In English vocabulary, which of the following words means "that which cannot be corrected" or describes a person or habit that cannot be reformed?

Introduction / Context:
Vocabulary questions in competitive exams often test subtle differences between similar looking words. The phrase "that which cannot be corrected" is frequently expressed in English by a specific adjective used for deeply rooted habits or character traits that no amount of punishment or correction can change. Selecting the correct word requires understanding the precise meanings of all the given options, many of which may appear similar or share prefixes like "il" and "in".

Given Data / Assumptions:
- We are looking for a word meaning "that which cannot be corrected" or "not capable of being reformed".
- Four options are given: Incorrigible, Illegible, Illegal, Indelible.
- Each of these words is a legitimate English word but they differ in meaning.

Concept / Approach:
The key word is "corrected". In the context of a person or behaviour, "correct" means to reform or improve habits. The adjective "incorrigible" comes from this root and means a person whose habits cannot be corrected or reformed, often used for persistent offenders or stubborn behaviour. The other words relate to different ideas such as not being able to read handwriting, breaking the law, or marks that cannot be erased. Understanding each word's definition allows us to choose accurately.

Step-by-Step Solution:
Step 1: Consider "Incorrigible". It literally means incapable of being corrected or reformed, especially regarding behaviour or moral habits. Step 2: Consider "Illegible". This means that handwriting or text cannot be read, often due to poor writing or faint ink. It is related to reading, not moral correction. Step 3: Consider "Illegal". This means contrary to law, forbidden by law, or not legal. It does not carry the idea of being uncorrectable; rather it refers to legality. Step 4: Consider "Indelible". This means a mark or impression that cannot be erased or removed, such as indelible ink or an indelible memory. Step 5: Compare all meanings with the phrase "that which cannot be corrected". Only "incorrigible" directly matches this sense, especially for people or their habits.

Verification / Alternative check:
Check usage examples. One may say "an incorrigible liar" or "incorrigible criminal" to describe someone whose behaviour cannot be improved by punishment. In contrast, one would say "illegible handwriting" for writing that cannot be read, "illegal activities" for actions forbidden by law, and "indelible ink" for ink that cannot be removed from paper. Only "incorrigible" involves the idea of correction of conduct or character.

Why Other Options Are Wrong:
- "Illegible" relates to reading, not to correcting behaviour; it refers to writing that cannot be read clearly.
- "Illegal" refers to a violation of law, not to something beyond correction.
- "Indelible" refers to marks or impressions that cannot be erased, often on surfaces or in memory, and is closer to "permanent" than to "uncorrectable" behaviour.
Therefore, these do not match the target definition in the question.

Common Pitfalls:
Students may confuse "incorrigible" with "indelible" because both carry a sense of permanence. However, "indelible" focuses on physical or emotional marks, whereas "incorrigible" focuses on moral or behavioural traits. The similar looking prefixes "il" and "in" can also mislead learners into thinking that any negatively prefixed word might fit. Understanding the base words "correct", "legible", "legal", and "delete" helps avoid this confusion and leads to the correct choice.

Final Answer:
The correct word meaning "that which cannot be corrected" is Incorrigible.