Personally my brain stack overflows at about 3 or 4 levels of being aware 
that I am aware that ... I am aware. I think it would require infinite 
memory to truly be aware of an infinite number of steps in such a recursive 
relation.

Maybe the infinite hierarchy doesn't have to be thought/remembered in full. 
Royce notes that the simple intention to be completely self-aware is 
enough: the infinite hierarchy is logically implied in the intention. Oskar 
Becker has a slightly different approach: he says that as soon as one 
notices the endlessness of 1, 2, 3.... etc. one is already beyond the 
natural numbers and has reached the first transfinite ordinal (small 
omega=N) which collects all the natural numbers into one set. And then of 
course one can continue: omega+1, omega+2...etc. Noting the endlessness of 
that series one has grasped the next transfinite ordinal (omegaxomega). And 
so on...  Becker notes that Cantor's generation principles can be recast as 
principles inherent in the structure of self-awareness. I think that's a 
strong point.

On the other hand, if there is an absolute self-awareness grounding reality 
as such, then maybe it must have infinite memory... I don't know, this 
account is still too undeveloped to say anything more specific in it.


What about when a being is not contemplating its self-awareness and merely 
living in the moment? Is it not conscious at those times?

In my view one cannot live in the present without self-consciousness. Maybe 
this is lame argument, but I think this is suggested by the similarity of 
the words "present" and "self-presence"... I would say: the self-presence 
of the absolute self-awareness is just what the present is... It is 
difficult to explain this, but intuitively it feels right to me... And then 
there are is also the claim that there is no (phenomenal) consciousness 
without a measure of self-awareness... Kant originally advanced this idea, 
but many present-day cognitive scientists accept it as well...

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