Towards a definition.
The first property of consciousness is directness.
There are two reasons. First, it is the exact same situation as with axioms; to reason, you have to start somewhere. Second, the property of sensation that I find most mysterious is this very directness.
To come to a logical conclusion - any logical conclusion - requires two completely unjustifiable assumptions. First, the rules of logic. Second, the first premises from which you construct your conclusion.
To justify the first assumption would be circular. The only way to prove the rules is to use the rules, begging the question. The only way to justify the second is to use a more fundamental assumption, which would then itself have to be justified.
Now, if you're Richard Brown and think infinite regression isn't a fallacy, this isn't a problem. However, if infinite regression isn't a fallacy, then you can resolve the Barber Paradox. The deal is, if the barber is unshaven, he concludes he must shave himself, but then the barber would be shaving a man who shaves himself - seemingly unnecessary, so he concludes he should stop shaving. But then he finds he's unshaven... This goes on ad infinitum...which should be resolvable if infinite regression isn't a fallacy.
Of course the real problem with the barber paradox is not oscillation, but the fact that it's a misuse of language. It's not actually a sentence. It's nonsen(ten)[s]e. I would assume I'm wrong and assign a truth value to the Barber paradox, but it's not actually a valid proposition in the first place, but simply does a good job at masquerading as one.
The other interesting example of infinite regression is to posit that there is no fundamental particle, that every particle is made up of yet smaller particles. The first problem is that we empirically know this is false. Beyond a certain size, smaller particles would be physically meaningless. The second problem is that for this to make sense, the smaller particles have to be different than the larger particles.
Consider the opposite; "The electron is made up of four smaller particles each with exactly one quarter of an e, spin 1/8, and so on. These particles cannot be unbound without immediate decay. Further, these particles are made up of yet smaller particles..." Ockham's razor scrapes that right off. Like quarks, the smaller particles must be different so as to explain interactions with other particles, instead of simply repeating the mysteries but with more entities to explain.
Unfortunately, with an actual infinite number of different particles, every electron would have within it every possible combination of properties. (Though presumably a tau lepton would have them in a different order.) It would be possible to create any possible interaction at all. It would be impossible not to create every possible interaction, constantly, as none of them would be barred.
Also, one of the possible combinations is a black hole. Every particle would be a black hole, contradicting the original premise. (Also we wouldn't be around to make these fancy proofs.)
A final point is the fact that infinite series can be summed. This is done using limits - in other words, we don't actually sum an infinite number of terms, but rather we get arbitrarily close to the sum. Note that the method for finding a sum of an 'infinite' series is actually very finite. That is, while it can be represented using infinity, it is exactly equivalent to a finite representation. Infinite series that are only equivalent to an infinite series - series where you would actually have to add an infinite number of terms - are divergent, and their sums are undefined; to define any sum is to make a contradiction.
Ultimately, while for physics you can treat it as a number, infinity is not really a number. To be sure, it is a mathematical object, but not a number. A proof by infinite regression would actually be making a very large number of claims that don't actually form a proof, but get arbitrarily close to a proof. This sounds a whole lot like induction to me, not deduction.
To reason, to deduce, you have to start somewhere, and that thing cannot be from deduction.
Here's my favorite quiz;
You have two facts. Jeanne is on the Eiffel tower. The Eiffel tower is in Brasilia.
Thus you can work out what city Jeanne is in; Brasilia.
But here's the actual question; How do you know Jeanne is on the Eiffel tower? That is, not 'how did you learn that' but once learned, how do you know it? What is the process of knowing? How do you justify knowing that Jeanne is on the Eiffel tower, and using it as a premise, given that a (reliable) source has supplied you with this data?
Similarly, all physically relevant math is based on counting, lately defined by Peano Arithmetic.
How do you know 1 comes after 0? How do you know what 0 is? The Peano Axioms don't supply these - they are assumed. Similarly, if the concept of 0 somehow depends on an actual justification, I can always bring it back until 'you just know' something, which means I've found a true fundamental axiom.
Here's the fact. You know these things directly. There is no proof or logic or process or justification. However we are to define 'you' as a coherent entity, knowing that Jeanne is on the Eiffel tower is known to you directly; it is an integral part of your identity. Even if it turns out there is some process, some instrument between you and Jeanne, then it just means you know that instrument directly instead.
Consider the opposite. Assume we know everything indirectly. The problem is the previous 'sentence' is no more a proposition than the Barber Paradox. It has dropped some implied words, so let me first expand a parallel example.
"Food is important." This is invalid. Food cannot be intrinsically important. It cannot value itself. It especially can't value itself for nutritional value without someone to eat it - that is, without an extrinsic entity. In reality, the full concept is, "Food is necessary for my survival, and my survival is important to me." It is necessary for and important to. Without a subject to modify, both words are meaningless.
Note, however, the chain ends at 'important to me.' This is axiomatic. It's true by virtue of being assumed. "Assume we know everything indirectly, through (x)." The concept is self-contradictory. Since infinite regress is a fallacy, this indirectness has to stop somewhere, at an (x). And whatever this unknown is, we know it directly.
And it is this I find so fascinating and mysterious about consciousness. It is the set of all things we know directly at any given moment. (It may be other things as well, but it's always this one thing.) Similarly, anything we start to know directly will become conscious. That's what it is; that's what I'm defining it as.
As always, this is a rational logic definition which follows an emotional logic definition. The rational definition should essentially confirm and clarify the emotional definition, though it's allowed a few unintuitive consequences. Since definitions are essentially assumptions, this cannot be challenged deductively, except by self-contradiction. (It seems to me, however, that the negation - consciousness isn't direct - has already been shown invalid by self-contradiction.) However, a better definition may be possible, which would match the emotional definition with greater fidelity. Nevertheless, my definition would not die, nor would whatever properties I unearth change; it would simply lose the label 'consciousness.'