On Wednesday, March 2, 2022 at 10:58:31 AM UTC-5, Fritz Feldhase wrote:
> On Wednesday, March 2, 2022 at 3:22:08 PM UTC+1, Dan Christensen wrote:
>
> > As I have shown, there are infinitely many binary functions on N that satisfy
> >
> > 1. x^2 = x*x
> > 2. x^(y+1) = x^y * x

> But we are not interested in these infinitely many variants of "power".
>
> Actually, we would like to define ^ (on IN) such that it is total. (Just like + and *.)

If you don't mind arithmetic on N and on R being inconsistent with each other, you could pick any value for 0^0 that you find convenient.

Dan

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

You dont even have a definition of domain in DC Proof.

Dan Christensen schrieb am Mittwoch, 2. März 2022 um 15:08:06 UTC+1:
> On Wednesday, March 2, 2022 at 8:22:44 AM UTC-5, Mostowski Collapse wrote:
>
> > Dan Christensen schrieb am Dienstag, 1. März 2022 um 15:25:15 UTC+1:
> > > What is 0^0? Your teachers in school probably told you it was undefined on the natural numbers. Among mathematicians, however, the value of 0^0 depends on the context. 0^0 could be undefined, or it could be 1. It would be undefined if you are talking about 0 as a real number. And it would have a value of 1, as a handy shortcut if you are talking about 0 as a natural number for certain applications, e.g. the binomial theorem.
> > >
> > > I am a bit troubled by this inconsistency. There are, after all, no inconsistencies between real and natural number arithmetic other than in the case of 0^0. In my very humble opinion, your teachers were right, but they probably didn’t give you the full story. Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers, as it is on the real numbers.
> > >
> > > Full text: https://www.dcproof.com/ZeroExponentsOnN.htm
> > >
> > Where are the specifications and proofs?
> If you want formal proofs, see my blog posting at https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> > If 0^0 were really undefined in DC Proof, then
> > you could prove that this is undefined as well:
> >
> > forall x (x e R => 0^(x-x) is undefined)
> >
> To prove, for example, that f(x) is undefined, you simply have to prove that x is not in the domain of f.
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Fritz Feldhase <franz.fritschee.ff@gmail.com> wrote:
> On Wednesday, March 2, 2022 at 10:31:48 AM UTC+1, Alan Mackenzie wrote:

>> Please don't do this again.

> What can I say? Fuck you?

No. You could apologise for uncalled for abuse. Otherwise, you're just
one of the many, many people who have caused Usenet to degrade from the
useful discussion forum it was twenty years ago to the sewer it is
today.

--
Alan Mackenzie (Nuremberg, Germany).

On Wednesday, March 2, 2022 at 5:18:06 PM UTC+1, Alan Mackenzie wrote:
> Fritz Feldhase <franz.fri...@gmail.com> wrote:
> >
> > What can I say? Fuck you?
> >
> No. You could apologise for uncalled for abuse. Otherwise, you're just
> one of the many, many people who have caused Usenet to degrade from the
> useful discussion forum it was twenty years ago to the sewer it is
> today.

I appreciate your opinion. On the other hand, imho Dan is rather stubborn (and henc effectively /dense/) sometimes.

On Wednesday, March 2, 2022 at 5:06:47 PM UTC+1, Dan Christensen wrote:
> On Wednesday, March 2, 2022 at 10:58:31 AM UTC-5, Fritz Feldhase wrote:
>
> [...] we are not interested in these infinitely many variants of "power".
> >
> > Actually, we would like to define ^ (on IN) such that it is total. (Just like + and *.)
> If you don't mind arithmetic on N and on R being inconsistent with each other, you could pick any value for 0^0 that you find convenient.

They aren't "inconsistent".

We may extend "^" for IR in a way such that it is identical with "^" on IN c IR.

| [...] Knuth goes on to write that the debate ended with the result that 0^0 should be undefined, and then he states, "No, no, ten thousand times no!"

Agree. :-)

| In keeping with the honored pedagogical technique of "First tell 'em what you are going to tell 'em, then tell 'em, then tell 'em what you told 'em," we summarize.
| If you are dealing with limits, then 0^0 is an indeterminate form, but if you are dealing with ordinary algebra, then 0^0 = 1.

