the "set theoretic" approach.

On Wednesday, August 24, 2022 at 5:46:04 AM UTC+2, Dan Christensen wrote:
> On Tuesday, August 23, 2022 at 11:41:33 PM UTC-4, Fritz Feldhase wrote:
> > On Wednesday, August 24, 2022 at 4:54:28 AM UTC+2, Dan Christensen wrote:
> > >
> > > Why should I incorporate in DC Proof?
> > >
> > Because it is "necessary" from a formal point of view.
> >
> Debatable. I can see maybe P( { (y,z) } ).

Huh?! WTF?!

Ok, if you don't like the double usage of "(" and ")" here, why not use "<" and ">" for denoting ordered pairs?

Actually, it's quite common to write <x, y> instead of (a, b). This might even avoid "confusion". No?

Hence we would have, say, ordered_pair(<x, y>) or first_component(<x, y>) (pi_1(<x, y>), etc.

Might indeed be a good idea in the context of DC Proof!

Oh boy, did you really ask about use cases for pairs as first
class citizens when you wrote this?

Dan Christensen schrieb am Mittwoch, 24. August 2022 um 01:41:30 UTC+2:
> Maybe a silly question, but why we would it ever be necessary to formally
> equate a single object to an ordered list of two or more objects. What
> important insights would be missed if we never allowed such constructs?
https://groups.google.com/g/sci.logic/c/xjV--M0JhXE/m/eGZIviRhBQAJ

Did you ever do these constructions:

1) Z from N, via equivalence class over NxN, we would have (2,5) ~ -3
Can you do that in DC Proof?

2) Q from Z, via equivalence class over Zx(N\{0}), we would have (3,6) ~1/2
Can you do that in DC Proof?

Dan Christensen schrieb am Mittwoch, 24. August 2022 um 05:46:04 UTC+2:
> On Tuesday, August 23, 2022 at 11:41:33 PM UTC-4, Fritz Feldhase wrote:
> > On Wednesday, August 24, 2022 at 4:54:28 AM UTC+2, Dan Christensen wrote:
> >
> > > You will never see P((y,z)) in math text books. It would only lead to confusion.
> > No, it wouldn't [snip abuse]
> > > Why should I incorporate in DC Proof?
>
> > Because it is "necessary" from a formal point of view.
> >
> Debatable. I can see maybe P( { (y,z) } ).
>
> Dan
> > Hint: We may define
> >
> > ordered_pair(z) :<-> ExEy(z = (x,y)).
> >
> > Now if, say,
> >
> > z0 := (1, 2) (*)
> >
> > Then it should be able to prove
> >
> > ordered_pair(z0)
> >
> > Moreover "identity theory" should allow to derive/conclude
> >
> > ordered_pair((1, 2)). (After all z0 = (1, 2) from (*).)
> >
> > Which FROM A FORMAL POINT OF VIEW clearly is NOT the same as
> >
> > ordered_pair(1, 2).
> >
> > Hint: ordered_pair(.) is an UNARY predicate, in some systems even written as ordered_pair^1(.), while ordered_pair(., .) would be a BINARY predicate, in some systems even written as ordered_pair^2(., .). Now neither "ordered_pair^1(x, y)" would be a wff, nor "ordered_pair^2(z)". Hence in a formal system it should not be allowed just to write ordered_pair(1, 2) instead of ordered_pair((1, 2)) (and vice versa).
> >
> > Moreover, from a semantic point of view, how would you express that, say, the set z0 = (1, 2) is an ordered pair without an UNARY predicate "ordered_pair(.)"?

On Wednesday, August 24, 2022 at 12:19:14 AM UTC-4, Fritz Feldhase wrote:
> On Wednesday, August 24, 2022 at 5:46:04 AM UTC+2, Dan Christensen wrote:
> > On Tuesday, August 23, 2022 at 11:41:33 PM UTC-4, Fritz Feldhase wrote:
> > > On Wednesday, August 24, 2022 at 4:54:28 AM UTC+2, Dan Christensen wrote:
> > > >
> > > > Why should I incorporate in DC Proof?
> > > >
> > > Because it is "necessary" from a formal point of view.
> > >
> > Debatable. I can see maybe P( { (y,z) } ).
> Huh?! WTF?!
>
> Ok, if you don't like the double usage of "(" and ")" here, why not use "<" and ">" for denoting ordered pairs?
>
> Actually, it's quite common to write <x, y> instead of (a, b). This might even avoid "confusion". No?
>

Instead of <x, y>, consider the natural bijection f(x,y) = {(x,y)}. Working on a formal existence proof.

Dan

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

On Thursday, August 25, 2022 at 4:16:19 AM UTC+2, Dan Christensen wrote:

> Instead of <x, y>, consider the natural bijection f(x,y) = {(x,y)}. Working on a formal existence proof.

