|Pioneer||Journal where reappraisal appeared||Title||Link to article containing reappraisal|
|Thoralf Skolem||Foundations of Science||"Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics"||13c, Section 3.2|
Thoralf Skolem (1887-1963) was a Swedish logician. Skolem's
extensions provide a modern illustration of the Leibnizian concept
of infinita terminata (bounded infinities). In 1933, Skolem
developed extensions of ℕ. Such an extension, say M, satisfies
the axioms of Peano Arithmetic (and in this sense is indistinguishable
from ℕ). Yet M is a proper extension, of which ℕ
is an initial segment (more precisely, ℕ is isomorphic to an
initial segment of M). Each element of the complement M ∖
ℕ is greater than each element of ℕ and in this sense can be
said to be infinite.
Depending on the background logical system, one can view Skolem's extensions as either potentially or actually infinite (of course in the former case neither ℕ nor M exists as a completed whole). The sense in which elements of M ∖ ℕ are 'infinite' is unrelated to the Aristotelian distinction, and provides a modern formalisation of the infinita terminata.
In his 1966 book Non-standard Analysis, page vii, Robinson
wrote: "The resulting subject was called by me Non-standard Analysis
since it involves and was, in part, inspired by the so-called
Non-standard models of Arithmetic whose existence was first pointed
out by T. Skolem."
More on infinitesimals
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