Number Theory


An Introduction to the $n$-Formal Sequents and the Formal Numbers

Authors: A. Zaganidis

In this work, we introduce the $n$-formal sequents and the formal numbers defined with the help of the second order logic. We give many concrete examples of formal numbers and $n$-formal sequents with the Peano's axioms and the axioms of the real numbers. Shortly, a sequent is $n$-formal iff the sequent is composed by some closed hypotheses and a $n$-formal formula (a close formula with one internal variable such that the formula is only true when we set that variable to the unique natural number $n$), and it does not exist some strict sub-sequent which are composed by some closed sub-hypotheses and some sub-$m$-formal formula with $m>1$. The definition is motivated by the intuition that the ``Nature's hypotheses'' do not carry natural numbers or "hidden natural numbers" except for the numbers $0$ and $1$, i.e., they can be used in a $n$-formal sequent. Moreover, we postulate at second order of logic that the ``Nature's hypotheses'' are not chosen randomly: the ``Nature's hypotheses'' are the only hypotheses which give the largest formal number $N_Z\cong 2^{1.0\times 10^4}-2^{2.4\times 10^6}$. The Goldbach's conjecture, the Polignac's conjecture, the Firoozbakht's conjecture, the Oppermann's conjecture, the Agoh-Giuga conjecture, the generalized Fermat's conjecture and the Schinzel's hypothesis H are reviewed with this new (second order logic) formal axiom. Finally, three open questions remain: Can we prove that a natural number is not formal? If a formal number $n$ is found with a function symbol $f$ where its outputs values are only $0$ and $1$, can we always replace the function symbol $f$ by a another function symbol $\tilde{f}$ such that $\tilde{f}=1-f$ and the new sequent is still $n$-formal? Does a sequent exist to make a difference between the definition of the $n$-formal sequents and the following weaker variant of that definition: we look at the explicit sub-formulas of $\phi$ which induce the $m$-formal formulas instead of looking at the explicit sub-formulas of $\phi_{n-formal}$ which are $m$-formal formulas?

Comments: 16 Pages. I have chosen the category Mathematics-Number Theory since most of the consequences of the present article are inside the number theory

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Submission history

[v1] 2017-04-03 11:20:53
[v2] 2017-06-02 04:16:44

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