Again, it seems that to a certain degree a function g(x, y) can work as a "stand in" for f((x, y)).

Actually, it's quite common in math to mention a function f: X x Y ---> Z, but to write f(x, y) = ... with x e X and y e Y instead of f(z) = ... with z e X x Y.

Of course, from a FORMAL point of view this is problematic, I'd say. Hmmm.... The only solution I can see is to "interpret" the expression f(x, y) (almost) as if we had written f((x, y)). And right, there is a joke: "A set-theorist is a person for whom all functions are unary". EXACTLY! :-)

See: https://math.stackexchange.com/questions/2339024/a-set-theorist-is-a-person-for-whom-all-functions-are-unary

But there are still some serious quirks involved when following this "practice" in a formal setting, I'd say.

______________________________________________________________________

Hint: That approach will NOT work for PREDICATES, after all things like

ordered_pair(p) :<-> ExEy(p = (x, y)) should be possible.

As well as things like

~((1, 1) e {{1, 0), (0, 1)) ,

(1, 1) = (1, 1)

AxAy((x, y) = (1, 1) -> ...)

etc.

Actually, ordered pairs are important "objects" (though just certain sets) in set theory. There we actually can (and usually do) define them the following way:

(x, y) := {{x}, {x, y}} .

Hence DC Proof should allow for "(., .)" as "term expressions".

Maybe just banning them from appearing as term in an expression of the form "f(term)".

On the other hand, it SHOULD be allowed to define, say, a function add: IR^2 x IR^2 with add(x, y) = x + y for all x e IR^2 and y e IR^2.

Then it shoukd be possible to write, say, add((1, 0), (0, 1)) = (1, 1).

Btw. when using infix notations for certain functions, we already follow the "common practice." For example we may define a function on IN the follwowing way: +: IN x IN --> IN. But instead of +((x, y)), or +(x, y) we usually just write x + y (!). Here clearly the "ordered pair" has "vanished" - at least when looking at the expression x + y. :-P (On the other hand, we still maintain that we will not necessarilly have x + y = y + x for all x, y e IN.)

Just some thoughts.

On Thursday, August 25, 2022 at 6:27:50 AM UTC+2, Fritz Feldhase wrote:

> Actually, it's quite common in math to mention a function f: X x Y ---> Z, but to write f(x, y) = ... with x e X and y e Y instead of f(z) = ... with z e X x Y.
>
> Of course, from a FORMAL point of view this is problematic, I'd say. Hmmm.... The only "solution" I can see is to "interpret" the expression f(x, y) (almost) as if we had written f((x, y)). And right, there is a joke: "A set-theorist is a person for whom all functions are unary". EXACTLY! :-)
>
> See: https://math.stackexchange.com/questions/2339024/a-set-theorist-is-a-person-for-whom-all-functions-are-unary
>
> But there are still some serious quirks involved when following this "practice" in a formal setting, I'd say.

In the context of set theory we might define the expressions

f(x)
f(x, y)

by, say,

f(x) := U{y : <x, y> e f}

and

f(x, y) := U{z : <<x, y>, z> e f}

Where .(.) is a binary operator (of FOPL) and .(., .) is a ternary operator..

Hence, no ordered-pair term involved in these expressions. Hence no problem with substition. I mean, "(x, y)" in "f(x, y)" isn't an ordered- pair term in this case.

Of course, you may treat these operators as undefined primitives too.

Important point: here "f(x, y)" is not just some "abbreviation" for "f((x, y))".

Dan Christensen wrote:

>> Actually, it's quite common to write <x, y> instead of (a, b). This
>> might even avoid "confusion". No?
>>
> Instead of <x, y>, consider the natural bijection f(x,y) = {(x,y)}.
> Working on a formal existence proof.

nothing. Here you have it, two nazis, both khazars. That's why the Bible
says you may not have them at your table. They are goys.

HILARIOUS： Look at Boris Johnson did before he shook Zelensky's hand
https://www.bitchute.com/video/9hKviSIfY5uq/

lol, those "himars" are a fucking joke.

Russian Air Defense shoots down HIMARS headed towards power station and
debris starts EXPLODING https://www.bitchute.com/video/izfTsCHZmsYl/

No progress so far, concerning Dan Christense "needlessly cumbersome" ?`
Roy Dickhoff R.I.P. corrected a paper because he used the word "clumsy":

* CONTRACTION-FREE SEQUENT CALCULI FOR INTUITIONISTIC LOGIC:
* A CORRECTION - ROY DYCKHOFF
* The Journal of Symbolic Logic
* Volume 83, Number 4, December 2018
* https://core.ac.uk/download/pdf/162928594.pdf

He wrote:

It follows that G4ip and G3ip are equivalent. This proof replaces that in Section
3 of [1]. It avoids complicated definitions of otherwise useless concepts like
“awkward”, “clumsy” and “sensible”, and the uses of these definitions.

