Monday, December 06, 2010

Cantor's Angel

The brief argument I gave here needs expanding. I wrote:
Every day I wake up from sleep, that 'little slice of death', and become
conscious. Imagine the following thought-experiment. I wake up an infinite
number of times. Could I have a conscious moment after that infinite sequence?
Is it possible that there could be a waking moment belonging to my consciousness
such that there are an infinite number of waking moments before that? Surely
not. I can't think of an argument to prove it, rather, it seems an irreducible
part of my idea of consciousness that I cannot conceive of an actual or
'completed' infinity of conscious moments.
Imagine the following thought-experiment. My soul is in hell, and I am being tormented by a demon dentist who is removing my teeth by drilling through their nerves in an exquisitely painful way. My agonising screams are echoing through the halls of the inferno. After all my teeth are removed, they are supernaturally replaced, and the whole process begins again. I understand that this process is to continue infinitely. (There is a colourful depiction of the infinity of hell by James Joyce here).

While I am waiting for the demon to replace my teeth, a Cantorean angel whispers to me. I must not despair. After this process has been repeated infinitely many times, my soul will enter a transfinite Cantorean paradise. I will still be conscious of every one of the infinite moments that has passed in hell. But those moments will now be behind me. They have all happened, infinitely many of them, an infinite number of teeth drilled out and replaced.

Now I ask. Does the pronouncement give me any hope? Surely not. I cannot hope ever to escape this infinite painful process; I have no hope. But if the consciousness in the Cantorean paradise were my consciousness, I would have such a hope. Therefore the consciousness in the Cantorean paradise cannot be my consciousness.

My consciousness is a set of conscious moments tied together by their belonging to a single consciousness. Any future moment must be such that I can hope or expect to experience it by the process of waiting. Thus no future moment of the same consciousness can be such that it is preceded by an infinite number of such moments belonging. For I cannot hope or expect the experience of such a moment. I would be waiting for and expecting something that will never happen to me. (I concede it is logically possible that such a moment could happen to someone else, who was remembering my conscious moments as if they were my own, but more on that later).


David Brightly said...

Quick comment: Surely what counts (sorry!) is not the cardinality of the set of moments of agony, but their measure, and this comes down to how they are situated relative to one another. An interval of one minute has the same number of moments as one of ten minutes, or even of an infinity of minutes, but I know which of these I'd rather be at the dentist's for.

Just as with Zeno, if each supernatural drilling out and replacement takes half the time of the preceding cycle, the whole nasty business is over in finite time. But we return to the question of what kinds of acts can be speeded up in this way.

Edward Ockham said...

OK I thought you would say that. But you appear to agree with my argument that there is no sense in there being a 'transfinite moment' of consciousness, at least not of my consciousness.

If you agree this, there is another step in the argument to consider.

David Brightly said...

If by 'moment of consciousness' you mean a time when I was conscious then I don't see a problem. But if you mean a remembered conscious episode, like waking up, then this would seem to require an infinite lifetime (ruling out those speed-ups). On the other hand, if having an infinity of memories just means that, for each n in (0, -1, -2, -3,...) one has a vivid recall of waking up on that day relative to today, I don't immediately see that there is anything inherent in the nature of consciousness to make this inconceivable---the memories might be artificially generated perhaps. Likewise a sequence ordered as w+1: Achilles distinctly remembers being 1/2^n behind the Tortoise at time 2-2/2^n and reaching him at time 2. But anyway, let me concede this point so that we can proceed to your paradox.

Edward Ockham said...

>>But anyway, let me concede this point so that we can proceed to your paradox.

There is an inherent weakness in my argument that I need to think about. It is essential to my argument that time is irrelevant. The passage of physical time and the ordering of conscious moments are things which have a natural correspondence but which are logically distinct. Think of Rip Van Winkle.

The problem with my argument about the infinity in Dentist Hell is that it is not clear whether the impossibility of 'getting to' the Cantorean Paradise is the infinite time (as I think you would say) or because of the nature of consciousness. My argument is that the for a set of conscious moments to belong to the same consciousness, they must form a 'chain' of finite length.

I am not even denying that there could be Cantorean consciousness that artificially 'remembered' all the infinite moments of a previous eternity. But I deny that this consciousness would be the same consciousness as the one which the remembered moments belonged to. A sort of artificial memory, if you like.

But then we are onto the difficult question of what unifies consciousness. And the question of whether, given that false memory is possible, my impression that I have lived for n years, was once a child, went to school and so on, is an illusion. (Or perhaps not an illusion, but these memories belong to another person).

At this stage it gets so difficult I have to stop. I need to find some literature on the subject.