Source: https://www.maa.org/book/export/html/116806

Fritz Feldhase <franz.fritschee.ff@gmail.com> wrote:
> On Wednesday, March 2, 2022 at 5:18:06 PM UTC+1, Alan Mackenzie wrote:
>> Fritz Feldhase <franz.fri...@gmail.com> wrote:

>> > What can I say? Fuck you?

>> No. You could apologise for uncalled for abuse. Otherwise, you're just
>> one of the many, many people who have caused Usenet to degrade from
>> the useful discussion forum it was twenty years ago to the sewer it is
>> today.

> I appreciate your opinion. On the other hand, imho Dan is rather
> stubborn (and henc effectively /dense/) sometimes.

Thanks! The word "dense" in English implies stupidity. My Collins
Dictionary gives "beschränkt", "unterbelichtet", and "blöd" as German
equivalents.

For what it's worth, my view is that 0^0 is undefined, for reasons given
by others here. It can only be firmed up by a process of limits, and the
answer you get varies according to what expression you take the limit of.

Still, Dan has actually produced something, DC Proof, for which respect
is due. I doubt many others here have done anything similar.

--
Alan Mackenzie (Nuremberg, Germany).

Alan Mackenzie explained :
> Fritz Feldhase <franz.fritschee.ff@gmail.com> wrote:
>> On Wednesday, March 2, 2022 at 5:18:06 PM UTC+1, Alan Mackenzie wrote:
>>> Fritz Feldhase <franz.fri...@gmail.com> wrote:
>
>>>> What can I say? Fuck you?
>
>>> No. You could apologise for uncalled for abuse. Otherwise, you're just
>>> one of the many, many people who have caused Usenet to degrade from
>>> the useful discussion forum it was twenty years ago to the sewer it is
>>> today.
>
>> I appreciate your opinion. On the other hand, imho Dan is rather
>> stubborn (and henc effectively /dense/) sometimes.
>
> Thanks! The word "dense" in English implies stupidity. My Collins
> Dictionary gives "beschränkt", "unterbelichtet", and "blöd" as German
> equivalents.
>
> For what it's worth, my view is that 0^0 is undefined, for reasons given
> by others here. It can only be firmed up by a process of limits, and the
> answer you get varies according to what expression you take the limit of.
>
> Still, Dan has actually produced something, DC Proof, for which respect
> is due. I doubt many others here have done anything similar.

I agree, he even explained that it is a teaching tool he uses to help
convey some of the ideas in set theory. He's not claiming to change the
math world with a new foundation.

On Wednesday, March 2, 2022 at 11:12:43 AM UTC-5, Mostowski Collapse (aka Jan Burse) wrote:

> Dan Christensen schrieb am Mittwoch, 2. März 2022 um 15:08:06 UTC+1:
> > On Wednesday, March 2, 2022 at 8:22:44 AM UTC-5, Mostowski Collapse wrote:
> >
> > > Dan Christensen schrieb am Dienstag, 1. März 2022 um 15:25:15 UTC+1:
> > > > What is 0^0? Your teachers in school probably told you it was undefined on the natural numbers. Among mathematicians, however, the value of 0^0 depends on the context. 0^0 could be undefined, or it could be 1. It would be undefined if you are talking about 0 as a real number. And it would have a value of 1, as a handy shortcut if you are talking about 0 as a natural number for certain applications, e.g. the binomial theorem.
> > > >
> > > > I am a bit troubled by this inconsistency. There are, after all, no inconsistencies between real and natural number arithmetic other than in the case of 0^0. In my very humble opinion, your teachers were right, but they probably didn’t give you the full story. Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers, as it is on the real numbers.
> > > >
> > > > Full text: https://www.dcproof.com/ZeroExponentsOnN.htm
> > > >
> > > Where are the specifications and proofs?
> > If you want formal proofs, see my blog posting at https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> > > If 0^0 were really undefined in DC Proof, then
> > > you could prove that this is undefined as well:
> > >
> > > forall x (x e R => 0^(x-x) is undefined)
> > >
> > To prove, for example, that f(x) is undefined, you simply have to prove that x is not in the domain of f.

