[seqfan] Thank yous, context, and future directions in Decimial Goedelization of WFFs
Jonathan Post
jvospost3 at gmail.com
Fri Oct 9 03:48:43 CEST 2009
I am enormously grateful to Charles R Greathouse IV for correcting m
sequences and examples A100200 Decimal Goedelization of antitheorems
from propositional calculus, in Richard Schroeppel's metatheory of
A101273; and A101248 Decimal Goedelization of contingent WFFs
(well-formed formulae) from propositional calculus...
The Venn diagram would show an ambient universe of the 10^N strings of
decimal digits of length N, within which are three mutually exclusive
proper subsets whose union is the Decimial Goedelization of WFFs in
Schroeppel's notation:
(1) A101273 = Theorems (tautologies);
(2) A100200 = Antitheorems (always false WFFs);
(3) A101248 = Contingent WFFs (neither theorem nor antitheorem).
We have a proper subset within A101273 which is the Goedelization of
theorems that van be proved from A140861 Decimal Goedelization of
Heyting's 11 axioms for intuitionistic propositional logic; and
analogous intuitionistic subsets within (2) and (3) above.
Now, analogous to Schroeppel's conjecture, I conjecture that a power
law approximates the number of integers in each of these sequences,
where the number with N digits is approximately N to the power of some
real number D differing for each sequence.
The count as shown in the Greathouse corrections for N=1, 2, 3, 4 digits is:
| A101273(N) | = 0, 0, 4, 10, ?
| A100200 (N) | = 0, 0, 2, 9, 23, ?
| A101248(N) | = 2, 6, 30, ?
It would be interesting to see the Greathouse code that corrected my
error-prone hand-made first cut of these, and what the parameters of
the power laws are, if they are valid as seems reasonable.
I am intrigued by looking at the graphs of these seqs so far, and look
forward to eventually publishing a refereed paper which could properly
acknowledge Schroeppel and Greathouse, in the enumeration of what in a
sense is the fractal structure of formal representations of
propositional calculus.
Thank you, and thank njas for saying that he'd like to see the seqs
corrected, and to the Associate Editor who noted the errors.
-- Jonathan Vos Post
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