Obviously the enumerating notation of sets like $\{ 1,2,3,4,5,\ldots \}$ is not unique because it is not clearly defined how to continue the dots "$\ldots$". For example the above defined set could also be $\{ 1,2,3,4,5,23,42,128,\ldots \}$.
Unfortunately this example is not convincing for each student because one can argue, that in the above example nearly each person thinks that the given set is the set of the natural numbers. So I was looking for a more convincing example and came up with the set $\{2,\, 4,\, 6,\, 8,\, \ldots\}$. This set may be:
- the set of even numbers beginning at 2: $\{2,\, 4,\, 6,\, 8,\, \ldots\} = \{2,\, 4,\, 6,\, 8,\, 10,\, 12,\, 14,\, \ldots\}$
- the set of non prime numbers together with 2: $\{2,\, 4,\, 6,\, 8,\, \ldots\} = \{2,\, 4,\, 6,\, 8,\, 9,\, 10,\, 12,\, \ldots\}$
- the set of all numbers with 2 and 3 as the only prime divisors: $\{2,\, 4,\, 6,\, 8,\, \ldots\} = \{2,\, 4,\, 6,\, 8,\, 12,\, 16,\, 18,\, \ldots\}$
Do you have a better example? I will accept the answer with an example which is most convincing for students, that the enumerating notation of sets is not unique. (I know this is subjective, but I hope that's okay).
Of course one can get more ambiguity with shorter initial segment, but when the initial segment is too short (two elements), then there is no obvious candidate.
Three elements might be ok. $\{1, 2, 3, …\} = \mathbb{N}$ or $\{1, 2, 3, …\} = \{1, 2, 3, 5, 8, …\}$. Another one: $\{1, 3, 5, 7, …\} = $ all odd numbers or odd numbers with at most two divisors.