Let us not solve Riemann Hypothesis. Let us instead bring the child like wonder – one which I had trying to decipher a pattern among prime numbers or to prove the infinitude of primes by myself. So we shall tackle problems involving how to think of primes when they become large… REALLY large? Or why cannot we predict the next prime number? We will consider the big picture and try to dissolve the problem in an outer framework as Grothendieck might’ve expected in his creation of architecture of mathematics.

1. Euclid’s proof of infinitude of primes

Well, Euclid dealt the first blow by proving – constructively? – that there are infinitude of prime numbers. I already wrote a paper trying to express the fact that it is a tautological proof.  Here is Euclid’s proof:


Let us suppose that p 1 , p 2 , p 3 , … p n   are prime numbers. Multiply them together and add 1, calling this number a new integer q . If q is a prime number, then we have a new prime. If q is not a prime, it must be divisible by a prime number r . But r cannot be p 1 or any other from our original list of prime numbers, because if you divide q by any of p 1 , p 2 , p 3 , … p n you will get a remainder 1, which means that q is not divisible by any of these prime numbers. So r is a new prime. Whichever way you choose to look at it, either you have found a new prime q, or if q is not a prime, than you have found that it has a new prime for a prime factor.

The paper I wrote was a very informal one and during writing of it I knew nothing of Brouwer or Kleene or even the details of intuitionist. But I raised two objections:

    1. How to define prime? Because the definition of primality is predicatively dependent on their meanings? I used the car example. If there are infinite chasis then there must be infinite cars, for no car can be constructed out of the form of chasis. But by definition, prime numbers include 2 and not 1 and it seems we’ve put a seed that describes the so-called pseudo-pattern.  Primality is an adjective and denotes the characteristics of a number. Just like ‘redness’.  Now if one knows there are infinite number of apples all red, then ipso facto ‘redness’ is infinite. I have not studied set theory but it is about one set containing another. Set of natural numbers embed prime numbers. Intuitively, if there are infinite numbers, and if prime numbers belong to sub-set of natural numbers, then it follows automatically that there are infinitude of primes. Common sense. Yet, Euclid’s proof is is a convolution of eye-wash to give an illusion of ‘proof’ when it is circular in nature.
    2. My second objection was how do we even count? If you scroll up and check then you shall see that we already assume a finite collection of primes exist. I looked at translation of elements but Euclid defines terms in terms of another resulting in vicious circle. What is divisibility? Euclid uses line segment and other primitive analogies but modern problems do not go away.However the purpose of this brainstorming is not to debate about circularity of the proof nor about the constructive nature of the proof. What I am more interested in what happens when numbers become large…. REALLY large?

2. Behavior of ‘large’ primes numbers

Currently, the question posed above exists in the following form in How to understand numbers that become really large?

But we are interested in the algorithmic compressibility of large numbers.  Or to state a bit formally the Kolmogorov complexity of primes.

3. Glimpse of a ‘large’ prime number

To give a taste of just exactly what kind of numbers we are dealing with here is a link. At naked glance, basically it’s a page full of random digits. But the number is a prime. Sure if you are Ulam you may seek patterns such as Ulam spiral:

Stare you may at this mandala of dotmatrix pattern but God’s face won’t show up. So although it gives us a sense of capturing of large numbers it doesn’t elucidate anything else.

4. Conceiving the inconceivable

Magritte, a favorite painter of mine, Belgian, wanted to paint ‘unthought of thoughts’ and as the image shows it’s his way of attempting the impossible which we shall emulate. albeit logically.

Since it is an informal treatise of brainstorming here are some concepts that bothers me.  How to transcend or go beyond predicative? Or as I asked:Does the concept of predicativity need to be formalized to go beyond Feferman-Schutte ordinal?
There is also the problem of Ω which Cantor declared as the ultimate logical limit and may even be called “God”. Absolute logic of Cantor. Also, Woodin’s pursuit of L: Ultimate Logic. Also related problems in mathematics, in spite of Peano’s axiom and Lambda calculus, is the fact of entailment. Does R resolve the paradox of entailment?

And of course, in terms of set theory there is Kunen’s inconsistency theorem is at top of Cantor’s attic. But why? Why is Kunen inconsistency at the top of Cantor’s upper attic?

So we are just putting scraps of notes and puzzles here and there and finally we will attempt to combine and connect all the thoughts. Another problem is mathematical formulation of Indra’s net, paradox in relevant logic, etcetera.

