[FOM] Follow-up to Tennant
Dean Buckner
Dean.Buckner at btopenworld.com
Sun Feb 16 12:17:37 EST 2003
Neil Tennant's entertaining parable about the crocodiles neatly illustrates
the difficulty of reconciling a theory of plural reference with set theory.
Given the pair axiom, and the existence of the empty set, then ("Kaboom")
"the hierarchy of hereditarily pure sets mushrooms up in the Platonic
portion" of the crocodiles mind. {} exists, and {{}} exists, and {{{}}}
exists, and ...
Everything in Omega exists. Yet Omega need not exist. So what exactly
_are_ we asserting to exist when we say that Omega also exists?
Let me connect this to the thought about plural reference. A term that has
plural reference, such as "the people at the bar" is not a term for a set.
That is because we can assert an identity between the referents of a plural
term, and objects referred to individually by name, as in "the people at the
bar are Alice, Bob and Carol". But we cannot write
{a,b,c} = a, b and c
for obvious reasons. (It is ill-formed, for a start).
Because of this identity, the assertion of existence using a plural term
amounts to nothing more than the assertion that individual objects exist.
Thus
Alice exists and Bob exists
the people at the bar are Alice and Bob
:. the people at the bar exist
Consequently we do not need axiom of pairs in a theory of plural reference.
The argument above follows by substitution salva veritate, just as in the
following argument
Alice exists
the person at the bar is Alice
:. the person at the bar exists
Nor for the same reason do we require the Axiom of Infinity. As the Axiom
of Pairs is to 2 individuals, so Axiom of Infinity is to infinitely many
individuals. If every object in Omega exists, and in "Omega" were a term of
plural reference, it would follow, without AxInf, that Omega exists.
Conclusion (I think this point has been misunderstood): a name for a set
refers to a single object, and is not a term of plural reference. That was
all I really wanted to say.
As should be plain, in any case from
The set S contains a large number of objects
The expression "a large number of objects" applies to a large number of
objects. The expression "The set S" does not, otherwise the whole sentence
would assert that S contains itself. Similarly, "the set containing all the
numbers between 1 and 100" refers to something different from "all the
numbers between 1 and 100". Yes?
Dean Buckner
London
ENGLAND
Work 020 7676 1750
Home 020 8788 4273
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