> You dont even have a definition of domain in DC Proof.

Wrong again, Jan Burse. On the main menu of DC Proof, to obtain a menu of its axioms of set theory, click on Sets. To invoke the Function Axiom and obtain the definition of a function, select the Function option. You will then be prompted for the number of variables. If you enter 1, the following statement will appear on the main screen:

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

Here, we have

dom = domain of the function
cod = codomain of the function
gra = graph of the function

I hope this helps.

Dan

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

onsdag 2. mars 2022 kl. 22:03:34 UTC+1 skrev Alan Mackenzie:
> Fritz Feldhase <franz.fri...@gmail.com> wrote:
> > On Wednesday, March 2, 2022 at 5:18:06 PM UTC+1, Alan Mackenzie wrote:
> >> Fritz Feldhase <franz.fri...@gmail.com> wrote:
>
> >> > What can I say? Fuck you?
>
> >> No. You could apologise for uncalled for abuse. Otherwise, you're just
> >> one of the many, many people who have caused Usenet to degrade from
> >> the useful discussion forum it was twenty years ago to the sewer it is
> >> today.
>
> > I appreciate your opinion. On the other hand, imho Dan is rather
> > stubborn (and henc effectively /dense/) sometimes.
> Thanks! The word "dense" in English implies stupidity. My Collins
> Dictionary gives "beschränkt", "unterbelichtet", and "blöd" as German
> equivalents.
>
> For what it's worth, my view is that 0^0 is undefined, for reasons given
> by others here. It can only be firmed up by a process of limits, and the
> answer you get varies according to what expression you take the limit of.
>
> Still, Dan has actually produced something, DC Proof, for which respect
> is due. I doubt many others here have done anything similar.
> --
> Alan Mackenzie (Nuremberg, Germany).

I agree that in some of math 0^0 is undefined. But to make the binomial theorem without exceptions it is wise to define 0^0 as 1.
It's like in earlier definitions of a prime number they had to make an exception for the number 1. Nowadays they use a definitions which avoid that.
KON

On Wednesday, March 2, 2022 at 11:42:36 AM UTC-5, Fritz Feldhase wrote:
> On Wednesday, March 2, 2022 at 5:06:47 PM UTC+1, Dan Christensen wrote:
> > On Wednesday, March 2, 2022 at 10:58:31 AM UTC-5, Fritz Feldhase wrote:
> >
> > [...] we are not interested in these infinitely many variants of "power".
> > >
> > > Actually, we would like to define ^ (on IN) such that it is total. (Just like + and *.)
> > If you don't mind arithmetic on N and on R being inconsistent with each other, you could pick any value for 0^0 that you find convenient.
> They aren't "inconsistent".
>

Does there exist a number Ummm..... it depends. Are we talking about arithmetic on N or on R?

Sure sounds inconsistent to me.

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, March 2, 2022 at 11:42:36 AM UTC-5, Fritz Feldhase wrote:
> On Wednesday, March 2, 2022 at 5:06:47 PM UTC+1, Dan Christensen wrote:
> > On Wednesday, March 2, 2022 at 10:58:31 AM UTC-5, Fritz Feldhase wrote:
> >
> > [...] we are not interested in these infinitely many variants of "power".
> > >
> > > Actually, we would like to define ^ (on IN) such that it is total. (Just like + and *.)
> > If you don't mind arithmetic on N and on R being inconsistent with each other, you could pick any value for 0^0 that you find convenient.
> They aren't "inconsistent".
>

What is 0^0? Ummm..... it depends. Are we talking about arithmetic on N or on R?

Sure sounds inconsistent to me.

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 pretended :
> On Wednesday, March 2, 2022 at 11:42:36 AM UTC-5, Fritz Feldhase wrote:
>> On Wednesday, March 2, 2022 at 5:06:47 PM UTC+1, Dan Christensen wrote:
>>> On Wednesday, March 2, 2022 at 10:58:31 AM UTC-5, Fritz Feldhase wrote:
>>>
>>> [...] we are not interested in these infinitely many variants of "power".
>>>>
>>>> Actually, we would like to define ^ (on IN) such that it is total. (Just
>>>> like + and *.)
>>> If you don't mind arithmetic on N and on R being inconsistent with each
>>> other, you could pick any value for 0^0 that you find convenient.
>> They aren't "inconsistent".
>>
>
> What is 0^0? Ummm..... it depends. Are we talking about arithmetic on N or on
> R?
>
> Sure sounds inconsistent to me.

