[FOM] feasibility (was PA Incompleteness)
Vaughan Pratt
pratt at cs.stanford.edu
Wed Oct 17 03:30:41 EDT 2007
From: Vladimir Sazunov
> 2. Another approach is based on Parikh's formalization of feasible
> numbers and, as I believe, more adequate formalization which I
> mentioned in my recent posting to FOM
> http://www.cs.nyu.edu/pipermail/fom/2007-October/012042.html
>
> Here it is assumed that the "semi-set" (using the term of Vopenka) F of
> feasible numbers contains 0,1 and is closed under +, but is bounded by
> 2^1000 (Alternatively, forall n log log n < 10 is postulated what means
> that 2^1024.)
If we are talking about the "real" world here, may I conservatively
suggest 2^^6, a stack of six 2's? (Less conservatively, four 2's and a
3, but six 2's is simpler as well as offering a handy margin of error.
The notation 2^^6 adapts to ASCII Knuth's notation for the next
operation after exponentiation in the Grzegorczyk hierarchy.)
Rationale: it is plausible that there are fewer than 2^256 qubits in
what we understand by the "real world," which therefore would have at
most 2^(2^256) observable states, noting that 256 = 2^(2^3) < 2^^4,
whence a state count of six 2's or 2^^6.
2^1024 is tiny, being the square root of the number of memory states of
the MITS Altair 8800, which at $400 in 1975 was the first PC readily
available to consumers. Its 256 bytes of RAM gave it 2^2048 states for
its RAM alone, rather less than a stack of four 2's (2^65536 = 2^^4, the
number of states of RAM in the enhanced MAI JOLT kit I assembled in
September 1975).
For that money today you can buy a terabyte of hard drive offering well
over 2^^5 states.
Another two in the stack takes you to 2^^6, which should see you through
the foreseeable future of the universe.
Unless of course you were planning to reason about the universe with
either dynamic or temporal logic. In that case you would appear to be
looking at 2^^7 possible predicates on the states of the universe.
On the other hand the counterargument implied by
http://boole.stanford.edu/pub/coimbra.pdf suggests 2^^5 as a more
reasonable estimate of the number of such predicates, on the ground that
the 2^^6 states form a complete atomic Boolean algebra or CABA,
predicates on which are better understood as complete ultrafilters, i.e.
the qubits themselves as the atoms of the CABA, than simply as subsets
of its underlying set.
Number theory will no doubt demand larger numbers, feasibility be
damned, but for reasoning about the real world, 2^^6, a stack of six
2's, may well suffice.
Vaughan Pratt
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