[seqfan] Re: Thank yous, context, and future directions in Decimial Goedelization of WFFs

[Hagen von Eitzen]

Jonathan Post schrieb:
> 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.
>   
Hm, the union of the sequences, i.e. *all* well-formed formulas do 
follow a power law
be its recursive nature (i.e a WFF of length n is either one of two 
variables of length n
or  "not P" with P of length n-1 or "(P)" with P of length n-2 or "P 
operator Q" with
length(P)+length(Q) = n-1).
Next, if P is a theorem of length n, then "A and P" is a contingent WFF 
of length n+2,
if P is an anti-theorem of length n, then "A or P" is a contingent WFF 
of length n+2,
and similarly if P is any WFF of length n, then "(A equiv A) or P" is a 
theorem and
"(A xor A) and P" an antitheorem. Hence if there are approximately c * 
D^n  WFF's of length n
we already see that e.g. #{theorems of length n) is at least 
approximately c/D^6 * D^n


> 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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