Why did John Wells need three lexical sets--NORTH, FORCE and THOUGHT--for the same vowel /ɔː/?

Question

The standard Lexical sets for English were introduced by professor John Wells which are widespread. Each lexical set represents a vowel present in a number of words, for example: the THOUGHT vowel /ɔː/ is found in taught, sauce, hawk, jaw, broad. The GOAT vowel /əʊ/ is found in soap, joke, home, know, so, roll etc.
However, there are three lexical sets NORTH, FORCE and THOUGHT for the same vowel /ɔː/ especially in British English. He has included different words in each lexical set:

THOUGHT: taught, sauce, hawk, jaw, broad
NORTH: for, war, short, scorch, born, warm
FORCE: four, wore, sport, porch, borne, story

(All the above information comes from Wikipedia)
All those words sound the same to me (a non-native speaker of English who is learning British English). I can't detect any difference between them and looking the sets up on UCL website, there is no information on whether there is any difference between them or not.
Why did John Wells need three lexical sets for the same vowel when he could easily have incorporated those words in one lexical set (say for example THOUGHT)? Are those vowels--NORTH, FORCE and THOUGHT--different in British English? Can anyone provide the phonetic values/realizations of those vowels they have in British English?
Edit:
I emailed professor John Wells and asked him why there were three Lexical sets for the same vowel. I said to him "someone (I meant @Nardog) told me that the whole point of lexical sets is to make it easier to describe differences between [vowels in different] accents. He replied to me and said:

"Someone" is correct. And not only American English, but also Scottish English and various other varieties. Read the book!
John Wells

Nardog · Accepted Answer

Because they differ in (Wells's model of) General American.
The whole point of lexical sets is to make it easier to describe differences between accents. Since not only phonetic values but the distribution of phonemes vary across accents, it's often not enough to say e.g. "What is phoneme X in Received Pronunciation is realized as Y in this accent" when describing an accent. By using lexical sets, you can illustrate the features of an accent in a succinct way.
In Wells's model of GenAm, THOUGHT/NORTH/FORCE have /ɔ/, /ɔr/, and /or/ (GOAT + /r/), respectively. Note that Accents of English was published in 1982 (and written mostly in the 1970s). Obviously nowadays in the US, NORTH and FORCE are mostly merged, and THOUGHT is increasingly being merged with PALM.
Wells's are called "standard" lexical sets because they derive from the differences between RP and GenAm. This means some accents differ in ways that the standard sets cannot account for. For instance, some varieties of Scottish English use different vowels in fern, fir, and fur, which all belong to the NURSE set.

herisson · Answer

A lexical set does not represent a vowel.
It represents a set of words that are all pronounced with the same vowel phoneme in Wells's two reference accents of "Received Pronunciation" and "General American". These are artificial standards and as Nardog says, Accents of English was written several decades ago, so this is not exactly equivalent to "British English" and "American English".
This means any word in the NORTH set is pronounced with the same vowel phoneme as any other word in the NORTH set, any word in the KIT set is pronounced with the same vowel phoneme as any other word in the KIT set, and so on.
It does not mean that words in different lexical sets always have different vowel phonemes: that is not the case.
Neither accent uses a separate distinct vowel for each of Wells's lexical sets. But the sets that have identical vowels differ between the two accents.

Why did John Wells need three lexical sets--NORTH, FORCE and THOUGHT--for the same vowel /ɔː/?

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