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Imagine that we have a computer program that produces the conscious awareness of the present moment.

Let us assume that every time the program is run a counter is incremented.

Let us also assume that the program is encapsulated in an infinite loop.

Each time the program is run the computer "wakes up" and asks itself "what is the present value of the loop counter?"

The computer reasons that a priori it is equally likely to find itself in any loop with any value of the loop counter.

Therefore the prior probability that the loop counter has any particular value $n$ is $1/\infty=0$.

But the computer will find that it has some loop counter value.

There seems to be a contradiction here.

I think the problem is with the assumption that a computer program can produce conscious awareness in the first place.

I think this argument shows that any deterministic machine cannot be conscious.

Such determinism implies a single-valued time evolution whose (theoretically) infinite extent leads to this zero probability paradox.

I believe to avoid this paradox one needs a branching concept of time in the manner of the many-worlds quantum theory in which each branch into an alternate future is only of finite extent. Maybe the brain itself avoids the paradox by exploiting this many-worlds ontology in its operation.

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closed as off topic by David Z Apr 1 '13 at 22:47

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None of this makes any sense. No where has the computer's *'conscious awareness' come into play. The 'paradox' is simply that the probability of an occurrence approaches zero as the number of possibilities becomes unbounded. You could just as easily examine the probability of selecting a particular real number in any non-zero range... –  zhermes Apr 1 '13 at 22:34
    
The conscious awareness ensures that the computer is equally likely to "find itself" in any program loop n.For instance if one happened to come across such a computer from the "outside" so to speak one could not say that it is equally likely that the current program counter n has any value from 1 to infinity. Instead one would have to use a so-called improper prior probability weighting like the function 1/n which favours small values of n over large ones. Such a function implies that the log of n is uniformly distributed from 0 to infinity rather than n itself. –  John Eastmond Apr 1 '13 at 22:56
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may I suggest philosophy.stackexchange.com ? –  SF. Apr 2 '13 at 11:50
    
Thanks I'll do that. –  John Eastmond Apr 2 '13 at 21:13
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2 Answers 2

This argument (the question) is bogus. The same could be said of dropping a coin on a 1-dimensional number line. There are an uncountably infinite number of places on that line that the coin could land so the probability of it landing in any given place is 0 and yet the coin lands in one spot, "violating" probability.

There are many "paradoxes" involving discrete cases in continuous processes. None of them make for good arguments against probability or consciousness or anything else of that sort.

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You're right, of course, but this isn't a discrete case in a continuous process. This is a discrete case out of a countably infinite set. –  Jerry Schirmer Apr 1 '13 at 22:47
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A trite answer to this is that the value will have to be stored in memory somewhere, and therefore, since it is in memory, the variable in which the loop counter is being stored will have some maximum value, making your prior probability 1/MAX_INTEGER.

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I assume a Turing machine with an infinite tape. –  John Eastmond Apr 1 '13 at 23:01
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