Z-direction. Quantisation axis

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LagrangeEuler
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I'm a bit confused with this concept of quantisation axis. ##|+\rangle## and ##|-\rangle## spin. And operators ##\hat{S}^+=\hat{S}_x+i\hat{S}_y## and ##\hat{S}^-=\hat{S}_x-i\hat{S}_y## are rising and lowering operator of spin projection. Why z-axis. How is measured that?
 
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Why z-axis.
The coordinate system is arbitrary, you could use any other axis as well. The coordinate system is just a mathematical tool, not something physical.
How is measured that?
What?
 

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