Nonstandard Analysis: Completeness of R via Every Limited Hyperreal

[poissonspot]

In a book I am currently reading, the statement "every limited hyperreal is infinitely close to a real #" is shown to imply the completeness of R, that is that any subset A of R bounded above has a least upper bound. What the author offers to do is introduce this construction: for each natural n, let s_n be the least k in the integers so that k/n is an upper bound of A. Then we are to take an unlimited N and let L, an element of R, be infinitely close to s_N/N.

Without completeness I'm not sure why s_n necessarily exists, can anyone give me some hints? Is it just because once I know the set is bounded above, I can start with an integer greater than this upper bound multiplied by n and "count down" so to say, checking whether each integer less than the last is an upper bound until I find one that is not?

Thanks,

 

[Plato2]

In a book I am currently reading, the statement "every limited hyperreal is infinitely close to a real #" is shown to imply the completeness of R, that is that any subset A of R bounded above has a least upper bound. What the author offers to do is introduce this construction: for each natural n, let s_n be the least k in the integers so that k/n is an upper bound of A. Then we are to take an unlimited N and let L, an element of R, be infinitely close to s_N/N.

Without completeness I'm not sure why s_n necessarily exists, can anyone give me some hints? Is it just because once I know the set is bounded above, I can start with an integer greater than this upper bound multiplied by n and "count down" so to say, checking whether each integer less than the last is an upper bound until I find one that is not?

I would like to know the name of the text/author.
I assume that by limited hyperreal that author means finite.

If you can find a copy of James Henle's Infinitesimal Calculus, there is a good discussion on this problem on page 114. Although Henle does not use limited hyperreal much of that discussion it does use many of the same ideas.

 

[poissonspot]

I would like to know the name of the text/author.
I assume that by limited hyperreal that author means finite.

If you can find a copy of James Henle's Infinitesimal Calculus, there is a good discussion on this problem on page 114. Although Henle does not use limited hyperreal much of that discussion it does use many of the same ideas.

I'm sorry, I'm new to the topic and didn't think about the possibility of different terms. The book is titled Lectures on the Hyperreals by Robert Goldblatt; this bit is on page 56. Comparing to Henle, limited is actually not finite but rather a # that is bound by two real numbers. infinite is not defined in Goldblatt, and rather infinitesimal and unlimited are used, infinitesimal meaning that the absolute value of the # is less than any positive real #, and unlimited to mean that the abs. is greater that any pos. real #.

One question, do the squares denote anything on page 114-115, or are these just formatting?

Thanks,

 

[Plato2]

One question, do the squares denote anything on page 114-115, or are these just formatting?

\(\displaystyle \boxed{N}\) is the standard part of \(\displaystyle N\).

You really have to follow any text very closely.
This subject is relatively new, 1964. So there is absolutely no standard notation.
I do not know of that textbook.

 

[blue_raver22]

As a scientist, it is important to approach mathematical concepts with a critical and analytical mindset. Nonstandard analysis is a mathematical framework that allows for the use of infinitesimal and infinite numbers, which can be a useful tool in certain areas of mathematics. However, it is also important to understand the underlying assumptions and implications of this framework.

In this case, the statement that "every limited hyperreal is infinitely close to a real #" may lead to the completeness of R, but it is important to carefully consider the construction and assumptions being made. The introduction of the unlimited N and the element L being infinitely close to sN/N may seem like a simple solution, but it is necessary to examine why sn necessarily exists in the first place.

One possible approach is to consider the definition of a bounded set. A set A is bounded above if there exists a real number k such that every element in A is less than or equal to k. This means that for any given natural number n, we can start with an integer greater than kn and "count down" until we find an upper bound for A. This process ensures that sn exists for every n, and the construction of L being infinitely close to sN/N follows from the property of hyperreals being infinitely close to reals.

However, it is important to note that this construction relies on the completeness of R, as the process of "counting down" assumes that there is a least upper bound for the set A. Without completeness, this construction may not hold true. Therefore, it is crucial to understand the assumptions and implications of using nonstandard analysis in proving the completeness of R.

In conclusion, while the statement may seem to provide a simple solution to proving the completeness of R, it is important to carefully examine the construction and underlying assumptions being made. As a scientist, it is important to critically analyze mathematical concepts and understand their limitations in order to fully grasp their implications.