It has precursors, but Russell's paradox of 1901 shows that the obvious set theory, as proposed by Frege has to be modified in unpleasant ways. Frege's basic idea is to let us define the set of all objects having a given property, in more modern notation
giving the set of all x with the property 
. Thus the
set of all red dogs is denoted by 
, or if
the set of dogs is denoted dogs and the set of red objects as
reds, we can also write 
. This
notation for forming sets is very convenient and is much used in
mathematics. The principle is called comprehension.
Bertrand Russell in his 1901 letter to Gottlob Frege pointed out that forming the set
i.e. the set of all sets that are not members of themselves, leads
promptly to a contradiction. We get 
.
There are many ways of restricting set theory to avoid the
contradiction. The most commonly chosen is that of Zermelo, whose
set theory Z allowed only writing 
, where
A is a previously defined set. This turned out to be not quite
enough to represent mathematics and Fraenkel introduce a further axiom
schema of replacement giving a system now called ZF.
ZF is less convenient than Frege's inconsistent system because of the
need to find the set A, and the unrestricted comprehension schema is
often used when it is clear that the needed A could be found.
A more direct inconvenience for giving robots consciousness is the paradox discovered by Richard Montague [Montague, 1963] concerning a set of desirable axioms for knowledge of sentences.
We might denote by knows(person,sentence) the assertion that person knows sentence and consider this as holding at some time t in in some situation s. However, Montague's paradox arises even when there is only one knower, and we write Kp for the knower knowing the sentence p. Montague's paradoxes arise under the assumption that the language of the sentences p is rich enough for ``elementary syntax'', i.e. allows quantifiers and operations on sentences or on Gödel numbers standing for sentences.
The axioms are
and
Intuitively these axioms state that if you know something, it's true, if you know something, you know you know it, and you can do modus ponens. Added to this are schemas saying that you know some sentences of elementary logic.
From these, Montague constructed a version of the paradox of the liar. Hence they must be weakened, and there are many weakenings that restore consistency. Montague preferred to leave out elementary syntax, thus getting a form of modal logic.
I think it might be better to weaken (18) by introducing a hierarchy of introspective knowledge operators on the idea that knowing that you know something is knowledge at an introspective level.
Suppose that we regard knowledge as a function of time or of the
situation. We can slither out of Montague's paradox by changing the
axiom 
to say that if you knew something in the
past, you now know that you knew it. This spoils Montague's recursive
construction of the paradox.
None of this has yet been worked out for an AI system.