Unless we tackle the foundational issues regarding number theory I am afraid the task will not be solved at all.

5. Intuitively speaking…

What is a proof? I mean if I tell you to prove that 2+2=4, will you show me the paleontological treatise of Russell-Whitehead? Cannot 2+2=3 if we define 4 as 3 and vice versa? Who comes with these arbitrary laws?

You will stop my nonsense at this point immediately saying well they are symbolic representations of quantities.  I perfectly understand the mouthful but that is just a veil of circular talk. Sorites permits us to ask “what constitutes a quantity?”
Intuitively speaking, Riemann Hypothesis deals with prime numbers, zeta function, complex plane and induction.

Can induction be used to prove RH? Let’s back up a bit.  I was opposed to Euclid’s proof for a long time as I did not understand it. In deep depression I realized that the reason why average people don’t understand complex math problems is not because they are stupid but because of the carefully camouflaged circularity in mathematical argument.  If Euclid began by saying there there be prime numbers: p1,p2….. I would stop him right there.

But let’s accept Euclid. So that means there is an infinitude of prime numbers. Loosely, a number with so many digits that they can easily outnumber the particles in universe. Skewes. Now Hardy managed to give a description of Skewes number which I am reproducing here:

Hardy thought it ‘the largest number which has ever served any definite purpose in mathematics’, and suggested that if a game of chess was played with all the particles in the universe as pieces, one move being the interchange of a pair of particles, and the game terminating when the same position recurred for the 3rd time, the number of possible games would be about Skewes’ number.

But one caveat from that same article:

One last thing before we move on: the Skewes’ number we’ve been talking about is the smaller of the two. There’s another Skewes’ number that the mathematician demonstrated in 1955. The first number relies on something called the Riemann hypothesis being true – it’s a particularly complex bit of mathematics that remains unproven but is massively helpful when it comes to prime numbers. Still, if the Riemann hypothesis is false, Skewes found that the crossover point jumps all the way up to 10^10^10^963.

Let us however return to the “intuitive feel” for the problem. What do we know? Well we know that there is a concept of primeness, and we can manufacture limitless mathematical structures from that concept of primeness. But what happens when a new prime is found or discovered? It enables us to build bigger and bigger building blocks or numbers. What of dimensions? Say a person lives in 2-dimensional flatland; what will happen when he moves up to third dimension or fifth or seventh or eleventh….It’s as if one is unlocking or jail-breaking the universe from it’s hidden contents. It’s like every step up is an opening of new world or strata.  If each levels are prime numbers then as one reaches the ultimate prime found so far, then she will be propelled to another new world when a new prime, say, P, then P helps us to launch to new dimension where the world may be very different.

But these are just sci-fi thought experiments and not maths.

5.1. Shelah’s dream

In Logical Dreams set theorist Saharon Selah casually mentioned about his “dream”. User Junkie replies quoting Shelah here ( I copy-pasted the image as I am still learning how to LaTex the posts here in wordpress).


5.2 Saharon Shelah for Dummies

I know forcing is a relatively difficult concept to master which requires hours of devotion and I do not know much about it. But doesn’t the fact that what we intuitively thought has a semblance of his dream?  Consider that the universe is a compilation of marble like particles. And on each particle you write down a symbol. It could be anything. Chinese characters, Egyptian hieroglyphics, Mayan glyphs….. Then it immediately becomes clear that there could be a finite list or infinite list. It’s finite because we are humans and we have finite collection of symbols. But they can be infinite. I can just create a new one, such as,  |! , to signify “a 3rd degree murder” where the line is a symbol or sword or spear or stick and the exclamation mark a danger sign. Thus logical symbols are also creative outputs. As technology evolves, we can compress many complex ideas in a single glyph. Korean glyphs already have such encapsulation as well as ideograms and Mayan glyphs. What if we were to create our own number glyphs where arbitrary symbols will encapsulate, say, a consistency strength of a proof or a large prime number. Now, can we create a system where we take all the symbols of the world, start permutations, and produce one proof of another.

In simpler terms, what are the odds that if a deck of cards with mathematical symbols were to be uncovered one by one, it’d spell out the proof of RH?

Now back to Shelah.  In short his dream is that RH is unprovable in Peano Arithmetic but possibly provable in some higher theory.

I am smelling a Godelian self-reference application somewhere if we were to pursue this.


