The well-ordering theorem, also known as Zermelo's theorem, says that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering.
A strict total order on a set X is a strict partial order on X in which any two elements are comparable. That is, a total order is a binary relation < on some set X, which satisfies the following for all a,b and c in X:
Not a < a (irreflexive).
If b < c and a < b, then a < c (transitive).
If a =/= b, then a < b or b < a (connected).
Zermelo's approach was to argue that each element of a set could be withdrawn from it sequentially and then, if necessary, put into correspondence with a set controlled by the < relation. That is, the members of X are ranked in accordance with Y, a set of numbers ordered by <.
In the point set (0,1] it is obvious that no smallest number can exist after 0.
That is, limn-->inf. rn = 0. Otherwise, rn --> rn+1. That is a "small number" always implies a yet smaller number. Or see the epsilon-delta proofs.
But then, the same applies to any real, not only 0, that appears in the point set [0,1].
This then means that if there is a smallest set after 0, that set must be empty -- which would violate the well ordering concept. Only if we permit well-ordering to mean that any two members of a set imply a strict ordering, are we out of the woods.
Consider:
x1 < x2 -->
x2 < x3 -->
.
.
.
xn-1 < xn
That is, redefining well-ordering and using ordinary mathematical induction should help.
Of course, it has been argued that the axiom of choice proves well-ordering (in first order logic). I suppose that would mean if we have a set of ever smaller elements, we should be able to pick out one by some metaphysical method and call that the smallest. Rather than go to that length, I prefer a modified definition of well-ordering. In fact, all we really need to say is that any two members of a set of numbers obey the relation <, as in a < b or b < a. That is, ALLx e R(xa < xb or the converse).
A strict total order on a set X is a strict partial order on X in which any two elements are comparable. That is, a total order is a binary relation < on some set X, which satisfies the following for all a,b and c in X:
Not a < a (irreflexive).
If b < c and a < b, then a < c (transitive).
If a =/= b, then a < b or b < a (connected).
Zermelo's approach was to argue that each element of a set could be withdrawn from it sequentially and then, if necessary, put into correspondence with a set controlled by the < relation. That is, the members of X are ranked in accordance with Y, a set of numbers ordered by <.
In the point set (0,1] it is obvious that no smallest number can exist after 0.
That is, limn-->inf. rn = 0. Otherwise, rn --> rn+1. That is a "small number" always implies a yet smaller number. Or see the epsilon-delta proofs.
But then, the same applies to any real, not only 0, that appears in the point set [0,1].
This then means that if there is a smallest set after 0, that set must be empty -- which would violate the well ordering concept. Only if we permit well-ordering to mean that any two members of a set imply a strict ordering, are we out of the woods.
Consider:
x1 < x2 -->
x2 < x3 -->
.
.
.
xn-1 < xn
That is, redefining well-ordering and using ordinary mathematical induction should help.
Of course, it has been argued that the axiom of choice proves well-ordering (in first order logic). I suppose that would mean if we have a set of ever smaller elements, we should be able to pick out one by some metaphysical method and call that the smallest. Rather than go to that length, I prefer a modified definition of well-ordering. In fact, all we really need to say is that any two members of a set of numbers obey the relation <, as in a < b or b < a. That is, ALLx e R(xa < xb or the converse).
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