Archimedean property concept

I want to know what the "big deal" about the Archimedean property is. Abbott states it is an important fact about how $\Bbb Q$ fits inside $\Bbb R.$

First, I want to know if the following statements are true:

The Archimedean property states that $\Bbb N$ isn't bounded above--some natural number can be found such that it is greater than some specified real number.

The Archimedean property also states that there is some rational $\frac1n, n \in\Bbb N$ such that it is less than some specified real number.

Secondly, what do the above statements imply about the connection between $\Bbb Q$ and $\Bbb R?$ Does it imply that $\Bbb R$ fills the gaps of $\Bbb Q$ and $\Bbb N;$ is it the proof for $\Bbb R$ completing $\Bbb Q?$

Lastly, what insights have you obtained from the Archimedean property?

I am sorry if some of my questions are unclear.

It has been pointed out above that your two versions of the Archimedean property should be equivalent--$\Bbb N$ has no upper bound if and only if $\left\{\frac1n:n\in\Bbb N\right\}$ has no positive lower bound. Your presentations of these two versions were misleading, however (and still are). More precisely:

• Given any real number $x$, there is a natural number $n$ greater than $x$. This natural number $n$ will of course depend on which $x$ we're looking at in the first place, but we will always be able to find some (infinitely-many, in fact) natural number greater than $x$, regardless of the $x$ we started with.
• Given any positive real number $y$, there is a natural number $n$ such that $\frac1n$ is less than $y$. Again, the natural number $n$ will depend on which $y$ we're looking at.

It's a good exercise to show that these are equivalent statements.

Now, let me try to explain what Abbott meant by "how $\Bbb Q$ fits inside $\Bbb R$." Specifically, I will show that the following are equivalent:

1. The Archimedean property holds.
2. Given any $a,b\in\Bbb R$ with $a<b,$ there is some $r\in\Bbb Q$ such that $a<r<b$. (More briefly, $\Bbb Q$ is dense in $\Bbb R$.)
3. Given any $a,b\in\Bbb R$ with $a<b,$ there is some $p\in\Bbb Z$ and some $n\in\Bbb N$ such that $a<\frac{p}{2^n}<b$.

It is clear that 3 $\implies$ 2. To see that 2 $\implies$ 1, take any $x>0,$ and find $r\in\Bbb Q$ such that $0<r<x$. We know that $r=\frac mn$ for some positive integers $m,n,$ so $\frac1n\le\frac mn=r<x,$ and so the Archimedean property holds.

To see that 1 $\implies$ 3, we proceed by contrapositive, so suppose that there exist $a,b\in\Bbb R$ with $a<b$ such that for all $p\in\Bbb Z$ and all $n\in\Bbb N,$ we have $\frac{p}{2^n}\le a$ or $\frac{p}{2^n}\ge b$. For each $n\in\Bbb N,$ let $p_n$ the greatest integer $p$ such that $\frac{p}{2^n}\le a.$ (By the Well-ordering property of $\Bbb N,$ such a $p_n$ exists.) From this, it follows by hypothesis that $$\frac{p_n}{2^n}\le a<b\le\frac{p_n+1}{2^n}$$ for all $n\in\Bbb N$. But then $$0<b-a\le\frac{p_n+1}{2^n}-\frac{p_n}{2^n}=\frac1{2^n}$$ for all $n\in\Bbb N.$ Since $n<2^n$ for all $n\in\Bbb N,$ then it follows that $0<b-a<\frac1n$ for all $n\in\Bbb N,$ and so the Archimedean property fails. Thus, by contrapositive, 1 $\implies$ 3.

In a group with a linear order, if $a$ and $b$ are two positive elements, $a$ is said to be infinitesimal with respect to $b$, if no positive integer multiple of $a$ is greater than $b$. Then the linear order is called Archimedean if there are no infinitesimal elements. The first two statements you mentioned are easy consequences of $\mathbb{R}$ being Archimedean as a linear order:

If $r$ is any positive real number, since $1$ is not infinitesimal with respect to $r$ by the Archimedean property, there exists a natural number $n$ such that $n.1=n> r$. This shows $\mathbb{N}$ is not bounded above.

The second statement is also true for positive real numbers: Suppose we're given a positive real number $r$. Then by the previous property, there exists a natural number $n$ such that $1/r < n$. Therefore $1/n<r$.

But to make sense of the rest of your question, we have to use the notion of Archimedean normed fields. If $F$ is a field with an absolute value $|\cdot|$, then $F$ is called Archimedean if, given any element $x\in F$, some positive integer multiple of $x$ has absolute value greater than 1: $\exists n\in \mathbb{N}, \ |nx|>1$.

This definition of Archimedean implies the earlier version. Indeed, suppose the normed Archimedean field $F$ is also a linear order, with $|x|=x$ for positive elements. Let $a,b\in \mathbb{F}$ be positive. Then letting $x=a/b$, there must exist an $n$ such that $|nx|>1$. Then by the triangle inequality $|x|+|x|+\cdots +|x| \text{ (n times)} \geq |nx| > 1$. Therefore $1<n|x|=n|a/b|=na/b$, since $a/b$ is positive. Then we have $na>b$, so $a$ is not infinitesimal with respect to $b$. This shows $F$ is Archimedean in the previous sense, so somewhat more indirectly, $\mathbb{R}$ being Archimedean as a normed field also implies the statements you gave.

