Welcome. We present a paper in which we create a family of first-order-arithmetic interpretations in which natural numbers are in association with time states. This allows to construct dynamic processes in which one acceleration characterizes at least one arithmetic statement (for example, the Goldbach Conjecture) , a characterization which is lost in an instant of time, obtaining a temporal singularity.
Dynamic processes associated to Natural Numbers [1]
QUOTATIONS
1. I have sometimes thought that the profound mystery which envelops our conceptions relative to prime numbers depends upon the limitations of our faculties in regard to time, which like space may be in essence poly-dimensional and that this and other such sort of truths would become self-evident to a being whose mode of perception is according to superficially as opposed to our own limitation to linearly extended time. (J.J. Sylvester from “On certain inequalities relating to prime numbers”, Nature 38 (1888) 259-262, and reproduced in Collected Mathematical Papers, Volume 4, page 600, Chelsea, New York, 1973)
Remark: In [1] we have created a bi-parametric family
of First Order Arithmetic interpretations corresponding to a movement (
) of movements (
) in such a way that at least one arithmetical statement is characterized by means of an acceleration for every movement
with temporal singularity at the movement
.
2. The
system (First Order Arithmetic) contains a closed well formed formula which is true in the model
but it is not a theorem of
Remark: In [1] we prove that the model
3. The discrete and the continuous are complementary because a requirement for discreteness or apartness of movements in time is the existence of a ‘between’ . (Brouwer, from “On the foundations of mathematics”)
Remark: The consideration of different ‘betweens’ (or equivalently, different processes of counting natural numbers) allows to construct a discontinuous characterization of some arithmetical statement. All of this in the precise terms of [1] (section 3.2).
4. How then do assertions arise which concern, not all natural, but all real numbers, i.e., all values of a real variable? Brouwer shows that frequently statements of this form in traditional analysis, when correctly interpreted, simply concern the totality of natural numbers. In cases where they do not, the notion of sequence changes its meaning: it no longer signifies a sequence determined by some law or other, but rather one that is created step by step by free acts of choice, and thus remains in statu nascendi. This ‘becoming’ selective sequence represents the continuum, or the variable, while the sequence determined ad infinitum by a law represents the individual real number falling into the continuum. The continuum no longer appears, to use Leibniz language, as an aggregate of fixed elements but as a medium of free ‘becoming’. (H. Weyl from Philosophy of Mathematics and Natural Science. Princeton University Press, 1949, pg. 52).
Remark: See Time and symbolism: two hidden faces of the prime numbers
