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"Can you prove RH (Riemann Hypothesis) in an extension of C, the canonical 2OL formal system for complex numbers (with 'prime' mentioned in RH being a new unary predicate constant symbol with axioms)? [If you can, please write down the new axioms.]"

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Khong Dong

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Oct 15, 2022, 1:13:39 PM10/15/22
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https://qr.ae/pvJiJ5

Khong Dong

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Oct 15, 2022, 1:41:07 PM10/15/22
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On Saturday, 15 October 2022 at 11:13:39 UTC-6, Khong Dong wrote:

> https://qr.ae/pvJiJ5

Obviously, (new) axioms for Gaussian "primes" would be ill-advised.

Khong Dong

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Oct 16, 2022, 3:22:50 PM10/16/22
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I've provided a fuller (though still partial) answer to the Quora question there.

Khong Dong

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Oct 18, 2022, 8:12:47 PM10/18/22
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On axiom A4, made a comment that:

This is naturally a logical axiom, but in time will serve as mathematical quantum coherent-supposition on certain syntactical deductions (proofs) involving prime(z).

I plan to use this ("quantum") complex theory qC to make a full proof -- resolution -- about RH. But we'll see ...

Khong Dong

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Oct 20, 2022, 7:51:35 PM10/20/22
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I've made a note in axiom A5 that, face value, this axiom in conjunction with A4 is homologous with Pauli Exclusion Principle (QM).

Khong Dong

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Oct 23, 2022, 7:23:09 PM10/23/22
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On Saturday, 15 October 2022 at 11:13:39 UTC-6, Khong Dong wrote:

> https://qr.ae/pvJiJ5

In my (Quora) answer, I've added proven resolutions on RH and P-versus-NP problems.

Jeffrey Rubard

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Oct 23, 2022, 7:36:52 PM10/23/22
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The "Riemann hypothesis"?
"You ignorant fool."
Oh, I've *heard* of it before. I just don't really understand it.

Khong Dong

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Oct 23, 2022, 7:53:14 PM10/23/22
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_Honestly_ which particular axiom there, for instance, that you "don't really understand it"?

Or are you just a full-of-hate sci.logic poster?

Khong Dong

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Oct 23, 2022, 8:13:17 PM10/23/22
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What about the "Proof on P-versus-NP problem" there, Jeffrey Rubard?

(It looks _extremely_ short and simple, imho).

Khong Dong

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Oct 23, 2022, 11:48:20 PM10/23/22
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Imho, an interesting observation here is that while the Riemann Hypothesis and P-versus-NP problems seem to be unrelated, they both share the same root cause: The Euler Identity should have either the least Mersenne prime identity or the least non-Mersenne prime counterpart — but has none.

This, on the surface, stands as a sharp contrast to the IUTT (Inter-universal Teichmuller Theory) which, by some accounts, could only be used to “prove” the alleged ABC conjecture.

Khong Dong

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Oct 24, 2022, 12:24:26 AM10/24/22
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On Sunday, 23 October 2022 at 21:48:20 UTC-6, Khong Dong wrote:
> On Sunday, 23 October 2022 at 17:23:09 UTC-6, Khong Dong wrote:
> > On Saturday, 15 October 2022 at 11:13:39 UTC-6, Khong Dong wrote:
> >
> > > https://qr.ae/pvJiJ5
>
> > In my (Quora) answer, I've added proven resolutions on RH and P-versus-NP problems.
> Imho, an interesting observation here is that while the Riemann Hypothesis and P-versus-NP problems seem to be unrelated, they both share the same root cause: The Euler Identity should have either the least Mersenne prime identity or the least non-Mersenne prime counterpart — but has none.

We're talking about complex primes -- complex numbers -- here of course.

Khong Dong

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Nov 9, 2022, 9:05:38 PM11/9/22
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On Saturday, 15 October 2022 at 11:13:39 UTC-6, Khong Dong wrote:

> https://qr.ae/pvJiJ5

In my answer I've added the section "Added the section "Formal proof of undecidability of RH in qC".

Finally, the expected proof of RH (Riemann Hypothesis) can rest in peace, so to speak.

P.S. The question and answer can also be seen in https://qr.ae/pvlXWp

Ross A. Finlayson

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Nov 10, 2022, 4:05:11 AM11/10/22
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That's just its own thing.

That's just what would be a concomitant fact.

That's just "whether infinity is effective" and "whether there's standard integers".

Those other things have nothing to do with it.

Ross A. Finlayson

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Nov 10, 2022, 7:33:42 AM11/10/22
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hat's at least two independent conditions under the road-forks under the coat-tails you're trying to ride into relevancy.

Not that there's anything wrong with that. For a lark.

Quit dragging it out. Take it off. Read the cards.

You have to fashion your own jacket with coat-tails for other people to ride. Then climb it.

Khong Dong

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Nov 29, 2022, 6:52:51 PM11/29/22
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So far there has been no successful counter argument against my proofs, in https://qr.ae/pvlXWp -- on RH and P-vs-NP.

I'm confident the proofs will withstand the test of time.

Khong Dong

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Dec 22, 2022, 6:42:08 PM12/22/22
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Made a note in Section "Formal proof of undecidability of RH in qC" (https://qr.ae/pvlXWp):

<quote>

Note: To the the extend that

- the Zeta function ζ is at best only partially defined [it being not a wff (well-formed-formula)]
- 2 as a numeral symbol is subjectively interpreted as denoting the least natural prime number

the Riemann Hypothesis, if it were a wff, could be be stated (rephrased) as:

All the non-trivial zeros z of the ζ function are of the form:

z = 1/ω + (0+yi)

y being real.

</quote>

Khong Dong

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Jan 30, 2023, 2:23:33 AM1/30/23
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On Saturday, 15 October 2022 at 11:13:39 UTC-6, Khong Dong wrote:
> https://qr.ae/pvJiJ5

It's really in here

https://qr.ae/pvlXWp

and, prior to axiom A6, I've enhanced the definition of prime(z) with a cuspidal expression specific to complex-number primes.

Through general inference-cuspidation, we could see how mathematical primality is defined throughout different (kinds of) formal systems: FOL, SOL, etc.

Khong Dong

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Sep 16, 2023, 7:05:54 PM9/16/23
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Per the section "Formal proof of undecidability of RH in qC" of my Quora answer, it seems interesting to hear the followings from a renowned set theorist on the questionable proof of Riemann Hypothesis ("On the Mathematical Necessity of the Infinite" by Hugh Woodin (https://www.youtube.com/watch?v=KI4yrWzRSWI&t=4721s) 1:35:00 - 1:39:37):

"Take the Riemann Hypothesis. That has the same, if you reformulate it [...], syntactic form of [the] consistency of the PD frame. There are a lot of number theorists who believe the Riemann Hypothesis is true, but I've never heard [of] anyone [who] just declares it's true."

"[...]"

"Does that mean if we show that you can't prove the Riemann Hypothesis, we're going to say it's true? I don't think so."

My proof of the (eternal, absolute) undecidability of RH rhymes with his "I don't think so": Syntactically one can't prove the Riemann Hypothesis one way or the other; hence, semantically, one can't abductively say it's true either.

Khong Dong

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Sep 19, 2023, 12:14:56 AM9/19/23
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So, accidently, inadvertently and to nobody's fault, I got "cheated" out: The Clay Institute should have awarded the prize for each of the following scenarios (assuming consistency):

- RH is provable.
- ~RH is provable.
- Neither RH nor ~RH is provable.

They can still change their mind, right? :-)

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