Today at lunch I was discussing interesting facets of second-order logic, such as the (known) fact that first-order logic cannot, in general, distinguish finite models from infinite models. The conversation branched out, as such things do, to why you would *want* a cognitive agent to think about finite numbers that were unboundedly large, as opposed to boundedly large.

So I observed that:

- Although the laws of physics as we know them don't allow any agent to survive for infinite subjective time (do an unboundedly long sequence of computations), it's possible that our model of physics is mistaken. (I go into some detail on this possibility below the cutoff.)
- If it
*is*possible for an agent – or, say, the human species – to have an infinite future, and you cut yourself off from that infinite future and end up stuck in a future that is merely very large, this one mistake outweighs all the finite mistakes you made over the course of your existence.

And the one said, "Isn't that a form of Pascal's Wager?"

I'm going to call this the Pascal's Wager Fallacy Fallacy.

You see it all the time in discussion of cryonics. The one says, "If cryonics works, then the payoff could be, say, at least a thousand additional years of life." And the other one says, "Isn't that a form of Pascal's Wager?"

The original problem with Pascal's Wager is not that *the purported payoff is large.* This is *not* where the flaw in the reasoning comes from. That is not the problematic step. The problem with Pascal's original Wager is that *the probability is exponentially tiny *(in the complexity of the Christian God) and that *equally large tiny probabilities offer opposite payoffs for the same action* (the Muslim God will damn you for believing in the Christian God).

However, what we have here is the term "Pascal's Wager" being applied *solely because the payoff being considered is large* – the reasoning being perceptually recognized as an instance of "the Pascal's Wager fallacy" as soon as someone mentions a big payoff – without any attention being given to whether the probabilities are in fact small or whether counterbalancing anti-payoffs exist.

And then, once the reasoning is perceptually recognized as an instance of "the Pascal's Wager fallacy", the other characteristics of the fallacy are automatically inferred: they *assume* that the probability is tiny and that the scenario has no specific support apart from the payoff.

But infinite physics and cryonics are both possibilities that, leaving their payoffs entirely aside, get significant chunks of probability mass purely on merit.

Yet instead we have reasoning that runs like this:

- Cryonics has a large payoff;
- Therefore, the argument carries even if the probability is tiny;
- Therefore, the probability is tiny;
- Therefore, why bother thinking about it?

(Posted here instead of Less Wrong, at least for now, because of the Hanson/Cowen debate on cryonics.)

Further details:

Pascal's Wager is actually a serious problem for those of us who want to use Kolmogorov complexity as an Occam prior, because the size of even the finite computations blows up much faster than their probability diminishes (see here).

See Bostrom on infinite ethics for how much worse things get if you allow non-halting Turing machines.

In our current model of physics, time is infinite, and so the collection of real things is infinite. Each time state has a successor state, and there's no particular assertion that time returns to the starting point. Considering time's continuity just makes it worse – now we have an uncountable set of real things!

But current physics also says that any finite amount of matter can only do a finite amount of computation, and the universe is expanding too fast for us to collect an infinite amount of matter. We cannot, on the face of things, expect to think an unboundedly long sequence of thoughts.

The laws of physics *cannot *be easily modified to permit immortality: lightspeed limits and an expanding universe and holographic limits on quantum entanglement and so on all make it *inconvenient* to say the least.

On the other hand, many *computationally simple* laws of physics, like the laws of Conway's Life, permit indefinitely running Turing machines to be encoded. So we can't say that it requires a *complex* miracle for us to confront the prospect of unboundedly long-lived, unboundedly large civilizations. Just there being *a lot more to discover* about physics – say, one more discovery of the size of quantum mechanics or Special Relativity – might be enough to knock (our model of) physics out of the region that corresponds to "You can only run boundedly large Turing machines".

So while we have no particular reason to *expect *physics to allow unbounded computation, it's not a *small, special, unjustifiably singled-out possibility* like the Christian God; it's a *large *region of what various possible physical laws will allow.

And cryonics, of course, is the *default* extrapolation from known neuroscience: if memories are stored the way we now think, and cryonics organizations are not disturbed by any particular catastrophe, and technology goes on advancing toward the physical limits, then it is possible to revive a cryonics patient (and yes you are the same person). There are negative possibilities (woken up in dystopia and not allowed to die) but they are exotic, not having equal probability weight to counterbalance the positive possibilities.

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