Comments on: Lazy Eight and the Turing Oracle http://www.rudyrucker.com/blog/2006/06/21/lazy-eight-and-the-turing-oracle/ Sat, 22 Nov 2008 08:19:45 +0000 http://wordpress.org/?v=2.6.1 By: Rudy http://www.rudyrucker.com/blog/2006/06/21/lazy-eight-and-the-turing-oracle/#comment-17738 Rudy Mon, 10 Nov 2008 21:41:25 +0000 http://www.rudyrucker.com/wordpress/?p=68#comment-17738 John, this is a tricky topic, you might look at the Appendix to my <a target="blank" href="http://www.rudyrucker.com/lifebox" rel="nofollow">Lifebox </a>tome for more info. I'll make one response now, but it would not be fruitful to try and work all this out in a comment thread, so I probably won't respond on this topic again. A turing oracle tells you if a given computation will halt or, which is equivalent, if a given S is provable from a given theory T. The complexity of the theory T doesn't matter nor does the size of the axioms, you're mistaken in thinking that makes a difference. Any normal mathematical theory's axioms can be listed by a Turing machines, so the notion of provability can be formulated in terms of TMs as well. It is however true, that if you add a new "Turing Oracle" primitive to your language, you can get a richer set of theories and a new type of Turing machine that's allowed to consult the Turing oracle, so, as you say, you get into a regress that goes on as long as you like. Finally, a Truth Machine is, as I say, a much stronger concept than a Turing Oracle. Due to undecidability, there will be sentences that are true but not provable or disprovable. John, this is a tricky topic, you might look at the Appendix to my Lifebox tome for more info. I’ll make one response now, but it would not be fruitful to try and work all this out in a comment thread, so I probably won’t respond on this topic again.

A turing oracle tells you if a given computation will halt or, which is equivalent, if a given S is provable from a given theory T. The complexity of the theory T doesn’t matter nor does the size of the axioms, you’re mistaken in thinking that makes a difference. Any normal mathematical theory’s axioms can be listed by a Turing machines, so the notion of provability can be formulated in terms of TMs as well.

It is however true, that if you add a new “Turing Oracle” primitive to your language, you can get a richer set of theories and a new type of Turing machine that’s allowed to consult the Turing oracle, so, as you say, you get into a regress that goes on as long as you like.

Finally, a Truth Machine is, as I say, a much stronger concept than a Turing Oracle. Due to undecidability, there will be sentences that are true but not provable or disprovable.

]]> By: John Reynolds http://www.rudyrucker.com/blog/2006/06/21/lazy-eight-and-the-turing-oracle/#comment-17737 John Reynolds Mon, 10 Nov 2008 21:22:33 +0000 http://www.rudyrucker.com/wordpress/?p=68#comment-17737 I am a bit confused. First of all, it is my understanidng that there is actually a hierarchy of Turing Oracles, so that if one has an Oracle for the Halting Problem on a Turing Machine, that Oracle cannot solve its own Halting Problem; an Oracle at the next level of the Hhierarchy is needed for that purpose. Secondly, it seems to me that the Truth Machine is indeed much more powerful than any Turing Oracle (at any finite level of the Turing Oracle hierarchy); however, this is only true if we fix the number of bits in the axioms of the formal system wherein the Turing Oracle examines possible proofs. Am I mistaken in this belief? If we allow the size of the Axiomatic System to be unbounded then isn't the Turing Oracle able to establish the truth of any proposition of Number Theory just as well as the Truth Machine? Of course, the problem is that human mathematical ingenuity is only able to create axiomatic systems of limited length, and the indefinite extension of this ability is presumably what a Truth Machine provides. Am I correct? ==John R I am a bit confused. First of all, it is my understanidng that there is actually a hierarchy of Turing Oracles, so that if one has an Oracle for the Halting Problem on a Turing Machine, that Oracle cannot solve its own Halting Problem; an Oracle at the next level of the Hhierarchy is needed for that purpose.

Secondly, it seems to me that the Truth Machine is indeed much more powerful than any Turing Oracle (at any finite level of the Turing Oracle hierarchy); however, this is only true if we fix the number of bits in the axioms of the formal system wherein the Turing Oracle examines possible proofs. Am I mistaken in this belief? If we allow the size of the Axiomatic System to be unbounded then isn’t the Turing Oracle able to establish the truth of any proposition of Number Theory just as well as the Truth Machine?

Of course, the problem is that human mathematical ingenuity is only able to create axiomatic systems of limited length, and the indefinite extension of this ability is presumably what a Truth Machine provides. Am I correct?

==John R

]]> By: Steve H http://www.rudyrucker.com/blog/2006/06/21/lazy-eight-and-the-turing-oracle/#comment-7140 Steve H Thu, 22 Jun 2006 12:47:06 +0000 http://www.rudyrucker.com/wordpress/?p=68#comment-7140 Hmmmm.....argh.....whoa. I just grew another wrinkle on my brain trying to visualize that. Suppose there was an interface of some kind that let you perceive the lazy-8 state without actually having to unroll spacetime? Maybe that's what psychics are doing? Hmmmm…..argh…..whoa. I just grew another wrinkle on my brain trying to visualize that. Suppose there was an interface of some kind that let you perceive the lazy-8 state without actually having to unroll spacetime? Maybe that’s what psychics are doing?

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