I've just started P.Halmos' book "Finite Vector Spaces", and at the first chapter, after defining the axioms of a field, there is an exercide that asks: if we consider the non negative integers, can we redefine addition and multiplication so that they form a field? Can't we, since $\mathbb{N}$ is countable just like $\mathbb{Q}$ that is a field, use some sort of coding to represent the field $\mathbb{Q}$ with it's operations with the natural numbers?

  • $\begingroup$ Yes, that would work. Any infinite set, or any finite set of prime power size $>1$, can be given the structure of a field. $\endgroup$ – Noah Schweber Oct 26 '16 at 23:56
  • $\begingroup$ I was always under the impression that they were. $\endgroup$ – Ninosław Brzostowiecki Oct 26 '16 at 23:59
  • $\begingroup$ Can someone give an example of how that would look like? $\endgroup$ – Fede Poncio Oct 27 '16 at 0:00

Yes. In general, if you have two sets that are in bijection with each other, and one of them has any sort of extra structure (e.g. is a group, a field, a vector space), then you can use the bijection to define the corresponding structure on the other set. This is sometimes referred to as "transport of structure."

  • $\begingroup$ can you be more specific on how you do that? give some examples? $\endgroup$ – Mano Plizzi Oct 27 '16 at 0:14
  • $\begingroup$ Suppose you want to put a group structure on $\mathbb{N}$. Since $\mathbb{N}$ has the same cardinality as $\mathbb{Z}$, there's some bijection $\phi$ between them. Then you can interpret $\mathbb{N}$ as a group where $a + b$ is interpreted as $\phi^{-1}(\phi(a) + \phi(b))$. $\endgroup$ – Nitin Oct 27 '16 at 0:22
  • $\begingroup$ Here's a simple example. The set {0,1} has an addition operation: define "+" by 0+0=0, 0+1=1, 1+0=1, 1+1=0. The set {A,B} is in bijective correspondence with the set {0,1}. A corresponds to 0 and B corresponds to B. Then you can define "addition" on the set {A,B} as follows: A+A=A, A+B=B, B+A=B, B+B=A. $\endgroup$ – Not a grad student Oct 27 '16 at 0:22
  • $\begingroup$ Here is an analogy that may make this helpful. Pretend that there is some country that uses an alphabet just like the English alphabet, except their letters are all sideways. Can they form words that correspond to ours? Yes. They would make their words the same but just using the sideways letters. This is because there is a bijection between the English alphabet and theirs. $\endgroup$ – Not a grad student Oct 27 '16 at 0:25
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    $\begingroup$ Your question was "if we consider the non negative integers, can we redefine addition and multiplication so that they form a field?" The answer to your question is yes as you just explained. The field constructed out of the nonnegative integers will be isomorphic to $\mathbb{Q}$. $\endgroup$ – Not a grad student Oct 27 '16 at 0:51

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