Five to the zero is one
Four to the zero is one
Three to the zero is one
Two to the zero is one
One to the zero is one
Zero to the zero is ??

Zero to the five is zero
Zero to the four is zero
Zero to the three is zero
Zero to the two is zero
Zero to the one is zero
Zero to the zero is ??

It doesn't appear that the reals are needed for there to be an
inconsistency as these discrete values also point to it.

On 3/2/2022 6:21 AM, Walton Drage wrote:
> sergio wrote:
>
>>>>>> I̶'l̶l̶ l̶e̶t̶ y̶o̶u̶ d̶o̶ t̶h̶e̶ m̶a̶t̶h̶ y̶o̶u̶r̶s̶e̶l̶f̶...
>>>>>
>>>>> completely nonsense.
>>>>
>>>> S̴̡͈͎̬͍̥͈̣͑̾͋e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ m̷̢͈̣̠̃̔̓́͛͒͂̽y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ p̷̛̻̺̝͂̓̎́̕͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕. Ṫ̸̡̤̉̐̽h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ U̷̢̱͒̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜i̶̢͑̄̈́̾̾̅̕c̷̯̬̮̼͆͊͋̆̊͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽ḇ̷̛̠͂̾̄̓u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝
>>>> n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝. I̴̡̛̲͙͖̭͑͛͐ ẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ẉ̸̣̘͙̍̕ m̷̢͈̣̠̃̔̓́͛͒͂̽à̶̜̲̈́͂̓̎͗̽n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝.
>>>
>>>
>>> you are too stupid to talk. Come back with a functional brain. Idiot.
>>
>> what online tool are you using ?
>
> you must be kidding, made by himself, and I still see some buffer overflow
> in it. Use your brain etc.

no, there is an online tool, I found it once

On 3/3/2022 6:03 AM, FromTheRafters wrote:
> Dan Christensen pretended :
>> On Wednesday, March 2, 2022 at 11:42:36 AM UTC-5, Fritz Feldhase wrote:
>>> On Wednesday, March 2, 2022 at 5:06:47 PM UTC+1, Dan Christensen wrote:
>>>> On Wednesday, March 2, 2022 at 10:58:31 AM UTC-5, Fritz Feldhase wrote:
>>>>
>>>> [...] we are not interested in these infinitely many variants of
>>>> "power".
>>>>>
>>>>> Actually, we would like to define ^ (on IN) such that it is total.
>>>>> (Just like + and *.)
>>>> If you don't mind arithmetic on N and on R being inconsistent with
>>>> each other, you could pick any value for 0^0 that you find convenient.
>>> They aren't "inconsistent".
>>>
>>
>> What is 0^0? Ummm..... it depends. Are we talking about arithmetic on
>> N or on R?
>>
>> Sure sounds inconsistent to me.
>
> Five to the zero is one
> Four to the zero is one
> Three to the zero is one
> Two to the zero is one
> One to the zero is one
> Zero to the zero is ??
>
> Zero to the five is zero
> Zero to the four is zero
> Zero to the three is zero
> Zero to the two is zero
> Zero to the one is zero
> Zero to the zero is ??
>
For even more confusion, what about this:

Five to the five is 3125
Four to the four is 256
Three to the three is 27
Two to the two is 4
One to the one is 1
Zero to the zero is ??

> It doesn't appear that the reals are needed for there to be an
> inconsistency as these discrete values also point to it.

