[FOM] Crystal Ball Theory

Harvey Friedman friedman at math.ohio-state.edu
Mon Feb 23 00:20:48 EST 2004

A crystal ball has been found in the Nevada desert, together with a message,
and an attached keyboard. The message reads as follows.

"We Hyperaliens have briefly visited the Earth and left this device. It is
lead shielded, and any attempt to break the shield will completely destroy
its contents. 

"We Hyperaliens have noticed such rudimentary plans as have been described
on your splendid Earthly FOM email list, whose Archives we regularly
monitor. For example, in H. Friedman's posting, A Newtonian computing
design, of 2/13/04, 12:16AM.

"That primitive posting of H. Friedman was rather naive in terms of the
required physical processes, and of course, H. Friedman did not even see how
to carry through the mathematics involved, even for his simplified physical

"We Hyperintelligent Hyperaliens have adjusted the working physics, have
carried through the associated mathematics, and have implemented the
relevant design. It has been thoroughly tested in our Hyperlaboratories.

"Just enter a Turing machine program on the attached keyboard, with at most
100 quadruples. Verify the correctness of your entry on the attached screen,
and then press ENTER.

"The Crystal Ball will, within a few moments, determine whether or not the
entered Turing machine eventually halts at the empty input tape, and provide
the answer on the screen as HALTS or DOES NOT HALT.

"We guarantee correctness for the first 1,000 queries. If you humans put
this to good use, untold mysteries will be resolved for you, and you will
also reap benefits in the form of increased wealth for all of your people.

"We Hyperintelligent Hyperaliens have been working on a version of the
Crystal Ball that will accept at most 200 quadruples. If you humans put this
next version to good use, there will be an order of magnitude greater
consequences for you, in mysteries and wealth, than for the present Crystal


This fantastic finding from the Nevada desert has made untold worldwide
Governmental and private money available for research concerning the

1. How do we reliably test the Crystal Ball for legitimacy? What is the
theory behind legitimacy testing here? A handful of trials - some involving
the finite simple groups have already shown the Crystal Ball to be
infallible so far. These trials involve FLT, the sporadic simple groups,
etcetera, as discussed in my A Newtonian computing design, of 2/13/04,

2. What kinds of exciting/important questions can be appropriately packed
into at most 100 Turing machine quadruples? What techniques can be used to
reduce the number of quadruples needed?

3. Same as 2, but replacing 100 by 200. Same as 2, but replacing 100 by
numbers higher than 200.

Harvey Friedman

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