This property alone does not prove the completeness of $\mathbb{R}$. For instance, the same properties hold for $\mathbb{Q}$ in place of $\mathbb{R}$, but $\mathbb{Q}$ is not complete. Completeness is altogether another property of $\mathbb{R}$, which would take too long to explain here from scratch. The main point is that $\mathbb{R}$ is the completion of $\mathbb{Q}$ with respect to the ordinary absolute value. It turns out that there are other absolute value functions one can define on $\mathbb{Q}$, and if we repeat the same procedure that we use to make $\mathbb{R}$ out of $\mathbb{Q}$ ,using these absolute values in place of the usual one, we get new completions of $\mathbb{Q}$, the so called $p$-adic fields, which are not Archimedean.

Let's firmly fix a prime number $p$. Given a rational number $x$, we can factor $x$ into primes (with negative exponents possibly). If $r$ is the exponent of $p$ that occurs in the factorization of $x$, we define the $p$-adic absolute value of $x$ to be $|x|_p=p^{-r}$. It turns out that this function has the same basic properties as ordinary absolute value. In fact in place of the triangle inequality, something even stronger holds:

$$|x+y|_p \leq \max\{|x|_p,|y|_p\}.$$

This is the so called ultra-metric inequality, and ultimately what makes this absolute value non-Archimedean.

The $p$-adic absolute value may look strange at first, but just like the ordinary absolute value, it measures something about numbers. Whereas the ordinary absolute value $|x|$ quantifies the overall size of a rational number $x$, the $p$-adic absolute value $|x|_p$ measures how far $x$ is divisible by $p$. The higher the power of $p$ that $x$ is divisible by, the closer $x$ is to zero in the $p$-adic absolute value. So for example $250=2\times 5^3$ is much closer to $0$ in $5$-adic absolute value than $2$ is: $|250|_5 = 1/125,\ \ |2|_5 = 1$.

If we form the completion of $\mathbb{Q}$ with respect to this new absolute value, similar to how we form $\mathbb{R}$ from $\mathbb{Q}$ (taking equivalence classes of Cauchy sequences), we get a new way to "fill the gaps" of $\mathbb{Q}$. This is called the field $\mathbb{Q}_p$ of $p$-adic numbers.

Unlike $\mathbb{R}$, the field $\mathbb{Q}_p$ is not Archimedean. This is a consequence of the ultra-metric inequality. To see this, suppose $x\in \mathbb{Q}_p$ is any $p$-adic number such that $|x|_p \leq 1$, then we have:

$$|2x|_p =|x+x|_p \leq \max\{|x|_p,|x|_p\}=|x|_p \leq 1.$$

Using $|2x|_p\leq 1$ and $|x|_p\leq 1,$

$$|3x|_p=|2x+x|_p \leq \max\{|x|_p,|2x|_p\} \leq 1,$$

and similarly $|4x|_p=|3x+x|_p\leq \max\{|3x|_p,|x|_p\} \leq 1$, and so by induction $|nx|_p\leq 1$ for all $n$. Thus $\mathbb{Q}_p$ is non-Archimedean.

We can also see that in $\mathbb{Q}_p$, the natural numbers $\mathbb{N}$ are bounded above: for any $n\in \mathbb{N}$, $|n|_p \leq 1$, since the exponent $r$ of $p$ appearing in the prime factorization of $n$ is non-negative: $|n|_p = 5^{-r} \leq 5^0 =1$.

This also shows that for $n\in \mathbb{N}$, $|1/n|_p \geq 1$, so we can not find $1/n$ that are very small in $p$-adic absolute value. So both of the facts that you mentioned in your question are false for $\mathbb{Q}_p$.

Now there is a surprising theorem of Ostrowski which says there are no other completions of $\mathbb{Q}$! Any other absolute values that you may define on $\mathbb{Q}$ will be equivalent either to the ordinary absolute value, or to one of the $p$-adics, and after carrying out the completion process, you will get either $\mathbb{R}$, or one of the $p$-adic fields $\mathbb{Q}_p$. This is the sense in which $\mathbb{R}$ is the Archimedean completion of $\mathbb{Q}$.

Finally, let me say that although the field $\mathbb{Q}_p$ is at first a very strange place to play in, it's used by number theorists all the time. Arithmetic in $\mathbb{Q}_p$ is sometimes interpreted as doing congruence arithmetic modulo all powers of p at the same time.

Hopefully, the notion of $\mathbb{R}$ being Archimedean is clearer once we are aware of the existence of these non-Archimedean completions. The $p$-adic numbers have many more strangely beautiful properties than can't be mentioned all in one place.

Your statements, strictly interpreted, are not true. You need to change the order of quantifiers.

You say "The Archimedean property states that $\Bbb{N}$ isn't bounded above--some natural number can be found such that it is greater than any real number." There is no natural number "such that it is greater than any real number." Rather, for any positive real number $x$, there is a natural number $n$ such that $n > x$.

Similarly, you say "The Archimedean property also states that there is some rational $1/n,n∈ℕ$ such that it is less than any real number." Rather, for any positive real number $x$, there is a positive integer $n$ such that $1/n < x$.

Be careful out there.