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Is Omega Impossible? Can we even ask?

-8 mwengler 24 October 2012 02:47PM

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

 

  1. Can god make a square circle?
  2. 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 
  1. 2+2 = 5
  2. 1+1 = 2
  3. 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
  1. Omega is an infallible intelligence that does not lie
  2. 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
  1. Omega is an infallible intelligence
  2. 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?  

 

Comments (51)

Extremely Counterfactual Mugging or: the gist of Transparent Newcomb

5 Bongo 09 February 2011 03:20PM

Omega will either award you $1000 or ask you to pay him $100. He will award you $1000 if he predicts you would pay him if he asked. He will ask you to pay him $100 if he predicts you wouldn't pay him if he asked. 

Omega asks you to pay him $100. Do you pay?

This problem is roughly isomorphic to the branch of Transparent Newcomb (version 1, version 2) where box B is empty, but it's simpler.

Here's a diagram:

Comments (79)

Omega can be replaced by amnesia

15 Bongo 26 January 2011 12:31PM

Let's play a game. Two times, I will give you an amnesia drug and let you enter a room with two boxes inside. Because of the drug, you won't know whether this is the first time you've entered the room. On the first time, both boxes will be empty. On the second time, box A contains $1000, and Box B contains $1,000,000 iff this is the second time and you took only box B the first time. You're in the room, do take both boxes or only box B?

This is equivalent to Newcomb's Problem in the sense that any strategy does equally well on both, where by "strategy" I mean a mapping from info to (probability distributions over) actions.

I suspect that any problem with Omega can be transformed into an equivalent problem with amnesia instead of Omega.

Does CDT return the winning answer in such transformed problems?

Discuss.

 

Comments (43)
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