On 3/3/2022 7:58 AM, sergio wrote:
> On 3/2/2022 6:21 AM, Walton Drage wrote:
>> sergio wrote:
>>
>>>>>>> I̶'l̶l̶ l̶e̶t̶ y̶o̶u̶ d̶o̶ t̶h̶e̶ m̶a̶t̶h̶ y̶o̶u̶r̶s̶e̶l̶f̶...
>>>>>>
>>>>>> completely nonsense.
>>>>>
>>>>> S̴̡͈͎̬͍̥͈̣͑̾͋e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ m̷̢͈̣̠̃̔̓́͛͒͂̽y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ p̷̛̻̺̝͂̓̎́̕͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜
>>>>> t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕.
>>>>> Ṫ̸̡̤̉̐̽h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝
>>>>> U̷̢̱͒̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜i̶̢͑̄̈́̾̾̅̕c̷̯̬̮̼͆͊͋̆̊͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽ḇ̷̛̠͂̾̄̓u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝
>>>>> n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝. I̴̡̛̲͙͖̭͑͛͐
>>>>> ẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ẉ̸̣̘͙̍̕ m̷̢͈̣̠̃̔̓́͛͒͂̽à̶̜̲̈́͂̓̎͗̽n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝
>>>>> ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝.
>>>>
>>>>
>>>> you are too stupid to talk. Come back with a functional brain. Idiot.
>>>
>>> what online tool are you using ?
>>
>> you must be kidding, made by himself, and I still see some buffer
>> overflow
>> in it. Use your brain etc.
>
>
>
>   no, there is an online tool, I found it once

I found the same/similar tool (actually a web page where you could cut
and paste to/from) some time ago as well. Also, easy enough to write
once you find your way through the Unicode.

Michael Moroney wrote:

>>> in it. Use your brain etc.
>>   no, there is an online tool, I found it once
>
> I̴̡̛̲͙͖̭͑͛͐ f̴̡̱̣͇̙̹͆́́͘͝͠ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ḑ̶̙͇̰̗̦̠̽ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕à̶̜̲̈́͂̓̎͗̽m̷̢͈̣̠̃̔̓́͛͒͂̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝/s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕i̶̢͑̄̈́̾̾̅̕m̷̢͈̣̠̃̔̓́͛͒͂̽i̶̢͑̄̈́̾̾̅̕l̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝à̶̜̲̈́͂̓̎͗̽r̸̡̹͚̣͔͍̮̩͌͐͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕l̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝ (à̶̜̲̈́͂̓̎͗̽c̷̯̬̮̼͆͊͋̆̊͝t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅu̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝à̶̜̲̈́͂̓̎͗̽l̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝l̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝y̷̢̛̪͈͒̆͝ à̶̜̲̈́͂̓̎͗̽ ẉ̸̣̘͙̍̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ḇ̷̛̠͂̾̄̓ p̷̛̻̺̝͂̓̎́̕͝à̶̜̲̈́͂̓̎͗̽g̶͎̲̲͎͍͈̱͔̭̾e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ ẉ̸̣̘͙̍̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̎̋̂̎͐̅͐̂͝�
�̢͕̯̱̥͔r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ y̷̢̛̪͈͒̆͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝ c̷̯̬̮̼͆͊͋̆̊͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝l̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝ḑ̶̙͇̰̗̦̠̽ c̷̯̬̮̼͆͊͋̆̊͝u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅ
> à̶̜̲̈́͂̓̎͗̽n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ḑ̶̙͇̰̗̦̠̽ p̷̛̻̺̝͂̓̎́̕͝à̶̜̲̈́͂̓̎͗̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅỏ̵̝̫̹̱̲̻̗͉̓͒͘̕/f̴̡̱̣͇̙̹͆́́͘͝͠r̸̡̹͚̣͔͍̮̩͌͐͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕m̷̢͈̣̠̃̔̓́͛͒͂̽) s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕m̷̢͈̣̠̃̔̓́͛͒͂̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅi̶̢͑̄̈́̾̾̅̕m̷̢͈̣̠̃̔̓́͛͒͂̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽g̶͎̲̲͎͍͈̱͔̭̾ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ à̶̜̲̈́͂̓̎͗̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ ẉ̸̣̘͙̍̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝l̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝l̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝. A̸͕̘̻̠͎͍̾̄̉̈́̐͐̚͝ͅl̷̡̧̢̙͈͍͈͈̉̄͋̔̉̆͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕, e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝à̶̜̲̈́͂̓̎͗̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕y̷̢̛̪͈͒̆͝ e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝n̸͙̯͓̓̄̆̀̓̈̾̕͜�
�̢͔̣͈ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝g̶͎̲̲͎͍͈̱͔̭̾h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ ẉ̸̣̘͙̍̕r̸̡̹͚̣͔͍̮̩͌͐͝i̶̢͑̄̈́̾̾̅̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝
> ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜c̷̯̬̮̼͆͊͋̆̊͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ y̷̢̛̪͈͒̆͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝ f̴̡̱̣͇̙̹͆́́͘͝͠i̶̢͑̄̈́̾̾̅̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ḑ̶̙͇̰̗̦̠̽ y̷̢̛̪͈͒̆͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝r̸̡̹͚̣͔͍̮̩͌͐͝ ẉ̸̣̘͙̍̕à̶̜̲̈́͂̓̎͗̽y̷̢̛̪͈͒̆͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕r̸̡̹͚̣͔͍̮̩͌͐͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝g̶͎̲̲͎͍͈̱͔̭̾h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ U̷̢̱͒̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜i̶̢͑̄̈́̾̾̅̕c̷̯̬̮̼͆͊͋̆̊͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝.