LoL

Fritz Feldhase schrieb am Donnerstag, 25. August 2022 um 06:43:06 UTC+2:
> On Thursday, August 25, 2022 at 6:27:50 AM UTC+2, Fritz Feldhase wrote:
>
> > Actually, it's quite common in math to mention a function f: X x Y ---> Z, but to write f(x, y) = ... with x e X and y e Y instead of f(z) = .... with z e X x Y.
> >
> > Of course, from a FORMAL point of view this is problematic, I'd say. Hmmm... The only "solution" I can see is to "interpret" the expression f(x, y) (almost) as if we had written f((x, y)). And right, there is a joke: "A set-theorist is a person for whom all functions are unary". EXACTLY! :-)
> >
> > See: https://math.stackexchange.com/questions/2339024/a-set-theorist-is-a-person-for-whom-all-functions-are-unary
> >
> > But there are still some serious quirks involved when following this "practice" in a formal setting, I'd say.
> In the context of set theory we might define the expressions
>
> f(x)
> f(x, y)
>
> by, say,
>
> f(x) := U{y : <x, y> e f}
>
> and
>
> f(x, y) := U{z : <<x, y>, z> e f}
>
> Where .(.) is a binary operator (of FOPL) and .(., .) is a ternary operator.
>
> Hence, no ordered-pair term involved in these expressions. Hence no problem with substition. I mean, "(x, y)" in "f(x, y)" isn't an ordered- pair term in this case.
>
> Of course, you may treat these operators as undefined primitives too.
>
> Important point: here "f(x, y)" is not just some "abbreviation" for "f((x, y))".

On 8/25/2022 1:19 PM, Hug Acquarone wrote:

> lol, those "himars" are a fucking joke.

The real joke is watching how the Nazi 卐Ru⚡︎⚡︎ian卐 soldiers WHINE when
HIMARS does its job and the bridges and ammo dumps go BOOM and they
can't get supplies! Did you see the video of a 卐Ru⚡︎⚡︎ian卐 soldier
videoing an exploding ammo dump when a large explosion blew out the
window he was videoing through?
>
> Russian Air Defense shoots down HIMARS

They stopped the missile with 1000 卐Ru⚡︎⚡︎ian卐 missiles in an ammo
dump! That will teach Ukraine!

> headed towards power station

Tell us again, why are the 卐Ru⚡︎⚡︎ians卐 shelling that nuke site again?
Don't they know that radiation doesn't respect borders, not even
imaginary 卐Ru⚡︎⚡︎ian卐 borders?

> and debris starts EXPLODING

It's so enjoyable to watch those 卐Ru⚡︎⚡︎ian卐 ammo dumps go BOOM!
Especially at night! 卐Ru⚡︎⚡︎ian卐 ammo dumps make such awesome
fireworks shows!

Michael Moroney wrote:

> On 8/25/2022 1:19 PM, Hug Acquarone wrote:
>
>> lol, those "himars" are a fucking joke.
>
> The real joke is watching how the Nazi 卐Ru⚡︎⚡︎ian卐 soldiers WHINE when
> HIMARS does its job and the bridges and ammo dumps go BOOM and they
> can't

how come "the russians are nazis" when they fight against the nazis, to remove them from the face of the earth, merely from the capitalist khazar west. You still don't know what nazis are, after so many wars and sufferings. Here's what the nazis of "uKraine" did to the polish people, hitler time. Don't delete the link, you fucking stupid.

https://en.wikipedia.org/wiki/Massacres_of_Poles_in_Volhynia_and_Eastern_Galicia#Atrocities

Atrocities

Attacks on Poles during the massacres in Volhynia and Eastern Galicia were marked with extreme sadism and brutality. Rape, torture and mutilation were commonplace. Poles were burned alive, flayed, impaled, crucified, disembowelled, dismembered and beheaded. Women were gang raped and had their breasts sliced off, children were hacked to pieces with axes, babies were impaled on bayonets and pitchforks or bashed against trees.

The atrocities were carried out indiscriminately and without restraint. The victims, regardless of their age or gender, were routinely tortured to death. Norman Davies in No Simple Victory gives a short but shocking description of the massacres:

Villages were torched. Roman Catholic priests were axed or crucified. Churches were burned with all their parishioners. Isolated farms were attacked by gangs carrying pitchforks and kitchen knives. Throats were cut. Pregnant women were bayoneted. Children were cut in two. Men were ambushed in the field and led away. The perpetrators could not determine the province's future. But at least they could determine that it would be a future without Poles.[117]

On 9/1/2022 3:30 PM, Damen Serpico wrote:
> Michael Moroney wrote:
>
>> On 8/25/2022 1:19 PM, Hug Acquarone wrote:
>>
>>> lol, those "himars" are a fucking joke.
>>
>> The real joke is watching how the Nazi 卐Ru⚡︎⚡︎ian卐 soldiers WHINE when
>> HIMARS does its job and the bridges and ammo dumps go BOOM and they
>> can't
>
> how come "the russians are nazis"

