|
%I #23 Sep 21 2017 09:03:19
%S 151,252,1431,1831,2432,2832,3141,3181,3242,3282,11511,12512,14151,
%T 15141,15331,15910,21521,22522,24252,25242,25332,25920,31531,32532,
%U 33151,33252,91051,91510,92052,92520,114311,118311,124312,128312,141431,141831,142431
%N Decimal Goedelization of antitheorems from propositional calculus, in Richard C. Schroeppel's metatheory of A101273.
%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. 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.
%C Besides A101273, there can also be the subsequence of theorems that can be proved within the more restricted Intuitionist logic; the 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.
%C Comment from Ed Brims (ejbrims(AT)hotmail.com), Aug 18 2008: Quite interestingly, this sequence is quoted (in its original incorrect form) in the novel "The End of Mr Y" by Scarlett Thomas, towards the end of Chapter 12.
%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.
%D Scarlett Thomas, "The End of Mr Y", Harcourt Books.
%H Charles R Greathouse IV, <a href="/A100200/b100200.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>. See this for table of functions (and their notations) which corresponds to English conjunctions such as "and," "or," "not," etc. that takes one or more truth values as input and returns a single truth value as output. This includes ones used in A101273.
%H Eric Weisstein et al., <a href="http://mathworld.wolfram.com/GoedelNumber.html">Goedel Number.</a>
%F Well-formed formulas that are always false (i.e. negations of theorems) represented as decimal integers in Richard C. Schroeppel's metatheory of A101273.
%e 1431 corresponds to the sentence which violates the Law of the Excluded Middle: A ^ -A.
%e 33252 corresponds to the sentence, with square brackets being implicit parentheses, [ --B ] xor B.
%e 118311 corresponds to this sentence asserting the equivalence of the 3rd variable with its negation, C = -C.
%Y Cf. A101273, A101248.
%K nonn,base
%O 1,1
%A _Jonathan Vos Post_, Dec 27 2004
%E Sequence and example corrected by _Charles R Greathouse IV_, Oct 06 2009
%E Entries replaced by values from b-file. - _N. J. A. Sloane_, Oct 04 2010
|
