The
Logical Roots of Indeterminacy
GILA SHER
I. Indeterminacy as Relativity to Logical Frameworks
In 1915, Leopold Lowenheim proved a remarkable theorem:
(L)
If the domain is at least denumerably infinite, it is no longer the case that a first
order fleeing equation is satisfied for arbitrary values of the relative coefficients.
(Lowenheim 1915, p. 235)
In contemporary terminology the theorem says that if a formula
@
of firstorder
logic with identity is finitely valid but not valid, then for every cardinal A
2
No,
@
is not Avalid (i.e., if
[email protected]
is satisfiable in an infinite model, then for every
infinite cardinal A,
[email protected]
is satisfiable in a model of cardinality A).' It follows
from this theorem, Lowenheim pointed out, that "[all1 questions concerning
the dependence or independence of Schroder's, Miiller's. or Huntington's class
axioms are decidable (if at all) already in adenumerable domain." (19 15, p. 240).
In a series of articles, Thoralf Skolem (1920, 1922, 1929, 1941, 1958) pre
sented a new version of Lowenheim's theorem and offered a new kind of proof
for it. We can formulate Skolem's result as
(LS)
Let T be a countable 1storder theory (where a theory is a set of 1storder
sentences). Then, if T has a model, T has a countable model; in particular:
(i) T has a model in the natural numbers; (ii) If !?I is a rnodel of T, then there is
a countable submodel
'LL'
of
'LL,
such that
a'
is a model of T.
Skolem's theorem was extended by Tarski to
(LST)
Let T be a set of sentences in a language. L of cardinality
K
2
No. Then, if
T has an infinite model (a model with an infinite universe), T has a model of
cardinality A for every A
1
K.
Skolem regarded LS as signaling the'.unaioidable relativity of mathemati
cal notions to logical frameworks. Skolem's view is sometimes referred to as
Skolem's paradox on the basis of passages such as this:
So far as I know, no one has called attention to this peculiar and apparently paradoxical
state of affairs. By virtue of the [settheoretical] axioms we can prove the existence of
higher cardinalities, of higher number classes, and so forth. How can it be, then, that the
The Logical Roots of Indeterminacy
101
entire domain B
[tk
~iverse
of an "LS model" of set theory] can already beenumerated
by means of the
fin^;^
positive integers? (Skolem 1922, p. 295)
However, the "pamdox" is swiftly explained away:
The explanation is not difficult to find. In the axiomatization "set" does not mean an
arbitrarily defined collection; the sets are nothing but objects that are connected with one
another through certain relations expressed by the axioms. Hence there is no contradic
tion at all if a set M of the domain B is nondenumerable in the sense of the axiomatization;
for this means merely that
within
B there occurs no onetoone mapping
0
of M onto
Zo
(Zermelo's number sequence). Nevertheless there exists the possibility of numbering
all objects in B, and therefore also the elements of M, by means of the positive integers;
of course, such an enumeration too is a collection of certain pairs, but this collection is
not a "set" (that is, it does not occur in the domain B). (Ibid.)
LS
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 Spring '07
 FITELSON
 Ontology, Speak, Mathematical logic, Model theory, Firstorder logic, indeterminacy, logical frameworks, GILA SHER

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