Is Omega Impossible? Can we even ask?

mwengler

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Is Omega Impossible? Can we even ask?

by mwengler

24th Oct 2012

2 min read

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EDIT: I see by the karma bombing we can't even ask. Why even call this part of the site "discussion?"

Some of the classic questions about an omnipotent god include

Can god make a square circle?
Can god create an immovable object? And then move it?

Saints and philosophers wrestled with these issues back before there was television. My recollection is that people who liked the idea of an omnipotent god would answer "omnipotence does not include the power to do nonsense" where they would generally include contradictions as nonsense. So omnipotence can't square a circle, can't make 2=3, can't make an atom which is simultaneously lead and gold.

But where do the contradictions end and the merely difficult to conceive begin? Can omnipotence make the ratio of the diameter to the circumference of a circle = 3, or 22/7? Can omnipotence make sqrt(2)=1.4 or 2+2=5? While these are not directly self-contradictory statements, they can be used with a variety of simple truths to quickly derive self-contradictory statements. Can we then conclude that "2+2=5" is essentially a contradiction because it is close to a contradiction? Where do we draw the line?

What if were set some problem where we are told to assume that

2+2 = 5
1+1 = 2
1+1+1+1+1 = 5

In solving this set problem, we can quickly derive that 1=0, and use that to prove effectively anything we want to prove. Perhaps not formally, but we have violated the "law of the excluded middle," that either a statement is true or its negation is. Once you violate that, you can prove ANYTHING using simple laws of inference, because you have propositions that are true and false.

What if we set a problem where we are told to assume

Omega is an infallible intelligence that does not lie
Omega tells you 2+2=5

Well, we are going to have the same problem as above, we will be able to prove anything.

Newcomb's Problem

In Newcomb's box problem, we are told to assume that

Omega is an infallible intelligence
Omega has predicted correctly whether we will one box or two box.

From these assumptions we wind up with all sorts of problems of causality and/or free will and/or determinism.

What if these statements are not consistent? What if these statements are tantamount to assuming 0=1, or are within a few steps of assuming 0=1? Or something just as contradictory, but harder to identify?

Personally, I can think of LOTS of reasons to doubt that Newcomb's problem is even theoretically possible to set. Beyond that, I can think that the empirical barrier to believing Omega exists in reality would be gigantic, millions of humans have watched magic shows performed by non-superior intelligences where cards we have signed have turned up in a previously sealed envelope or wallet or audience member's pocket. We recognize that these are tricks, that they are not what they appear.

To question Omega is not playing by the mathematician's or philosopher's rules. But when we play by the rules, do we blithely assume 2+2=5 and then wrap ourselves around the logical axle trying to program a friendly AI to one-box? Why is questioning Omega's possibility of existence, or possibility of proof of existence out-of-bounds?

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Is Omega Impossible? Can we even ask?

by mwengler

24th Oct 2012

2 min read

-12

EDIT: I see by the karma bombing we can't even ask. Why even call this part of the site "discussion?"

Some of the classic questions about an omnipotent god include

Can god make a square circle?
Can god create an immovable object? And then move it?

What if were set some problem where we are told to assume that

2+2 = 5
1+1 = 2
1+1+1+1+1 = 5

What if we set a problem where we are told to assume

Omega is an infallible intelligence that does not lie
Omega tells you 2+2=5

Well, we are going to have the same problem as above, we will be able to prove anything.

Newcomb's Problem

In Newcomb's box problem, we are told to assume that

Omega is an infallible intelligence
Omega has predicted correctly whether we will one box or two box.

From these assumptions we wind up with all sorts of problems of causality and/or free will and/or determinism.

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mwengler14y-10

At the time I added the edit, I had two comments and 6 net downvotes. I had replied to the two comments. It is around 25 hours later now. For me, 25 hour gaps in my responses to lesswrong will be typical, I'm not sure a community which can't work with that is even close to optimal. So here I am commenting on comments.

Of course you're free to downvote and I'm free to edit. Of course we are both free as is everyone else, to speculate whether the results are what we would want, or note. Free modulo determinism, that is.

As far as I know, this is the first thread in which it has ever been pointed out that Omega doesn't need to be infallible or even close to infallible in order for the problem to work. A newcomb's problem set with a gratuitous infallible predictor is inferior to a newcomb's problem set with a currently-implementable but imperfect prediction algorithm. Wouldn't you agree? When I say inferior, I mean both as a guide to the humans such as myself trying to make intellectual progress here, and as a guide to the coders of FAI.

As far as I am concerned, a real and non-trivial improvement has been proposed to the statement of Newcomb's problem as a result of my so-called "discussion" posting. An analagous improvement in another argument would be 1) Murder is wrong because my omniscient, omnipotent, and all-good god says it is. 2) I don't think an omniscient omnipotent all-good god is possible in our universe 1) well obviously you don't need such a god to see that Murder is wrong.

Whether my analogy seems self-aggrandizing or not, the value to the discussion of taking extraneous antecedents out of discussed problems I hope will be generally understood.

ArisKatsaris14y80

As far as I know, this is the first thread in which it has ever been pointed out that Omega doesn't need to be infallible or even close to infallible in order for the problem to work. [...] As far as I am concerned, a real and non-trivial improvement has been proposed to the statement of Newcomb's problem as a result of my so-called "discussion" posting.

I'll note here that the lesswrong wiki page on Newcomb's problem has a section which says the following:

Irrelevance of Omega's physical impossibility
Sometimes people dismiss Newcomb's proble

... (read more)

6benelliott14y

No, in maths you want to pick the simplest possible thing that embodies the principle you want to study, needless complications are distracting. Throwing in a probabilistic element to something that works fine as a deterministic problem is needless.

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