scholarly journals A SOUNDNESS ANALYSIS OF ZENO’S OF ELEA DICHOTOMY

2021 ◽  
pp. 27-42
Author(s):  
Igor Berestov
Keyword(s):  
Finite Time ◽  
Infinite Sequence ◽  
Sequence Of Movements

We are studying three basic interpretations of the Dichotomy aporia, in which Zeno tries to prove the impossibility of movement. In all these interpretations, the key assumption is the dubious statement about the impossibility of performing an infinite sequence of actions in a finite time. However, we show that in the two interpretations of the Dichotomy it is possible to get rid of the dubious key assumption, replacing it with the seemingly much more reliable assumption that covering the distance is representable as a sequence of displacements. Our approach is based on the thesis proved by P. Benacerraf that completing an infinite sequence of movements in an interpretation of the Dichotomy is not sufficient to arrive to the end of the distance.

High and low Kleene degrees of coanalytic sets

1983 ◽  
Vol 48 (2) ◽  
pp. 356-368 ◽  
Author(s):  
Stephen G. Simpson ◽  
Galen Weitkamp
Keyword(s):  
Characteristic Function ◽  
Cross Section ◽  
Finite Time ◽  
Infinite Sequence ◽  
Turing Machines ◽  
Jump Operator ◽  
Halting Problem ◽  
Computing Machine ◽  
Definition Of ◽  
Countably Infinite

We say that a set A of reals is recursive in a real y together with a set B of reals if one can imagine a computing machine with an ability to perform a countably infinite sequence of program steps in finite time and with oracles for B and y so that decides membership in A for any real x input to by way of an oracle for x. We write A ≤ yB. A precise definition of this notion of recursion was first considered in Kleene [9]. In the notation of that paper, A ≤yB if there is an integer e so that χA(x) = {e}(x y, χB, 2E). Here χA is the characteristic function of A. Thus Kleene would say that A is recursive in (y, B, 2E), where 2E is the existential integer quantifier.Gandy [5] observes that the halting problem for infinitary machines such as , as in the case of Turing machines, gives rise to a jump operator for higher type recursion. Thus given a set B of reals, the superjump B′ of B is defined to be the set of all triples 〈e, x, y〉 such that the eth machine with oracles for y and B eventually halts when given input x. A set A is said to be semirecursive in y together with B if for some integer e, A is the cross section {x: 〈e, x, y 〉 ∈ B′}. In Kleene [9] it is demonstrated that a set A is semirecursive in y alone if and only if it is

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