## Touching the variance from Kac’es, Erdos’es and Poincare’s dreams

First, note the CLT.

Second, see the claim from Kac, …it may be appropriate to quote a statement of Poincaré, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.“.

Third, again Kac, “Consider the integers divisible by both p and q [p and q both prime]. To be divisible by p and q is equivalent to being divisible by pq and consequently the density of the new set is 1/pq. Now, 1/pq = (1/p)(1/q), and we can interpret this by saying that the “events” of being divisible by p and q are independent. This holds, of course, for any number of primes, and we can say using a picturesque but not very precise language, that the primes play a game of chance! This simple, nearly trivial, observation is the beginning of a new development which links in a significant way number theory on the one hand and probability theory on the other.”.

Fourth, Hardy-Ramanujan thorem.
Fifth,  this one (from me, I called this variance limit):

$\lim_{n \to +\infty}{({{1}\over{2}} + {{1}\over{3}}+ ... + {{1}\over{p}})-({{1}\over{2*3}} + {{1}\over{2*5}}+ ... + {{1}\over{p_1*p_2}})+ ({{1}\over{2*3*5}} + {{1}\over{2*3*7}}+ ... + {{1}\over{p_1*p_2*..*p_3}})-..}$

Sixth,  shown simultaneously and fast by different people that it’s 1. First idea was right then, assuming  $\lim_{n\to \infty}1-\prod_{p\leq n}\left(1-\frac{1}{p}\right)$ (taken from JoeBlow’s message).

Seventh, I therefore conjecture that prime key set problem described by me in earlier posts is closely related to this and those  primes can be understood as probabilities of independent actions (infinite number of actions). Due to their character, we then would have normal distribution of certain situations, which requires more analysis, though. and I don’t want to go deeper into it right now.

What I could say right now, though, is that  if probabilities of infinite amount of situations are described by these primes, then the relation between the primes would be worth analyzing, the analysis itself should be treated in more objective Goedel sense, the ordering of primes should also be of significant importance. The fact that we already know that primes are out there, already, existing, makes the distribution of probabilities settled before anything else.

Moreover,  if we take a closer look at RSA or EC, we can notice the relation of points on curves, which is shaped by primes. We then notice that those things may be used for customizing information for one entity (different information for different people). The resemblance is interesting enough that we could hazard a guess that if we choose one primes out of the inifinte number of primes, we could then be able to have a certain function that leads us through our own “prime path”. Thus, potential reference to determinism.
Had that been true, we’d be seeing variance as planned. And not because two states of dices are different in a sense that a dice is irregular. But in sense that every action is given own ordering number, thus the very claim. That would make us finally focus on decision quality only. As Goedel had it and I here agree.

In this sense, induction on the quality of decision due to code coverage, also as Goedel suggested, could be done via finding contradicitons. That is simpler in the sense, that we only find contradictions and decrease the amount of required analysis. In the context of life, we could – for instance – elaborate on the importance of actions, schedule importance, settle who we are, what our goal is, and why. We could also analyze the quality of our goals in the context of our current analytical potential.

Clearly, such a set $T$ cannot be finite. Consider the set $\mathcal{P}$ of all subsets $P$ of primes, such that $\sum_{p\in P} \frac {1} {p} \leq 1$, endowed with inclusion, as order relation. Consider a chain $\mathcal{C}$ in this poset. Then $M = \bigcup_{P\in \mathcal{C}} P$ is a subset of primes, and $\sum_{p\in M} \frac {1} {p} \leq 1$, hence $M \in \mathcal{P}$. By Zorn’s lemma, $\mathcal{P}$ must contain maximal elements $T$, and then clearly we must have $\sum_{p\in T} \frac {1} {p} = 1$, since otherwise we may augment $T$.
Also as a corollary to Zorn’s lemma, any element sits under some maximal element. This means any set $P$ of primes such that $\sum_{p\in P} \frac {1} {p} < 1$ may be augmented to a maximal set, where we have equality, and this in infinitely many ways. So that’s all one can say about the structure of such sets $T$.