Interesting number paradox
Encyclopedia
The interesting number paradox is a semi-humorous paradox
that arises from attempting to classify natural number
s as "interesting" or "dull". The paradox states that all natural numbers are interesting. The "proof" is by contradiction
: if there were uninteresting numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, producing a contradiction. This contradiction continues as the next smallest number becomes the "smallest uninteresting number," therefore becoming "interesting."
.
Proof by Contradiction: Assume that you have a non-empty set of natural numbers that are not interesting. Due to the well-ordered property of the natural numbers, there must be some smallest number in the set of uninteresting numbers. Being the smallest number of a set one might consider not interesting makes that number interesting after all: a contradiction.
or an antinomy
of definition. Any hypothetical partition
of natural numbers into interesting and dull sets seems to fail. Since the definition of interesting is usually a subjective, intuitive notion of "interesting", it should be understood as a half-humorous application of self-reference
in order to obtain a paradox
. (The paradox is alleviated if "interesting" is instead defined objectively: for example, the smallest integer that does not, as of November 2011, appear in an entry of the On-Line Encyclopedia of Integer Sequences
was 12407.) Depending on the sources used for the list of interesting numbers, a variety of other numbers can be characterized as uninteresting in the same way.
However, as there are many significant results in mathematics that make use of self-reference (such as Gödel's Incompleteness Theorem), the paradox illustrates some of the power of self-reference, and thus touches on serious issues in many fields of study.
This version of the paradox applies only to well-ordered sets with a natural order, such as the natural number
s; the argument would not apply to the real numbers.
One proposed resolution of the paradox asserts that only the first uninteresting number is made interesting by that fact. For example, if 39 and 41 were the first two uninteresting numbers, then 39 would become interesting as a result, but 41 would not since it is not the first uninteresting number. However, this resolution is invalid, since the paradox is proved by contradiction: assuming that there is any uninteresting number, we arrive to the fact that that same number is interesting, hence no number can be uninteresting; its aim is not in particular to identify the interesting or uninteresting numbers, but to speculate whether any number can in fact exhibit such properties.
An obvious weakness in the proof is that what qualifies as "interesting" is not defined. However, assuming this predicate is defined with a finite, definite list of "interesting properties of positive integers", and is defined self-referentially
to include the smallest number not in such a list, a paradox arises. The Berry paradox
is closely related, since it arises from a similar self-referential definition. As the paradox lies in the definition of "interesting", it applies only to persons with particular opinions on numbers: if one's view is that all numbers are boring, and one finds uninteresting the observation that 0 is the smallest boring number, there is no paradox.
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
that arises from attempting to classify natural number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s as "interesting" or "dull". The paradox states that all natural numbers are interesting. The "proof" is by contradiction
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
: if there were uninteresting numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number, producing a contradiction. This contradiction continues as the next smallest number becomes the "smallest uninteresting number," therefore becoming "interesting."
Proof
Claim: There is no such thing as an uninteresting natural numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
.
Proof by Contradiction: Assume that you have a non-empty set of natural numbers that are not interesting. Due to the well-ordered property of the natural numbers, there must be some smallest number in the set of uninteresting numbers. Being the smallest number of a set one might consider not interesting makes that number interesting after all: a contradiction.
Paradoxical nature
Attempting to classify all numbers this way leads to a paradoxParadox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
or an antinomy
Antinomy
Antinomy literally means the mutual incompatibility, real or apparent, of two laws. It is a term used in logic and epistemology....
of definition. Any hypothetical partition
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
of natural numbers into interesting and dull sets seems to fail. Since the definition of interesting is usually a subjective, intuitive notion of "interesting", it should be understood as a half-humorous application of self-reference
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
in order to obtain a paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
. (The paradox is alleviated if "interesting" is instead defined objectively: for example, the smallest integer that does not, as of November 2011, appear in an entry of the On-Line Encyclopedia of Integer Sequences
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
was 12407.) Depending on the sources used for the list of interesting numbers, a variety of other numbers can be characterized as uninteresting in the same way.
However, as there are many significant results in mathematics that make use of self-reference (such as Gödel's Incompleteness Theorem), the paradox illustrates some of the power of self-reference, and thus touches on serious issues in many fields of study.
This version of the paradox applies only to well-ordered sets with a natural order, such as the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s; the argument would not apply to the real numbers.
One proposed resolution of the paradox asserts that only the first uninteresting number is made interesting by that fact. For example, if 39 and 41 were the first two uninteresting numbers, then 39 would become interesting as a result, but 41 would not since it is not the first uninteresting number. However, this resolution is invalid, since the paradox is proved by contradiction: assuming that there is any uninteresting number, we arrive to the fact that that same number is interesting, hence no number can be uninteresting; its aim is not in particular to identify the interesting or uninteresting numbers, but to speculate whether any number can in fact exhibit such properties.
An obvious weakness in the proof is that what qualifies as "interesting" is not defined. However, assuming this predicate is defined with a finite, definite list of "interesting properties of positive integers", and is defined self-referentially
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
to include the smallest number not in such a list, a paradox arises. The Berry paradox
Berry paradox
The Berry paradox is a self-referential paradox arising from the expression "the smallest possible integer not definable by a given number of words". Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G...
is closely related, since it arises from a similar self-referential definition. As the paradox lies in the definition of "interesting", it applies only to persons with particular opinions on numbers: if one's view is that all numbers are boring, and one finds uninteresting the observation that 0 is the smallest boring number, there is no paradox.
See also
- Grelling–Nelson paradox
- Richard's paradoxRichard's paradoxIn logic, Richard's paradox is a semantical antinomy in set theory and natural language first described by the French mathematician Jules Richard in 1905. Today, the paradox is ordinarily used in order to motivate the importance of carefully distinguishing between mathematics and metamathematics...
- Church-Turing thesis
- Gödel's incompleteness theorem
- Hardy–Ramanujan number1729 (number)1729 is the natural number following 1728 and preceding 1730.1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan...
(An example of an interesting number) - List of paradoxes
- 224 (number), the smallest natural number which does not have its own Wikipedia article.
- Unexpected hanging paradoxUnexpected hanging paradoxThe unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or prediction paradox is a paradox about a person's expectations about the timing of a future event The unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or...