get out of here, unprepared lying bitch. Fancy words or lines,
in crappy javascript, but the above at least, you won't find anywhere.

sergio wrote:

>>>>>
>>>>> S̴̡͈͎̬͍̥͈̣͑̾͋e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ m̷̢͈̣̠̃̔̓́͛͒͂̽y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ p̷̛̻̺̝͂̓̎́̕͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕. Ṫ̸̡̤̉̐̽h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣̓̄̆̀̓̈̾̕͜
͈e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ i̶̢͑̄̈́̾̾̅̕s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ U̷̢̱͒̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜i̶̢͑̄̈́̾̾̅̕c̷̯̬̮̼͆͊͋̆̊͝ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽ḇ̷̛̠͂̾̄̓u̸̲͓̫̙̘͉̾̈̎̇̃̽̍̃͝͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝
>>>>> n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝m̷̢͈̣̠̃̔̓́͛͒͂̽s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕i̶̢͑̄̈́̾̾̅̕f̴̡̱̣͇̙̹͆́́͘͝͠t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅe̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝. I̴̡̛̲͙͖̭͑͛͐ ẉ̸̣̘͙̍̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜ḑ̶̙͇̰̗̦̠̽e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝ h̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕ẉ̸̣̘͙̍̕ m̷̢͈̣̠̃̔̓́͛͒͂̽à̶̜̲̈́͂̓̎͗̽n̸̢͙̯͓̝͔̣͈̓̄̆̀̓̈̾̕͜y̷̢̛̪͈͒̆͝ ỏ̵̝̫̹̱̲̻̗͉̓͒͘̕t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝s̸̗͙̤͍̩̤͍͍̝̖̃̐̽̌̕ t̵̨̺̺̬̝̝̠̣̔͌̍̔̃͗́̚ͅh̷͓̗̜͚̤̣͉̦̐̈͊͗͒̓͋̑͆̕e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝ à̶̜̲̈́͂̓̎͗̽r̸̡̹͚̣͔͍̮̩͌͐͝e̵̢͕͕̯̱̥͔̎̋̂̎͐̅͐̂͝.
>>>>
>>>>
>>>> you are too stupid to talk. Come back with a functional brain. Idiot.
>>>
>>> what online tool are you using ?
>>
>> you must be kidding, made by himself, and I still see some buffer overflow
>> in it. Use your brain etc.
>
> no, there is an online tool, I found it once

I doubt. You may find some fancy word or line in javascript, which is
crap. But no abundant stuff to suit what you want to implement.
The above I bet you wouldn't find online on internet.

Crack-pottery is alive in this group.

One crackpot has a proof of FLT
another has written a calculus book based on algebra alone.
another thinks he has refuted the Halting Problem
another thinks ha has a program that can write mathematical proofs

It is a crazy world arounds here!!!!

