According to Wikipedia:
In inferential statistics, the term "null hypothesis" usually refers to a general statement or default position that there is no relationship between two measured phenomena, or no association among groups. ... The statement that is hoped or expected to be true instead of the null hypothesis is the alternative hypothesis
Source: https://en.wikipedia.org/wiki/Null_hypothesis
In my situation, the default position is that there is a correlation between two variables, and the statement that is hoped to be true is that there is no correlation (if some other variable is controlled for).
Questions:
1) Can I state the null hypothesis as "correlation is different from zero", and the alternate hypothesis as "correlation is equal to zero"?
2) Assuming I can do that, then consider situation when I obtain a correlation coefficient of say 0.05 with p-value 0.8. Would it be right to say the null hypothesis is rejected because the correlation coefficient is not different from zero with p-value of 0.8?