It is <i>essential </i> that all voltages used in your circuit have a common reference-point, i.e. meaning that the 0V points for all voltage rails <i>should </i> be tied together.
BTW: You used the term ‘ground’ in your post, (and it’s very common to refer to the 0V point as ground too), but in fact they can be two different things.
Sometimes DC electrical circuits will have their 0V point isolated from ground, or just have one connection to ground at a single location to prevent a troublesome condition known as ‘ground loops’ (also known as ‘earth loops’
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Note also that if you are using a computer PSU for your voltage rails, BEWARE! If the PSU is not sufficiently loaded (i.e. only running your Picaxe), then the voltage will almost certainly be too high and will damage your chip. This is because computer PSU’s are designed to provide currents significantly greater than the comparatively microscopic current that a Picaxe requires.
PS: Just as I post this, I’ve noticed that you have received some other replies also.
So I hope you find this information of some use.
Cheers