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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 :)

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    For the first one: http://math.stackexchange.com/questions/263284/given-real-numbers-define-integers/263296#2632962012-12-29

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