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I want to prove the following statements:

  • Is the function "sin" definable in the structure $(\Bbb{R},<,+,\cdot,0,1)$, that is does there exists a formula $\phi=\phi(x_0,x_1)$ such that for all $a,b\in\Bbb{R}$: $sin(a)=b$ iff $(\Bbb{R},<,+,\cdot,0,1)$ realizes $\phi[a,b]$ ?!
  • Is $\Bbb{Z}$ a definable subset of $(\Bbb{R},<,+,\cdot,0,1)$ ?
  • Is $\Bbb{Q}$ a definable subset of $(\Bbb{R},<,+,\cdot,0,1)$ ?

I have no idea how to solve this questions. I think i have to prove that i have to show that in a extension of the structure the subsets are not definable or i have to give a formula. Can someone help me?! Thanks

  • 0
    Note that $\sin$ definable implies $\mathbb Z$ defineable ($x\in\mathbb Z\iff \sin(\pi x)=0$ where $\pi$ is the smallest positive zero of $\sin$) implies $\mathbb Q$ defineable ($x\in\mathbb Q\iff \exists a,b\colon a,b\in \mathbb Z, b\ne0, bx=a$).2012-12-08
  • 3
    These are not statements. They are questions. You probably want to answer them, not prove them.2012-12-09

2 Answers 2