Because no matter how complex, convoluted, succinct, algorithmically compressible, or a large finite number of symbols are used to prove RH, the number of notations and symbols to be used in that infinite Turing tape will be far,far, faaaaar less than the large primes which are not discovered yet. You see, this is the danger of Euclid’s proof. He did deal the first blow, and it seems that no matter how we try to come up with a proof of RH theory, unless self-reference is used, we cannot make any statements about that RH, because the number of digits in a prime can be soooo large as if to encode Britannica.

Suppose this is a prime number spread out: 231248233489278482947239797523497593427999………………..

I basically typed some random numbers from keystrokes. It may or may not be prime. But that’s moot.  We are not interested in the number itself rather the ellipsis. Can a prime be so large as if to have an ellipsis where the entire length of universe is just a speck in that stream?

And that’s exactly where the problem lies.  May be it’s meaningless. Indescribable or a badly worded problem.  Think about it. There are layers of planes, let’s call them astral planes, and as you move upward from one plane to another you realize there is an infinite planes.  But then you want to know if the framework that supports the plane is in 1-1 correspondence, then if the planes extend to infinity, will there be a 1-1 correspondence between the astral planes and the framework that supports it. While it may appear true if trillions of digits are in accordance, it could also be false.  Assuming that is true (say, due to sheer statistical evidence)  we cannot claim that it is true for all astral planes. Because the next plane that is discovered has already opened up and exploded another new world ie new numbers and that cyclically involves more numbers or astral planes and then when one reaches the highest plane, since there can be still higher plane, one needs to step out of that system or construct higher theory. And when one invents new language or mathematical manipulation of symbols, no matter what higher theory one comes up with, it cannot talk prove anything about RH in PA because, those metatheories would require further proofs.

But I want to get back at the card-flipping thought experiment. Suppose you are really lucky and throw random symbols on a canvas that stretches to infinity in all direction, and poof! those symbols fall into perfect places magically to create the proof of Riemann Hypothesis. And it proves that RH is true. So what exactly happens when RH is true? That means that all non-trivial, zeros of the zeta function all lie on the critical line.

But whatever proof one uncovers, that proof is still within a system that can be considered marginally small in terms of sheer amount of digits in a prime number.

This is a subtle point. Because although people are hard-nosed skeptical about quantum theories and what not, my major motivation to asking that large number question in the philosophy forum was if “laws of logic” “break down” “at any point” when numbers come large. There is of course, Poincare Recurrence Theorem but that was a sore subject and got subdued, edited and vanished because of so power the minority mods wield in such forum.

Now I can boldly proclaim in my own turf:  CAN A NUMBER BECOME SO LARGE AS FOR THE LAWS OF PHYSICS TO BREAK DOWN WHEN ANYTHING APPROACHES SUCH BOUNDARY? Okay, it’s really not far-fetched because straight from that question this is what I also found:

It all started after reading The Unimaginable Mathematics of Borges’ Library of Babel and the review of it here. The problem arises when one sets to catalog the books as the number of the different books become approximately 10^10^6 (yet smaller than googoolplex), justifying the term “unimaginable”. Susan Stepney points out in the review that when one wants to catalogue the number of books in Library:

[…] the problem of finding a “short” description of the book to put in the catalogue: there are not enough short descriptions. For the Vast majority of the books in the Library, the shortest description (that distinguishes it from other books) is the book itself. Most books cannot be “compressed” to a short description.

And then comes the punchline:

Or,as Bloch puts it, the Library is its own catalogue.

Library is its own catalogue(!). Some concept there. Or like Ω ={Ω} like a Quine atom of Non-well founded set theory.

A contains B. B contains A. Paradox! Ah! My terrain. Law of Excluded Middle and the whole enchilada.

Here is a question: How to axiomatically build up from an axiom that is self-referential or paradoxical?

Another problem I am wrestling with is w-consistency. If T is This is true. And if ~T is This is not true, then : T is ω-consistent if it is not ω-inconsistent.

Apparently, there exists a generalization of ω-consistency to Γ-consistency which Leon Henkin used. I do not understand the latter at all as I couldn’t get my hands on sufficient resource but I was thinking more in terms of Indra’s net or.. if you will permit, an ω-sphere.

It’s an abstract generalization of ω-consistency. Basically, like de Finetti’s theorem, it is a sphere that contains all possibilities including the possibility of a non-possibility or non ω-sphere.  Analogy would be again L, ultimate logic, Cantor’s God, or multiverse. This tool will help us model any model.