Hello friends of mathematics :) I have some problems with the topic "Is something definable in a structure". I can solve some problems for example the following questions:
- Is the relation definable in the structure $(\Bbb{Q},+,0,1)$
- Is the function "sin" definable in the structure $(\Bbb{R},<,+,\cdot,0,1)$
- Is $\Bbb{N}$ definable subset of $(\Bbb{Z},<,+,\cdot,0,1)$
The answers are No, No, Yes resp.
But now i want to solve the following problems:
- Is $\Bbb{Z}$ definable subset of $(\Bbb{R},<,+,\cdot,0,1)$
- Is $\Bbb{Q}$ definable subset of $(\Bbb{R},<,+,\cdot,0,1)$
- There is no qeuntifier-free formula that defines the set $2\Bbb{Z}$ in the structure $(\Bbb{Z},+,<,S,0)$ (with $S$ the sucessorfunction)
For the first problem i thought this: If we can define $\Bbb{Z}$ then we could find a polynomial with zeros in all integers. But such a polynomial doesn't exist. I don't know how to solve it on another way. Can someone help me? Thank you beforehand :)
For the non-definability of $\mathbb{Z}$ in the reals, there is good structure theory (keyword: o-minimal) for the definable subsets of $\mathbb{R}$. One can also argue using decidability/undecidability.
It is an old result of Tarski that there is a decision procedure for the elementary theory of real-closed fields. If $Z$ were definable, that decision procedure could be used to produce a decision procedure that would determine, for every sentence $\phi$, whether or not $\phi$ is true in the natural numbers.
For definability of $\mathbb{Q}$, again structural information about definable subsets of $\mathbb{R}$ will do it. Another approach is to use the old, and not easy, result of Julia Robinson that the natural numbers are definable in $\mathbb{Q}$. (Please See Definability and Decision Problems in Arithmetic Journal of Symbolic Logic, Vol. 14, No. 2 (June 1949).)
Hence if $\mathbb{Q}$ were definable in $\mathbb{R}$, then $\mathbb{Z}$ would be. This, as shown above, is not true.