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First proposed answer:

See link http://en.wikipedia.org/wiki/Continued_fraction

Basically what we do is to correspond each rational number canonically to a continued fraction.(see the wiki link). We can denote $\mathbb{Q}^n$ the rational number that can be represented canonically by a continued fraction of length $n$. The restriction map (corresponding a rational number in $\mathbb{Q}$ with continuous fraction $[a_0;a_1, ..]$ to the rational number in $\mathbb{Q}^n$ with continued fraction expression $[a_0; a_1, \ldots , a_n]$ ) will then give a projection from $\mathbb{Q}$ to $\mathbb{Q}^n$. Notice that this map weakly preserve the ordering. Real number is then the inverse limit of this system. That is because an irrational number can be expressed uniquely as a continued fraction.

We do not have these projection map ring homomorphism, but we do have them ring homomorphism with some perturbation. This is consistent with the way the set theorist view real number as $ \omega^ \omega$ as well.

Second proposed answer:

Let me propose another answer which share some spirit with the previous one. The advantage here is that the quotient structure is also ring. But the disadvantage is the thing obtained is not exactly the inverse limit. I let you decide which one of them is better.

Let $ R $ be any discrete subring of $ \mathbb{Q}$ (e.g. $ R =\mathbb{Z} $, $ R = \mathbb{Z}[1/2]$). We define the map $ p_R: \mathbb{Q} \rightarrow R$ by $ q \mapsto r$ where $r$ is the greatest element in $R$ such that $r \leq q$. It can be seen that this is a projection. If $ R_1 $ is a subring of $R_2$, it can be seen that the above system of maps induces a projection from $R_2$ to $R_1$. Hence we have an inverse system of subset of $\mathbb{Q}$. The inverse limit of this system is something, in fact it is a ring . To get the $\mathbb{R}$ we need to quotient away the sequences that converge to 0.

Seems like nothing new, I am repeating the Cauchy construction and further more the projection that I was considering was not even a ring homomorphism. Let me explain why I think this construction is more reasonable than it first appears.

Let us observe what did we actually do for the case of primes $p$. A way to view $ \mathbb{Z} / p^n$ is to think of it as we are identifying every integer divisible by $p^n$. If we think of prime in term of valuation then this means ignoring the small difference ( $p^n$ is small in this case). That harmonize very well with the construction above (and also the construction by continuous fraction) where we throw away some small thing.

For the complain that the projection is not a ring homomorphism. It is true that it is not a ring homomorphism but it is a ring homomorphism up to a small perturbation. In fact, we can choose $ R$ approximates $ \mathbb{Q}$ close enough such that the projection map behave like a ring homomorphism at a few places if we wish. And at the limit stage we recover back the field structure.

First proposed answer:

See link http://en.wikipedia.org/wiki/Continued_fraction

Basically what we do is to correspond each rational number canonically to a continued fraction.(see the wiki link). We can denote $\mathbb{Q}^n$ the rational number that can be represented canonically by a continued fraction of length $n$. The restriction map (corresponding a rational number in $\mathbb{Q}$ with continuous fraction $[a_0;a_1, ..]$ to the rational number in $\mathbb{Q}^n$ with continued fraction expression $[a_0; a_1, \ldots , a_n]$ ) will then give a projection from $\mathbb{Q}$ to $\mathbb{Q}^n$. Notice that this map weakly preserve the ordering. Real number is then the inverse limit of this system. That is because an irrational number can be expressed uniquely as a continued fraction.

We do not have these projection map ring homomorphism, but we do have them ring homomorphism with some perturbation. This is consistent with the way the set theorist view real number as $ \omega^ \omega$ as well.

Second proposed answer:

Let me propose another answer which share some spirit with the previous one. The advantage here is that the quotient structure is also ring. But the disadvantage is the thing obtained is not exactly the inverse limit. I let you decide which one of them is better.

Let $ R $ be any discrete subring of $ \mathbb{Q}$ (e.g. $ R =\mathbb{Z} $, $ R = \mathbb{Z}[1/2]$). We define the map $ p_R: \mathbb{Q} \rightarrow R$ by $ q \mapsto r$ where $r$ is the greatest element in $R$ such that $r \leq q$. It can be seen that this is a projection. If $ R_1 $ is a subring of $R_2$, it can be seen that the above system of maps induces a projection from $R_2$ to $R_1$. Hence we have an inverse system of subset of $\mathbb{Q}$. The inverse limit of this system is something, in fact it is a ring . To get the $\mathbb{R}$ we need to quotient away the sequences that converge to 0.

Seems like nothing new, I am repeating the Cauchy construction and further more the projection that I was considering was not even a ring homomorphism. Let me explain why I think this construction is more reasonable than it first appears.

Let us observe what did we actually do for the case of primes $p$. A way to view $ \mathbb{Z} / p^n$ is to think of it as we are identifying every integer divisible by $p^n$. If we think of prime in term of valuation then this means ignoring the small difference ( $p^n$ is small in this case). That harmonize very well with the construction above (and also the construction by continuous fraction) where we throw away some small thing.

