I assumed that, since the only number mentioned was an integer, that the usual order relation was intended. I really don't see why this has gone to, now, 16 posts!
Thank you, Thank you, HallsofIvy.
If the concept of well ordering is meaningless for a single digit, and the positive integers are well ordered, then there is no such thing as a single integer subset, so if a subset of the integers contains n, it has to contain a larger number → Archimedes Postulate <-> Euclids Postulate
And also Archimedes postulate (Euclid) implies well ordering.
Now wasn’t that worth 16 posts?
one can ask a similar, but related question:
suppose A is a singleton set: A = {a}. how many possible partial orders can be defined on A?
i humbly submit that there is exactly ONE:
x ≤ y iff x = y, for all x, y in A. we can make this a strict order by defining:
x < y iff x ≠ y.
under this strict ordering A is indeed well-ordered, with smallest element a.
oh, bother!
the natural numbers come with a CANONICAL well-ordering.
if one uses the construction, s(x) = x U {x}, then k < n iff k is an element of n.
for example:
2 = {0,1}, so 1 < 2, and 0 < 2.
the well-ordering of the natural numbers is "axiomatic", that is: it is INTRINSIC, and equivalent in strength to the axiom schema of induction. actually, ZF set theory says something a bit MORE: there exists an infinite well-ordered set (which may, or may not be, the natural numbers). this stops "just short" of the axiom of choice, in that it does not assert that EVERY set is well-ordered (but certainly implies a method of well-ordering any FINITE set, using an injection into the well-ordered infinite set).
the notion that singleton subsets do not exist is absurd, and violates the axiom of extensionality.
the set {2,2} is just the same as the set {2} (they have the same elements).
it seems to me that a more profitable question for you, and one that is far more likely to be better-received on these forums is:
under what constraints on a set S (with whatever algebraic structure and order structure is necessary for our purposes), are the euclidean algorithm and the archimedean property equivalent?
that might actually lead to an interesting and fruitful discussion.
It's hard to believe this isn't spam. Your original question was "What is the least member of {2}?" That is a set with one member. Whether you ask for "least member", "largest member" with respect to whatever order relation, it must be a member so there is only one possible answer!
But the concept of order is NOT meaningless for the set of integers. And even a single integer is a member of the set of integers. The smallest member of {2} is 2. In fact it is also the largest member of {2}, the "evenest" member of {2}, and the "*****est" member of {2} as long as "*****" is an attribute that 2 has- because 2 is the only member of the set so any answerable question about a member of {2} must be "2".
The well ordering principle does apply to sets of integer that have a "lower bound". And we don't need to talk about a "glb". If a set of integers has a lower bound, then it has a smallest member, the least member of the set. We usually apply the term "glb" to sets that do NOT have a smallest member. For example, the set of all positive rational numbers does not have a smallest member. It has glb 0 which is not in the set.