Axiom of Completeness

[Ceorl]

Hi,

I have an issue. Quote, "For example, three upper bounds for the set of negative real numbers are 0, 1, 97. Of course any positive real number is an upper bound for this set. However, the least upper bound is 0 because any number less than zero is not upper bound."

My problem is trying to understand what positive upper bounds are doing is a set of negative real numbers. I would say negative one is the upper bound.

It then goes to say, "The greatest lower bound number of the set of numbers of the form 1/n, where n is a natural number is 0 because first it is a lower bound and secondly any number greater than 0 is not lower bound."

Since n is a natural number I would say 1 is the lower bound.

What do I fail to see?

 

[stapel]

I have an issue. Quote, "For example, three upper bounds for the set of negative real numbers are 0, 1, 97. Of course any positive real number is an upper bound for this set. However, the least upper bound is 0 because any number less than zero is not upper bound."

My problem is trying to understand what positive upper bounds are doing is a set of negative real numbers. I would say negative one is the upper bound.

So -1 is an upper bound (an upper limit, a value which is no smaller than any other element of the set) for the negative reals, which include -0.5, -0.03, -0.0000009, etc?

An "upper bound" is just a value which is no less than any other element of the set. For the set {1, 2, 3}, 3 is the lowest (smallest, "least") number that is larger than or equal to every element in the set. But 4 is an upper bound, too, because each of 1, 2, and 3 is less than 4. It's just that 4 is not the least upper bound, so we don't usually use it as "the upper bound" (where "the" is taken to mean "the least").

In the same way, 0 is an upper bound for all negatives, since all negatives must be less than zero. When you're dealing with negative integers (so {...-6, -5, -4, -3, -2, -1}), then -1 would be the least upper bound. But for the reals, which include -0.0001, -0.000001, -0.000000001, etc, no negative number can possibly be an upper bound, because there will always be another negative between it and zero.

But, just as 4 is an upper bound for {1, 2, 3}, so also 97 is an upper bound for the negative reals. It's not a useful upper bound, but it is a bound, since no negative number is greater than 97.

It then goes to say, "The greatest lower bound number of the set of numbers of the form 1/n, where n is a natural number is 0 because first it is a lower bound and secondly any number greater than 0 is not lower bound."

Since n is a natural number I would say 1 is the lower bound.

So 1/2 is greater than 1? And 1/3? And 1/97? :shock:

 

[pka]

What do I fail to see?

You fail to understand that the completeness axiom applies applies to the real numbers.
If you are thinking about sets of rational numbers or a set of integers alone it does apply.
In other words, it applies only to subsets of real numbers in the set of real numbers.

 

[Ceorl]

So -1 is an upper bound (an upper limit, a value which is no smaller than any other element of the set) for the negative reals, which include -0.5, -0.03, -0.0000009, etc?

An "upper bound" is just a value which is no less than any other element of the set. For the set {1, 2, 3}, 3 is the lowest (smallest, "least") number that is larger than or equal to every element in the set. But 4 is an upper bound, too, because each of 1, 2, and 3 is less than 4. It's just that 4 is not the least upper bound, so we don't usually use it as "the upper bound" (where "the" is taken to mean "the least").

In the same way, 0 is an upper bound for all negatives, since all negatives must be less than zero. When you're dealing with negative integers (so {...-6, -5, -4, -3, -2, -1}), then -1 would be the least upper bound. But for the reals, which include -0.0001, -0.000001, -0.000000001, etc, no negative number can possibly be an upper bound, because there will always be another negative between it and zero.

But, just as 4 is an upper bound for {1, 2, 3}, so also 97 is an upper bound for the negative reals. It's not a useful upper bound, but it is a bound, since no negative number is greater than 97.


So 1/2 is greater than 1? And 1/3? And 1/97? :shock:

Thanks for your explanation to my query on the Axiom of Completeness in this matter.

 

[Ceorl]

Hi,

You're right! The Axiom of Completeness is something new to me. You've been helpful in this matter.

Thanks.