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Search: id:A101248
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| A101248 |
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Decimal Goedelization of contingent WFFs (well-formed formulae) from propositional calculus, in Richard Schroeppel's metatheory of A101273. Truth value depends on truth value of variables, but is neither always true (theorem) nor always false (antitheorem). |
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+0
4
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| 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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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 formulae 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 Schroeppel's metatheory of A101273.
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REFERENCES
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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.
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LINKS
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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. Goedel Number.
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EXAMPLE
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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
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CROSSREFS
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Cf. A101273, A100200.
Sequence in context: A038118 A038117 A038116 this_sequence A038115 A089604 A038114
Adjacent sequences: A101245 A101246 A101247 this_sequence A101249 A101250 A101251
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KEYWORD
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nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 23 2005
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EXTENSIONS
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Corrected sequence and examples Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 06 2009
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