Because 卐Ru⚡︎⚡︎ia卐 uses techniques straight out of Hitler's playbook!
First they launch an "Anschluss" and steal Crimea and march into the
Sudetenland a.k.a. Donbass, purportedly to protect the German, I mean
Russian speaking population. In February all the troops near the
Ukrainian border were just there for exercises, no invasion planned,
until the invasion started anyway. And in that invasion, 卐Ru⚡︎⚡︎ia卐
kills more Russian speakers in Mariupol alone (90% Russian speakers!)
than they claim Ukraine killed in Donbass. BTW if a Russian calls it an
invasion or war, they risk 15 years in the Gulag. (see Joseph Goebbels
and the "Big Lie") Or perhaps Novichok or polonium-flavored tea if
they're too powerful/dangerous.

> when they fight against the nazis, to remove them from the face of the earth,

No, 卐Ru⚡︎⚡︎ia卐 launched an attack against a peaceful neighbor. The
REAL Nazis were defeated in 1945, those still alive are in their 90s
now. All countries have their radical right, Ukraine doesn't have any
more than anyone else. (remember their leader is Jewish who lost family
members to the REAL Nazis!).

> You still don't know what nazis are, after so many wars and sufferings.

It's not modern Ukraine! But 卐Ru⚡︎⚡︎ia卐 is now a fascist dictatorship,
led my an insane man with delusions of restoring the Soviet Union.

> Here's what the nazis of "uKraine" did to the polish people, hitler time.

The Soviet Union was responsible for all sorts of atrocities before,
during and after WW2. As parts of the Soviet Union, both Russia and
Ukraine have responsibility.

> Don't delete the link, you fucking stupid.

I cannot and will not delete your post.

Now tell us what the Russians (as part of the Soviet Union) did to
Germans in Koenigsburg and Russia proper, the Taters in Crimea, Poles
outside the new borders of Poland etc. etc.

Michael Moroney wrote:

>> Don't delete the link, you fucking stupid.
>
> I cannot and will not delete your post.

you stinking sack of dirt. Put the link to the scientific data and the fossil records back, you idiot. Read the data, fucking retard, it has nothing to do with the "soviet union". Your country is a shithole. You are a criminal bombing kenseto with two atomic bombs.

https://en.wikipedia.org/wiki/Massacres_of_Poles_in_Volhynia_and_Eastern_Galicia#Atrocities

Atrocities

Attacks on Poles during the massacres in Volhynia and Eastern Galicia were
marked with extreme sadism and brutality. Rape, torture and mutilation
were commonplace. Poles were burned alive, flayed, impaled, crucified,
disembowelled, dismembered and beheaded. Women were gang raped and had
their breasts sliced off, children were hacked to pieces with axes, babies
were impaled on bayonets and pitchforks or bashed against trees.

The atrocities were carried out indiscriminately and without restraint.
The victims, regardless of their age or gender, were routinely tortured to
death. Norman Davies in No Simple Victory gives a short but shocking
description of the massacres:

Villages were torched. Roman Catholic priests were axed or crucified.
Churches were burned with all their parishioners. Isolated farms were
attacked by gangs carrying pitchforks and kitchen knives. Throats were
cut. Pregnant women were bayoneted. Children were cut in two. Men were
ambushed in the field and led away. The perpetrators could not
determine the province's future. But at least they could determine
that it would be a future without Poles.[117]

On 9/1/2022 5:58 PM, Dana Belluomi wrote:
> Michael Moroney wrote:
>
>>> Don't delete the link, you fucking stupid.
>>
>> I cannot and will not delete your post.
>
> you stinking sack of dirt. Put the link to the scientific data and the fossil records back, you idiot. Read the data, fucking retard, it has nothing to do with the "soviet union". Your country is a shithole. You are a criminal bombing kenseto with two atomic bombs.

No comment on why the only Nazis around appear to be the 卐Ru⚡︎⚡︎ians卐?
>
> https://en.wikipedia.org/

Wikipedia as a reference, nymshifter? Surely you can do better!

> Atrocities

The Soviet Union sure was responsible for plenty of atrocities!
They don't get much attention because the Soviet Union helped beat the
real Nazis, and the Soviets got away with a lot.

Meanwhile, Ukraine has come a long way since its Soviet days, especially
since they got rid of their 卐Ru⚡︎⚡︎ian卐 puppet. Meanwhile, 卐Ru⚡︎⚡︎ia
卐 has regressed into becoming not much more than a third world country
with nukes, ever since 卐Владольф Путлер卐 came along with his wet dream
of creating the Soviet Union V2.0.