Answer
The definition of n leads to a contradiction because it describes n in fewer than 12 words, even though n is defined as being indescribable in fewer than 12 words. This is Berry’s paradox — a contradiction arising from self-reference and definability.
Work Step by Step
This is a famous logical puzzle known as **Berry’s Paradox**.
---
### Given:
Let
\[
n = \text{“the smallest integer not describable}\\
\text{ in fewer than 12 English words.”}
\]
The problem: There are only **finitely many** strings of 11 or fewer English words, so only **finitely many integers** can be described that way. Thus, there must be some smallest integer not so describable — so \(n\) should exist.
But now…
---
### ❗ The Paradox:
The phrase
> “the smallest integer not describable in fewer than 12 English words”
**is itself a description of \(n\)** — and it **contains only 11 words**!
So we have:
- \(n\) **cannot** be described in fewer than 12 words (by its own definition),
- but we just **described it** in 11 words.
This creates a **contradiction**.
---
### 🌀 Why It Happens:
The contradiction arises because the **definition** refers to what is **definable**, creating a **self-referential** loop.
This is similar in spirit to:
- Russell’s paradox (set of all sets that don’t contain themselves),
- the liar paradox (“This sentence is false.”),
- and Gödel’s incompleteness theorems (limits of formal systems to fully describe themselves).
