Notes on “Skolem’s Paradox” and its Philosophical Implications
Branden Fitelson
04/18/07
1
Hunter on the “Paradox” and Its Implications
1.1
Hunter on the Upward LST and NumberTheoretic Concepts
I begin with some of the material from pages 205–208 of Hunter. Here, Hunter provides some clarification
of the technical results surrounding “Skolem’s Paradox,” and he offers his own resolution of the “para
dox.” First, he gives some results pertaining to the upward LST and numbertheoretic concepts (two kinds).
49.1
. No firstorder theory can have as its
only
model one whose domain is the set of natural numbers
N
.
Proof.
This “follows” from the key lemma [45.15] of the completeness theorem of
QS
. Every consistent
firstorder theory
T
has a Henkin model
M
0
T
(in addition to its
intended
model
M
T
, the domain of which
is, say,
N
), the domain of which is the set of closed terms of
T
. Closed terms are symbols, not numbers.
So, the domain of
M
0
T
is not
N
. [At least, we don’t
intend
to be talking about numbers when we talk about
Henkin models! Structuralists (like Quine) need not worry about this weak kind of ambiguity, so long as
there is a structurepreserving 1–1 mapping – an
isomorphism
– between the two domains. Our proof of
lemma 45.15 did not
construct
such an isomorphism. But, it seems that one should exist here, since both
sets are denumerable, and the models are
equivalent
. We’ll return to these issues in depth, below.]
49.2
. Let
T
be any firstorder theory. Fix the meanings of the logical connectives and quantifiers [the
logical
constants
— on which, see the
Stanford Encyclopedia of Philosophy
entry by our own John MacFarlane] in
the usual way.
And, let the axioms of
T
fix the meanings of the nonlogical symbols (the prdicates,
functions,
etc
.) of
T
— to the extent that they
can
fix these meanings. Even if
T
has denumerably many
axioms, the axioms of
T
cannot force us to interpret any predicate symbol in
T
as meaning “is a natural
number”, and they cannot force us to interpret any expression in
T
as the name of a natural number.
Proof.
Again, this “follows” from 45.15, which says that
T
will have an
unintended
model
M
0
T
≠
M
T
. As
such, any expression of
T
that is a name of a natural number (or means “is a natural number”) on the
intended interpretation of
T
will be the name of some closed term of
T
(or will be a property whose
extension is not any subset of
N
but some subset of the set of closed terms of
T
) on
M
T
.
So, we are
not forced to interpret any expression of
T
as the name of a natural number (or as meaning “is a natural
number”). The moral here is supposed to be that we cannot unambiguously define
numeral nouns
using
just the narrowly logical structure of any firstorder theory. But, again, this weak sense of ambiguity (the
“intuitive” lack of “
identity isomorphism
” between
M
0
T
and
M
T
) is one that needn’t bother a structuralist,
since 45.15 is consistent with there being an
isomorphism
between
M
T
and
M
0
T
(more on this below).
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 Spring '07
 FITELSON
 Logic, Platonism, Mathematical logic, Model theory, Firstorder logic

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