For the complain that the projection is not a ring homomorphism. It is true that it is not a ring homomorphism but it is a ring homomorphism up to a small perturbation. In fact, we can choose $ R$ approximates $ \mathbb{Q}$ close enough such that the projection map behave like a ring homomorphism (identity map) "locally". And at the limit stage we recover back the field structure.

I think this might help: http://en.wikipedia.org/wiki/Continued_fraction

Basically what we do is to correspond each rational number canonically to a continued fraction.(see the wiki link). We can denote $Q_n$ the rational number that can be represented canonically by a continued fraction of length $n$. The restriction map (corresponding a rational number in $Q$ with continuous fraction $[a_0;a_1, ..]$ to the rational number in $Q_n$ with continued fraction expression $[a_0; a_1, \ldots , a_n]$ ) will then give a projection from $Q$ to $Q_n$. Notice that this map weakly preserve the ordering. Real number is then the inverse limit of this system. That is because an irrational number can be expressed uniquely as a continued fraction.

This is the best we can perhaps hope for since as I mention in the comment above the main actor here is not the arithmetic rule but the ordering. This is consistent with the way the set theorist view real number as $ \omega^ \omega$ as well.

First proposed answer:

See link http://en.wikipedia.org/wiki/Continued_fraction

Basically what we do is to correspond each rational number canonically to a continued fraction.(see the wiki link). We can denote $\mathbb{Q}^n$ the rational number that can be represented canonically by a continued fraction of length $n$. The restriction map (corresponding a rational number in $\mathbb{Q}$ with continuous fraction $[a_0;a_1, ..]$ to the rational number in $\mathbb{Q}^n$ with continued fraction expression $[a_0; a_1, \ldots , a_n]$ ) will then give a projection from $\mathbb{Q}$ to $\mathbb{Q}^n$. Notice that this map weakly preserve the ordering. Real number is then the inverse limit of this system. That is because an irrational number can be expressed uniquely as a continued fraction.

We do not have these projection map ring homomorphism, but we do have them ring homomorphism with some perturbation. This is consistent with the way the set theorist view real number as $ \omega^ \omega$ as well.

Second proposed answer:

Let me propose another answer which share some spirit with the previous one. The advantage here is that the quotient structure is also ring. But the disadvantage is the thing obtained is not exactly the inverse limit. I let you decide which one of them is better.

Let $ R $ be any discrete subring of $ \mathbb{Q}$ (e.g. $ R =\mathbb{Z} $, $ R = \mathbb{Z}[1/2]$). We define the map $ p_R: \mathbb{Q} \rightarrow R$ by $ q \mapsto r$ where $r$ is the greatest element in $R$ such that $r \leq q$. It can be seen that this is a projection. If $ R_1 $ is a subring of $R_2$, it can be seen that the above system of maps induces a projection from $R_2$ to $R_1$. Hence we have an inverse system of subset of $\mathbb{Q}$. The inverse limit of this system is something, in fact it is a ring . To get the $\mathbb{R}$ we need to quotient away the sequences that converge to 0.

Seems like nothing new, I am repeating the Cauchy construction and further more the projection that I was considering was not even a ring homomorphism. Let me explain why I think this construction is more reasonable than it first appears.

Let us observe what did we actually do for the case of primes $p$. A way to view $ \mathbb{Z} / p^n$ is to think of it as we are identifying every integer divisible by $p^n$. If we think of prime in term of valuation then this means ignoring the small difference ( $p^n$ is small in this case). That harmonize very well with the construction above (and also the construction by continuous fraction) where we throw away some small thing.

For the complain that the projection is not a ring homomorphism. It is true that it is not a ring homomorphism but it is a ring homomorphism up to a small perturbation. In fact, we can choose $ R$ approximates $ \mathbb{Q}$ close enough such that the projection map behave like a ring homomorphism at a few places if we wish. And at the limit stage we recover back the field structure.

I think this might help: http://en.wikipedia.org/wiki/Continued_fraction

Basically what we do is to correspond each rational number canonically to a continued fraction.(see the wiki link). We can denote $Q_n$ the rational number that can be represented canonically by a continued fraction of length $n$. The restriction map will then give a projection from $Q$ to $Q_n$. Real number is then the inverse limit of this system. That is because an irrational number can be expressed uniquely as a continued fraction.

This is the best we can perhaps hope for since as I mention in the comment above the main actor here is not the arithmetic rule but the ordering. This is consistent with the way the set theorist view real number as $ \omega^ \omega$ as well.

I think this might help: http://en.wikipedia.org/wiki/Continued_fraction

Basically what we do is to correspond each rational number canonically to a continued fraction.(see the wiki link). We can denote $Q_n$ the rational number that can be represented canonically by a continued fraction of length $n$. The restriction map (corresponding a rational number in $Q$ with continuous fraction $[a_0;a_1, ..]$ to the rational number in $Q_n$ with continued fraction expression $[a_0; a_1, \ldots , a_n]$ ) will then give a projection from $Q$ to $Q_n$. Notice that this map weakly preserve the ordering. Real number is then the inverse limit of this system. That is because an irrational number can be expressed uniquely as a continued fraction.

This is the best we can perhaps hope for since as I mention in the comment above the main actor here is not the arithmetic rule but the ordering. This is consistent with the way the set theorist view real number as $ \omega^ \omega$ as well.