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No. 0 + 0 = 0. You can't get something from nothing, and as such nothing plus nothing cannot be something.
The idea that 0 + 0 = 0 * 0 = 0^0 = 1 is frequently used to "prove" that 0 + 0 = 1. This is false because 0 + 0 = 2*0, not 0*0, and 0*0 = 0^2. Furthermore, 0^0 is in and of itself an indeterminate form, as it is equal to 1/(0^0) which doesn't lead anywhere good.
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