On Wednesday, March 2, 2022 at 11:12:43 AM UTC-5, Mostowski Collapse wrote:
> You dont even have a definition of domain in DC Proof.
> Dan Christensen schrieb am Mittwoch, 2. März 2022 um 15:08:06 UTC+1:
> > On Wednesday, March 2, 2022 at 8:22:44 AM UTC-5, Mostowski Collapse wrote:
> >
> > > Dan Christensen schrieb am Dienstag, 1. März 2022 um 15:25:15 UTC+1:
> > > > What is 0^0? Your teachers in school probably told you it was undefined on the natural numbers. Among mathematicians, however, the value of 0^0 depends on the context. 0^0 could be undefined, or it could be 1. It would be undefined if you are talking about 0 as a real number. And it would have a value of 1, as a handy shortcut if you are talking about 0 as a natural number for certain applications, e.g. the binomial theorem.
> > > >
> > > > I am a bit troubled by this inconsistency. There are, after all, no inconsistencies between real and natural number arithmetic other than in the case of 0^0. In my very humble opinion, your teachers were right, but they probably didn’t give you the full story. Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers, as it is on the real numbers.
> > > >
> > > > Full text: https://www.dcproof.com/ZeroExponentsOnN.htm
> > > >
> > > Where are the specifications and proofs?
> > If you want formal proofs, see my blog posting at https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> > > If 0^0 were really undefined in DC Proof, then
> > > you could prove that this is undefined as well:
> > >
> > > forall x (x e R => 0^(x-x) is undefined)
> > >
> > To prove, for example, that f(x) is undefined, you simply have to prove that x is not in the domain of f.
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

On Tuesday, March 1, 2022 at 6:25:15 AM UTC-8, Dan Christensen wrote:
> What is 0^0? Your teachers in school probably told you it was undefined on the natural numbers. Among mathematicians, however, the value of 0^0 depends on the context. 0^0 could be undefined, or it could be 1. It would be undefined if you are talking about 0 as a real number. And it would have a value of 1, as a handy shortcut if you are talking about 0 as a natural number for certain applications, e.g. the binomial theorem.
>
> I am a bit troubled by this inconsistency. There are, after all, no inconsistencies between real and natural number arithmetic other than in the case of 0^0. In my very humble opinion, your teachers were right, but they probably didn’t give you the full story. Using simple arithmetic (no equations or proofs), I will try to make the case that 0^0 could be left undefined on the natural numbers, as it is on the real numbers.
>
> Full text: https://www.dcproof.com/ZeroExponentsOnN.htm
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

You got there two copies of literals, not one.

Furthermore in a richer context called "mathematics"
not "my dumb little machine" the continuous functions defined
both 0^x+ and x+ ^ 0 and n^0 and 0^n.

On Thursday, March 3, 2022 at 6:04:38 AM UTC-5, FromTheRafters wrote:
> Dan Christensen pretended :
> > On Wednesday, March 2, 2022 at 11:42:36 AM UTC-5, Fritz Feldhase wrote:
> >> On Wednesday, March 2, 2022 at 5:06:47 PM UTC+1, Dan Christensen wrote:
> >>> On Wednesday, March 2, 2022 at 10:58:31 AM UTC-5, Fritz Feldhase wrote:
> >>>
> >>> [...] we are not interested in these infinitely many variants of "power".
> >>>>
> >>>> Actually, we would like to define ^ (on IN) such that it is total. (Just
> >>>> like + and *.)
> >>> If you don't mind arithmetic on N and on R being inconsistent with each
> >>> other, you could pick any value for 0^0 that you find convenient.
> >> They aren't "inconsistent".
> >>
> >
> > What is 0^0? Ummm..... it depends. Are we talking about arithmetic on N or on
> > R?
> >
> > Sure sounds inconsistent to me.
> Five to the zero is one
> Four to the zero is one
> Three to the zero is one
> Two to the zero is one
> One to the zero is one
> Zero to the zero is ??
>
> Zero to the five is zero
> Zero to the four is zero
> Zero to the three is zero
> Zero to the two is zero
> Zero to the one is zero
> Zero to the zero is ??
>
> It doesn't appear that the reals are needed for there to be an
> inconsistency as these discrete values also point to it.

This in itself does not result in an inconsistency on the natural numbers. No matter what value you attach to 0^0, your will get:

1. For all x in N: x^2 = x*x
2. For all x, y in N: x^(y+1) = x^y * x

Also, changing the value of 0^0 will have no effect on any of other combinations of base and exponent values on N. Thus we may be justified in leaving 0^0 undefined (or indeterminate or underspecified) on N.

Dan

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