What is the least member of {2}?
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What is the least member of {2}?
With respect to which order?
It looks to me like there is only one member so there is just one possible "largest member" or "smallest member"!
Reply to post #2: n1>n2 or n2<n2 or n1=n2
Reply to post #2: The well-ordering principle specifies a least member, what is it?
Just out of curiousity, why is there so much space in your posts.
Sorry, this is not an answer. You mentioned three relations using only their notations, without saying what they are. The least element is defined for a partially ordered set. Please provide a partial (or total) non-strict (i.e., reflexive) order.Quote:
Originally Posted by Hartlw
WikiQuote:
Originally Posted by emakarov
In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. (Wiki)
The concept of the least element makes sense only when the order is specified. Without specifying the order, talking about the well-ordering principle or the least element is like talking about the number of car doors without specifying the car make and model.
Two comments.Quote:
Originally Posted by Hartlw
1. On the Wikipedia statement of well-ordering there is a link for "least element." I suggest you look at it.
2. "A set A with an order relation < is said to be well-ordered if every nonempty subset of A has a smallest element." - "Topology, a first course" Munkres, 1975, pg. 63.
-Dan
1) I looked at it. Don't see your point. Least element is not the same as smallest element?Quote:
Originally Posted by topsquark
2) Circular definition
Frankly, I'm amazed that there should be so much confusion about the ordering of the integers. It is foundational in every (the ones I have seen) text on algebra and analysis. Landau, for example, states that, for any x and y, either x<y, or x>y or x=y.
Emakorov speaks of a non-strict order: If he means either x>y or x less than or equal to y, the well-ordering principle states less than.
I suspect what you are getting at is that{2} has a glb and a lub. Perhaps each set of integers contains its greatest lower bound makes more sense as a principle than the well-ordering principle. Thanks for your time and effort.
If by "the least" you mean "the least with respect to the usual non-strict order ≤ on natural numbers," then the least element of {2} is 2. In fact, as HallsofIvy pointed out, the least element of {2} with respect to any order is 2 because 2 ≤ x for all x ∈ {2}. This is due to the reflexivity of ≤, which is one component of the definition of a non-strict partial order.
At first I asked which order you mean to preclude some tricks like providing a strict (non-reflexive) or some other unusual relation instead of the regular order. I kept on asking because I wanted to make sure you know what "the least element" and "the well-ordering principle" are.
Least clearly means (not <=, but <) total ordering. b&m waffles on this by stating any collection C of positive integers must contain some member m such that whenever c is in C, m<=c, which to me says it contains it's glb. So you can't use the well ordering principle to prove every set of integers contains its glb.
I was trying to track down a proof that assumes total order (literally least) in a proof, in which case it would make a difference. Haven't been able to do so.
Maybe you should explain to me what "least element" and the "well ordering principle" are, just to be sure I get it.Quote:
Originally Posted by emakarov
"Least" means en element that is less or equal to any other element. The important point is that "least" is an adjective, and it is applied to an element. "Least" does not mean an ordering, whether strict or non-strict.Quote:
Originally Posted by Hartlw
Waffles on what? This statement is true.Quote:
Originally Posted by Hartlw
Yes, the well-ordering principle does not apply to arbitrary subsets of integers because not all of them are well-ordered and not all of them even have the lub.Quote:
Originally Posted by Hartlw
I did not understand this. A proof of what? So, your initial question has been answered. What other questions do you have?Quote:
Originally Posted by Hartlw
Wikipedia seems to explain well what the greatest element is, and the least element is similar. The well-ordering principle is here.Quote:
Originally Posted by Hartlw
Ok. I agree. It's a matter of definition and wiki gives your definition. btw, i meant glb not lub, corrected in edit.
Perhaps one could say the concept of order is meaningless for a single integer.