cp's OEIS Frontend

A100200 Decimal Goedelization of antitheorems from propositional calculus, in Richard C. Schroeppel's metatheory of A101273.

%I A100200 #24 Feb 16 2025 08:32:55
%S A100200 151,252,1431,1831,2432,2832,3141,3181,3242,3282,11511,12512,14151,
%T A100200 15141,15331,15910,21521,22522,24252,25242,25332,25920,31531,32532,
%U A100200 33151,33252,91051,91510,92052,92520,114311,118311,124312,128312,141431,141831,142431
%N A100200 Decimal Goedelization of antitheorems from propositional calculus, in Richard C. Schroeppel's metatheory of A101273.
%C A100200 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 A100200 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 A100200 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 A100200 Goedel, K. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. New York: Dover, 1992.
%D A100200 Hofstadter, D. R. Goedel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 17, 1989.
%D A100200 Kleene, S. C. Introduction to Metamathematics. Princeton, NJ: Van Nostrand, p. 39, 1964.
%D A100200 Scarlett Thomas, "The End of Mr Y", Harcourt Books.
%H A100200 Charles R Greathouse IV, <a href="/A100200/b100200.txt">Table of n, a(n) for n=1..10000</a>
%H A100200 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PropositionalCalculus.html">Propositional Calculus</a>
%H A100200 Eric Weisstein's World of Mathematics, <a href="https://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 A100200 Eric Weisstein et al., <a href="https://mathworld.wolfram.com/GoedelNumber.html">Goedel Number.</a>
%F A100200 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 A100200 1431 corresponds to the sentence which violates the Law of the Excluded Middle: A ^ -A.
%e A100200 33252 corresponds to the sentence, with square brackets being implicit parentheses, [ --B ] xor B.
%e A100200 118311 corresponds to this sentence asserting the equivalence of the 3rd variable with its negation, C = -C.
%Y A100200 Cf. A101273, A101248.
%K A100200 nonn,base
%O A100200 1,1
%A A100200 _Jonathan Vos Post_, Dec 27 2004
%E A100200 Sequence and example corrected by _Charles R Greathouse IV_, Oct 06 2009
%E A100200 Entries replaced by values from b-file. - _N. J. A. Sloane_, Oct 04 2010