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%I #24 Feb 24 2023 03:10:53
%S 1,2,11,12,21,22,31,32,111,112,121,122,141,142,152,161,162,172,182,
%T 211,212,221,222,241,242,251,261,262,271,281,311,312,321,322,331,332,
%U 910,920,1111,1112,1121,1122,1141,1142,1151,1152,1161,1162,1171,1172,1181,1182
%N Decimal Goedelization of contingent WFFs (well-formed formulas) from propositional calculus, in Richard C. Schroeppel's metatheory of A101273. Truth value depends on truth value of variables, but is neither always true (theorem) nor always false (antitheorem).
%C Blocks of 1's and 2s are variables: A = 1, B = 2, C = 11, D = 12, E = 21, ... Not (also written -) = 3; And = 4; Xor = 5; Or = 6; Implies = 7; Equiv = 8; Left Parenthesis = 9; Right Parenthesis = 0. Operator binding strength is in numerical order, Not > And > ... > Equiv. The non-associative "Implies" is evaluated from Left to Right; A->B->C = is interpreted (A->B)->C.
%C Redundant parentheses are permitted, so long as they are balanced and centered on a valid variable or sentential formula and not on the null character. Besides A101273 (theorems = tautologies), A100200 (antitheorems = always false WFFs) there can also be the subsequence of theorems that can be proved within the more restricted intuitionistic logic; this sequence of well-formed formulas whose truth value is contingent on the truth values of their variables; and many others.
%C As with A101273, I conjecture that a power law approximates the number of integers in this sequence, where the number with N digits is approximately N to the power of some real number D. The union of A101273, A100200 and this sequence is the set of all WFFs in Richard C. Schroeppel's metatheory of A101273.
%D Goedel, K. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. New York: Dover, 1992.
%D Hofstadter, D. R. Goedel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 17, 1989.
%D Kleene, S. C. Introduction to Metamathematics. Princeton, NJ: Van Nostrand, p. 39, 1964.
%H Charles R Greathouse IV, <a href="/A101248/b101248.txt">Table of n, a(n) for n=1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PropositionalCalculus.html">Propositional Calculus</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Connective.html">Connective</a>.
%H Eric Weisstein et al. <a href="http://mathworld.wolfram.com/GoedelNumber.html">Gödel Number</a>.
%e 1 A
%e 2 B
%e 11 C
%e 12 D
%e 21 E
%e 22 F
%e 31 -A
%e 32 -B
%e 111 G
%e 112 H
%e 121 I
%e 122 J
%e 141 A^A
%e 142 A^B
%e 152 A xor B
%e 161 A V A
%e 162 A V B
%e 172 A->B
%e 182 A=B
%e 211 K
%e 212 L
%e 221 M
%e 222 N
%e 241 B^A
%e 242 B^B
%e 251 B xor A
%e 261 B V A
%e 262 B V B
%e 271 B->A
%e 281 B=A
%e 311 -C
%e 312 -D
%e 321 -E
%e 322 -F
%e 331 --A
%e 332 --B
%e 910 (A)
%e 912 (B)
%e 1111 O
%e 1112 P
%e 1121 Q
%e 1122 R
%e 1141 C^A
%e 1142 C^B
%e 1151 C xor A
%e 1152 C xor B
%e 1161 C V A
%e 1162 C V B
%e 1171 C->A
%e 1172 C->B
%e 1181 C=A
%e 1182 C=B
%Y Cf. A101273, A100200.
%K nonn,base
%O 1,2
%A _Jonathan Vos Post_, Jan 23 2005
%E Corrected sequence and examples _Charles R Greathouse IV_, Oct 06 2009
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