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I have a machine that flashes a light on or off once a second, at random. You can view streams of output from the machine as possible events. Now, suppose an in

I have a machine that flashes a light on or off once a second, at random. You can view streams of output from the machine as possible events. Now, suppose an infinite amount of time has passed (aleph-null, by definition). Have I seen every possible sequence? Well, take the set of aleph-null length sequences (all aleph-null of them [1]) and put them in an order. Now construct the following sequence: invert bit 1 from sequence 1, invert bit 2 from sequence 2, ... etc. Note that this sequence was never produced by the machine, since we enumerated all the sequences it produced, but they are all different to this one. So, by a standard diagonalisation, the set of events is higher cardinality. [1] How many aleph-null length sequences come out of the machine? Well, they have to be continuous, so the only obvious aleph-null length sequence is the total sequence the machine produces. However, you can drop an entry from the beginning of the sequence to get another aleph-null length sequence. And so on, making a total of aleph-null.

hugh3 16y ago
Interesting attempt, but I'm not convinced by the part of the argument where you divide the complete output of the flashy light box into aleph-null sequences, each of length aleph-null. Mostly, I'm unconvinced that an "event" which takes an infinitely long time to complete actually counts as an event. Usually when we talk about events they're localised in space and time.
- theli0nheart 16y ago
  Well, it's kind of a Catch-22. This proof is dependent upon the lemma that time is infinite. On the other hand, the set of events we're looking for is the one where time ends. Therefore by assuming the lemma we have no need for the proof. OTOH, if we're looking at another event that's not so confusing (one that's not the end of time), the proof holds water. It's a pretty sweet proof that I think is similarly used to prove that the set of real numbers is uncountable.
  - hugh3 16y ago
    No, what I'm saying is that an event which takes an infinitely long time to happen (ie an infinite series of one-second flashes) isn't a proper "event", which as used at least in relativity, is something that happens in finite space and time. But there's no need to argue about the definition of event. I'll say instead that I'm interested in knowing whether infinite time implies that all finite-length events must eventually happen (and indeed, happen an infinite number of times as the paper claims).
    - Robin_Message 16y ago
      Yes, sorry, there is definitely a theorem that if you have an infinite sequence, uniformly distributed over an alphabet, then the probability of seeing any finite sequence is 1. I think that, given some assumptions, you can model the universe as an infinite numerical sequence, so you should see all finite events. However, the assumptions here seem tricky. Uniform distribution of probability seems unlikely, and if the universe is anything like the game of life, there are garden of Eden states with no predecessor state, which an evolving system cannot reach, even in infinite time. Cardinality is indeed tricky. I had thought that all finite sequences would be greater than aleph-null, but apparently it isn't.

hugh3 16y ago

Interesting attempt, but I'm not convinced by the part of the argument where you divide the complete output of the flashy light box into aleph-null sequences, each of length aleph-null. Mostly, I'm unconvinced that an "event" which takes an infinitely long time to complete actually counts as an event. Usually when we talk about events they're localised in space and time.

theli0nheart 16y ago
Well, it's kind of a Catch-22. This proof is dependent upon the lemma that time is infinite. On the other hand, the set of events we're looking for is the one where time ends. Therefore by assuming the lemma we have no need for the proof. OTOH, if we're looking at another event that's not so confusing (one that's not the end of time), the proof holds water. It's a pretty sweet proof that I think is similarly used to prove that the set of real numbers is uncountable.
- hugh3 16y ago
  No, what I'm saying is that an event which takes an infinitely long time to happen (ie an infinite series of one-second flashes) isn't a proper "event", which as used at least in relativity, is something that happens in finite space and time. But there's no need to argue about the definition of event. I'll say instead that I'm interested in knowing whether infinite time implies that all finite-length events must eventually happen (and indeed, happen an infinite number of times as the paper claims).
  - Robin_Message 16y ago
    Yes, sorry, there is definitely a theorem that if you have an infinite sequence, uniformly distributed over an alphabet, then the probability of seeing any finite sequence is 1. I think that, given some assumptions, you can model the universe as an infinite numerical sequence, so you should see all finite events. However, the assumptions here seem tricky. Uniform distribution of probability seems unlikely, and if the universe is anything like the game of life, there are garden of Eden states with no predecessor state, which an evolving system cannot reach, even in infinite time. Cardinality is indeed tricky. I had thought that all finite sequences would be greater than aleph-null, but apparently it isn't.