Continu’ous time goes by Russell. (English) Zbl 1113.03012

Lück, Uwe

Notre Dame J. Formal Logic 47, No. 3, 397-434 (2006).

Summary: Russell and Walker proposed different ways of constructing instants from events. For an explanation of “time as a continuum,” S. K. Thomason [J. Philos. Log. 13, 85–96 (1984; Zbl 0556.03005)] favored Walker’s construction. The present article shows that Russell’s construction [B. Russell, Our knowledge of the external world. Chicago etc.: Open Court (1914; JFM 45.0122.03)] fares as well. To this end, a mathematical characterization problem is solved which corresponds to the characterization problem that Thomason solved with regard to Walker’s construction. It is shown how to characterize those event structures (formally, interval orders) which, through Russell’s construction of instants, become linear orders isomorphic to a given (or, deriving, to some – nontrivial ordered) real interval. As tools, separate characterizations for each of resulting (i) Dedekind completeness, (ii) separability, (iii) plurality of elements, (iv) existence of certain endpoints are provided. Denseness is characterized to replace Russell’s erroneous attempt. Somewhat minimal nonconstructive principles needed are exhibited, and some alternative approaches are surveyed.

MSC:

03A05	Philosophical and critical aspects of logic and foundations
01A60	History of mathematics in the 20th century
03E25	Axiom of choice and related propositions
06A05	Total orders

Keywords:

time as a continuum; axiom of choice; denseness; constructing instants from events; interval orders; Dedekind completeness; separability; plurality of elements

Citations:

Zbl 0556.03005; JFM 45.0122.03