Matt Valeriote (McMaster University)
A Paradoxical Function
29 September, 2016
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A Paradoxical Function
Matt Valeriote
McMaster University
29 September, 2016
Matt Valeriote (McMaster University)
A Paradoxical Function
29 September, 2016
2 / 12
What is a Function?
From The Philosophy of Logical Atomism
“Everything is vague to a degree you do not realize till you have tried to
make it precise.” – Bertrand Russell (1872-1970)
Definition (from Stewart)
A function f is a rule that assigns to each element x in a set A exactly one
element, called f (x), in a set B.
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Numbers from Sentences
Consider the following sentences. Each of them defines a number.
“The number of planets in the solar system.”
“The cube of the number of living people on the earth.”
“The number of people in this room.”
We can use phrases or sentences from the English language to specify
or define integers.
Some phrases and sentences may be ambiguous, but in principle, we
should be able to correctly unravel their meaning.
We can measure a sentence according to the size of the number it
defines, if any.
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The Berry Function
Definition
For n a positive integer, define B(n) to be the function with domain N, the
set of natural numbers, such that for n ∈ N,
B(n) =
the least natural number that cannot be defined using
n or fewer words.
Is it well defined?
The number of words in the Oxford English Dictionary is about
616,500.
So, for a given n, there are at most (616, 500)n n-word English
sentences.
Thus at most (616, 500)n numbers can be defined using at most n
words.
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Some Values of the Berry Function
Definition
B(n) =
the least natural number that cannot be defined using
n or fewer words.
Some values
B(0) = 0
B(1) = 21
B(2) = 101
B(3) =??
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B(13)
A contradiction
Let N = B(13).
Then, by definition, N is:
The least natural number that cannot be defined with thirteen or
fewer words.
The number of words in red is thirteen.
So, in fact, N can be defined using thirteen words.
Thus B(13) is not defined.
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Berry and Russell
This paradox was introduced
by the famous logician and
philosopher Bertrand Russell
early last century.
Amazing Fact
McMaster University is home to
the Bertrand Russell Archives. It
is a world center for the scholarly
study of Bertrand Russell. You
can drop by Mills Library to tour
the archives.
Matt Valeriote (McMaster University)
He is well known for another
important paradox, called
Russell’s Paradox, and so to
avoid confusion, the current
paradox is referred to as the
Berry Paradox.
It is named after the Oxford
University Librarian Mr. G.
G. Berry, who, according to
Russell, suggested it to him.
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Russell’s (other) Paradox
Sets that are not members of themselves
Consider T , the set of all teacups.
Clearly T ∈
/ T , since T is not a teacup.
How about N, the set of all non-teacups?
We have N ∈ N.
So, it appears as if some sets can be members of themselves.
The Paradox
Let R be the set of all sets that are not members of themselves.
Either R ∈ R or R ∈
/ R, but
neither is possible.
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Conclusion
Circularity
The rule used to define the function B seems to be well defined, but
there are some subtle difficulties with it.
The definition of B(n) depends on being able to interpret an english
sentence as a number, but
interpreting an english sentence as a number is tied in with the
definition of Berry’s function.
The Liar’s Paradox
A simpler version of the paradox inherent in the definition of B is
manifested in the following sentence, known as the Liar’s Paradox (or
Eubilides’ Paradox):
THIS SENTENCE IS FALSE.
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