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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 *infinitum terminatum* (bounded infinity). 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 an element of the complement M
∖ ℕ is 'infinite' is unrelated to the Aristotelian
distinction, and provides a modern formalisation of the *infinitum
terminatum*.

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."

See also

Fermat

Leibniz

Euler

Cauchy

Riemann

Cantor

Robinson

Nelson

Hrbacek

Kanovei

Infinitesimal topics

More on infinitesimals

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