Arithmetical Proof and Open Sentences

If the concept of proof （including arithmetic proof） is syntactically restricted to closed sentences （or their Godel numbers）, then the standard accounts of Godel＇s Incompleteness Theorems （and Lob＇s Theorem） are blocked. In these standard accounts （Godel＇s own paper and the exposition in Boolos＇ Computability and Logic are treated as exemplars）, it is assumed that certain formulas （notably so called ＂Godel sentences＂） containing the Godel number of an open sentence and an arithmetic proof predicate are closed sentences. Ordinary usage of the term ＂provable＂ （and indeed ＂unprovable＂） favors their restriction to closed sentences which unlike so-called open sentences can be true or false. In this paper the restricted form of provability is called strong provability or unprovability. If this concept of proof is adopted, then there is no obvious alternative path to establishing those theorems.

The Prime Sequence: Demonstrably Highly Organized While Also Opaque and Incomputable-With Remarks on Riemann’s Hypothesis, Partition, Goldbach’s Conjecture, Euclid on Primes, Euclid’s Fifth Postulate, Wilson’s Theorem along with Lagrange’s Proof of It and Pascal’s Triangle, and Rational Human Intelligence

The main design of this paper is to determine once and for all the true nature and status of the sequence of the prime numbers, or primes—that is, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and so on. The main conclusion revolves entirely around two points. First, on the one hand, it is shown that the prime sequence exhibits an extremely high level of organization. But second, on the other hand, it is also shown that the clearly detectable organization of the primes is ultimately beyond human comprehension. This conclusion runs radically counter and opposite—in regard to both points—to what may well be the default view held widely, if not universally, in current theoretical mathematics about the prime sequence, namely the following. First, on the one hand, the prime sequence is deemed by all appearance to be entirely random, not organized at all. Second, on the other hand, all hope has not been abandoned that the sequence may perhaps at some point be grasped by human cognition, even if no progress at all has been made in this regard. Current mathematical research seems to be entirely predicated on keeping this hope alive. In the present paper, it is proposed that there is no reason to hope, as it were. According to this point of view, theoretical mathematics needs to take a drastic 180-degree turn. The manner of demonstration that will be used is direct and empirical. Two key observations are adduced showing, 1), how the prime sequence is highly organized and, 2), how this organization transcends human intelligence because it plays out in the dimension of infinity and in relation to π. The present paper is part of a larger project whose design it is to present a complete and final mathematical and physical theory of rational human intelligence. Nothing seems more self-evident than that rational human intelligence is subject to absolute limitations. The brain is a material and physically finite tool. Everyone will therefore readily agree that, as far as reasoning is concerned, there are things that the brain can do and things that it cannot do. The search is therefore for the line that separates the two, or the limits beyond which rational human intelligence cannot go. It is proposed that the structure of the prime sequence lies beyond those limits. The contemplation of the prime sequence teaches us something deeply fundamental about the human condition. It is part of the quest to Know Thyself.

A Set-Theoretical Lemma That Implies an Abstract Form of Gdel's Theorem

We propose a simple set-theoretical lemma that implies Godel＇s Incompleteness Theorem. Also mentioned are some related consequences.