2014 Mar 6 04:06 PM MST | English5 Grammar4 Math7 [2014]3

Although I have already proven that a repetition of the word 'buffalo' $n$ times is grammatically correct, if $n \geq 2$, a year ago, I am now publishing this finding.


Repeating $n$ times, where $n \geq 2$, any word that can be a noun or verb forms a grammatically valid sentence.


A sentence containing 2 instances of the word is grammatically valid.

[S [N I] [V am]]

The first instance acts as a noun, and the second acts as an intransitive verb.

[subject verb]
[S [N buffalo] [V buffalo]]

A sentence containing 3 instances of the word is grammatically valid.

[S [N I] [VP [V am] [N here]]]

[subject verb object]
[S [N buffalo] [VP [V buffalo] [N buffalo]]]

The first instance acts as a noun, the second acts as a transitive verb, and the third acts as an object.

Adding 2 instances of the word to the first two examples allows it to remain valid–and potentially causes ambiguity–, but this process may be repeated.

This is accomplished by adding relative clauses, whose head may be omitted when the modified noun acts as an object.

[N person] -> [NP [N person] [CP who/Ø [IP [N I] [V am] ___]]]

Since the two previous examples have at least one noun, and adding this kind of relative clause introduces another noun, there will always be an available noun for a new relative clause.

In the first example, this process may be repeated, as shown below:

[[subject [Ø noun verb ___]] verb]
[S [NP [N buffalo] [CP whom/Ø [IP [N buffalo] [V buffalo] ___]]] [V buffalo]]
[[subject [[Ø noun [Ø noun verb ___]] verb ___]] verb]
[S [NP [N buffalo] [CP Ø [IP [NP [N buffalo] [CP Ø [IP [N buffalo] [V buffalo] ___]]] [V buffalo] ___]]] [V buffalo]]

In the second example, this process can be applied to either the subject or object, which results in ambiguity.

[[subject [Ø noun verb ___]] verb object] or
[subject verb [object [Ø noun verb ___]]]

It is proven that $n$ instances of the word is valid for

\(n = 2 + 2x, x \in \mathbb Z\_{\ge 0}\)
\(n = 3 + 2x, x \in \mathbb Z\_{\ge 0}\)
\(\therefore n = 2 + x, x \in \mathbb Z\_{\ge 0}\)

which shows what we wanted to prove:

\[n \geq 2, n \in \mathbb Z\]