A101248 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).

1, 2, 11, 12, 21, 22, 31, 32, 111, 112, 121, 122, 141, 142, 152, 161, 162, 172, 182, 211, 212, 221, 222, 241, 242, 251, 261, 262, 271, 281, 311, 312, 321, 322, 331, 332, 910, 920, 1111, 1112, 1121, 1122, 1141, 1142, 1151, 1152, 1161, 1162, 1171, 1172, 1181, 1182

Offset: 1

Author

Jonathan Vos Post, Jan 23 2005

Keywords

• nonn
• base

Comments

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.

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.

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.

Examples

			1 A
2 B
11 C
12 D
21 E
22 F
31 -A
32 -B
111 G
112 H
121 I
122 J
141 A^A
142 A^B
152 A xor B
161 A V A
162 A V B
172 A->B
182 A=B
211 K
212 L
221 M
222 N
241 B^A
242 B^B
251 B xor A
261 B V A
262 B V B
271 B->A
281 B=A
311 -C
312 -D
321 -E
322 -F
331 --A
332 --B
910 (A)
912 (B)
1111 O
1112 P
1121 Q
1122 R
1141 C^A
1142 C^B
1151 C xor A
1152 C xor B
1161 C V A
1162 C V B
1171 C->A
1172 C->B
1181 C=A
1182 C=B
		

References

• Goedel, K. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. New York: Dover, 1992.
• Hofstadter, D. R. Goedel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 17, 1989.
• Kleene, S. C. Introduction to Metamathematics. Princeton, NJ: Van Nostrand, p. 39, 1964.

Links

• Charles R Greathouse IV, Table of n, a(n) for n=1..10000
• Eric Weisstein's World of Mathematics, Propositional Calculus.
• Eric Weisstein's World of Mathematics, Connective.
• Eric Weisstein et al. Gödel Number.

Crossrefs

Cf. A101273, A100200.

Extensions

Corrected sequence and examples Charles R Greathouse IV, Oct 06 2009