'How would one formally define -- in the language of real numbers -- the unary predicate constant symbol "prime"?'

[Khong Dong]

Fyi.,: https://mathoverflow.net/q/431148/125063

[Khong Dong]

On Saturday, 24 September 2022 at 11:53:53 UTC-6, Khong Dong wrote:

> Fyi.,: https://mathoverflow.net/q/431148/125063

Just in case, this question having been inexplicably closed in MO, I've posted a similar question in https://qr.ae/pv9JXS (in Quora).

[Jeff Barnett]

On 9/24/2022 11:53 AM, Khong Dong wrote:
>
> Fyi.,: https://mathoverflow.net/q/431148/125063

Well the real field which you have defined has definitions for addition,
the identity element 0, and the unit 1 among a whole host of other
things so you could define N as containing 0 and if n in N, then n+1 in
N. You now have the natural numbers and can bring in Peano axioms as
definitions. The only sticking point is you need an induction axiom. If
the logic given with the real field has an adequate one, you are off to
the races. Otherwise you have some work to do. Unfortunately, when most
text books define the reals, they are very lax on what supporting logic
is assumed.

The above assumes you are talking about the subset of natural numbers
that we call prime. However, "prime" has other meanings such as in prime
ideal, etc.
--
Jeff Barnett

[Khong Dong]

On Sunday, 25 September 2022 at 00:31:38 UTC-6, Jeff Barnett wrote:
> On 9/24/2022 11:53 AM, Khong Dong wrote:
> >
> > Fyi.,: https://mathoverflow.net/q/431148/125063

> Well the real field which you have defined has definitions for addition,
> the identity element 0, and the unit 1 among a whole host of other
> things so you could define N as containing 0 and if n in N, then n+1 in
> N.

Well, talk is cheap: _Where is_ your _finite formal syntactical definition of "prime" _ for _reals_ ?

>
> The above assumes you are talking about the subset of natural numbers
> that we call prime. However, "prime" has other meanings such as in prime
> ideal, etc.

Good grief. The level of learned mathematical knowledge these days is so bankrupted!

> --
> Jeff Barnett

[Khong Dong]

Have begun with “The short answer” in the Quora thread (https://qr.ae/pv9JXS).

[Jeff Barnett]

On 9/25/2022 10:28 AM, Khong Dong wrote:
> On Sunday, 25 September 2022 at 00:31:38 UTC-6, Jeff Barnett wrote:
>> On 9/24/2022 11:53 AM, Khong Dong wrote:
>>>
>>> Fyi.,: https://mathoverflow.net/q/431148/125063
>
>> Well the real field which you have defined has definitions for addition,
>> the identity element 0, and the unit 1 among a whole host of other
>> things so you could define N as containing 0 and if n in N, then n+1 in
>> N.
>
> Well, talk is cheap: _Where is_ your _finite formal syntactical definition of "prime" _ for _reals_ ?

I'm not going to help you with the bookkeeping portion of your home
work; it will mean more to you if you do it yourself. If you had
attempted to do it, you would notice quickly that the question asked was
ill-formed: which definition of the reals; what supporting logical
system; and what theory is the question embedded in?

>> The above assumes you are talking about the subset of natural numbers
>> that we call prime. However, "prime" has other meanings such as in prime
>> ideal, etc.
>
> Good grief. The level of learned mathematical knowledge these days is so bankrupted!

From one who has so much trouble asking for help with a homework
problems????? Tsk, tsk, tsk, etc.............
--
Jeff Barnett

[Khong Dong]

Have fixed the previous error by introducing the prerequisite concept of "urelement".

[Khong Dong]

I've refined the definition of prime(z) to ensure that a prime is neither 0 nor 1.

[Khong Dong]

In “The long answer”, I’ve generalized the definition of prime(z) in the “quantum” complex plane qC — based on there being infinitely many urelements.

[Khong Dong]

Not just there "there being infinitely many urelements" but also there being infinitely many distinct _kinds_ of *urelements*.

[Khong Dong]

I've fixed typos in Quora's (math symbol) scripting many times, but it should be stable now. (Sorry for any inconvenience).

[Khong Dong]

I've mentioned elsewhere but I think it's a good question whether or not it's impossible to syntactically prove RH (Riemann Hypothesis) in this ("quantum") complex theory qC — coupled with the prime(z) definition thereof.

I think it'd be impossible, if there's no counter example, but that has to be proven of course.

[Khong Dong]

I've also commented that axiom A3 connotes a continuation of the order relation < from the reals to the ("quantum") complex plane qC, which is homologous with Riemann's analytic continuation.

[Khong Dong]

In Quora, I've added a section on “Relationship between [complex] prime(z), Euler’s Identity, and Euler's Platonic-solids”.

[Ross A. Finlayson]

Shut up, idiot.

Or though that's nice, but not, not the way you say.

Or Euler's identity, for e or pi, waste your breath, then eat dirt.

[Ross A. Finlayson]

Why don't you all go kick it.