cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

151, 252, 1431, 1831, 2432, 2832, 3141, 3181, 3242, 3282, 11511, 12512, 14151, 15141, 15331, 15910, 21521, 22522, 24252, 25242, 25332, 25920, 31531, 32532, 33151, 33252, 91051, 91510, 92052, 92520, 114311, 118311, 124312, 128312, 141431, 141831, 142431
Offset: 1

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Author

Jonathan Vos Post, Dec 27 2004

Keywords

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, 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.
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.

Examples

			1431 corresponds to the sentence which violates the Law of the Excluded Middle: A ^ -A.
33252 corresponds to the sentence, with square brackets being implicit parentheses, [ --B ] xor B.
118311 corresponds to this sentence asserting the equivalence of the 3rd variable with its negation, C = -C.

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.
  • Scarlett Thomas, "The End of Mr Y", Harcourt Books.

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. 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.
  • Eric Weisstein et al., Goedel Number.

Crossrefs

Cf. A101273, A101248.

Formula

Well-formed formulas that are always false (i.e. negations of theorems) represented as decimal integers in Richard C. Schroeppel's metatheory of A101273.

Extensions

Sequence and example corrected by Charles R Greathouse IV, Oct 06 2009
Entries replaced by values from b-file. - N. J. A. Sloane, Oct 04 2010

A101273 Theorems from propositional calculus, translated into decimal digits.

Original entry on oeis.org

171, 181, 272, 282, 1531, 1631, 2532, 2632, 3151, 3161, 3252, 3262, 11711, 11811, 12712, 12812, 14171, 14181, 14271, 14272, 15171, 15172, 16171, 16181, 17141, 17161, 17162, 17261, 17331, 17910, 18141, 18161, 18331, 18910, 21721, 21821, 22722, 22822, 24171
Offset: 1

Views

Author

Richard C. Schroeppel, Dec 19 2004

Keywords

Comments

Blocks of 1s and 2s are variables: A = 1, B = 2, C = 11, D = 12, E = 21, ... Not = 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.
This is a decimal Goedelization of theorems from a particular axiomatization of propositional calculus. This should be linked to the subsequences of theorems and antitheorems. - Jonathan Vos Post, Dec 19 2004 [This comment is referring to A100200 and A101248. - N. J. A. Sloane, May 19 2020]
Comment from Charles R Greathouse IV, May 17 2020: (Start)
Each positive integer represents a string of one or more symbols, as described above. Some represent well-formed formulas. Of those, some are theorems (A101273) while others are antitheorems (A100200) with the remaining wffs in A101248. The first few theorems are
171, A -> A
181, A <-> A
272, B -> B
282, B <-> B
1531, A XOR ~A,
with 1 = A, 7 = ->, etc. (End)
In short: any well-formed formula (wff) can be mapped to an integer. The sequence lists those integers that correspond to wff's that are theorems. - N. J. A. Sloane, May 19 2020

Examples

			Example: 17162 is the theorem A->AvB.

References

  • M. Davis, Computability and Unsolvability. New York: Dover 1982.
  • D. R. Hofstadter, Goedel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 18, 1989.
  • S. C. Kleene, Mathematical Logic. New York: Dover, 2002.

Links

Crossrefs

See A100200 and A101248 for further information.

Formula

It appears that the n-th term is very roughly n^c, for some c>1.

Extensions

Corrected and edited by Charles R Greathouse IV, Oct 06 2009

A344900 a(n) is the number of well-formed formulas (wffs) of zeroth-order logic containing n characters (see comments).

Original entry on oeis.org

1, 1, 13, 25, 37, 61, 97, 561, 1869, 4437, 9097, 17097, 54101, 194189, 583857, 1490017, 3371997, 8916485, 28974361, 94338361, 277239461, 728378813, 1938657473, 5839518033, 18961970605, 59883346869, 174804016553, 493085118121, 1460284207861, 4646560028141
Offset: 1

Views

Author

Christoph B. Kassir, Jun 01 2021

Keywords

Comments

The characters of zeroth-order logic include the tilde (~), ampersand (&), wedge (∨), horseshoe (⊃), triple bar (≡), left and right parentheses, and variables (upper-case letters with or without subscripts.) However, since the set of upper-case letters with or without subscripts is infinitely large, it is then, for the sentences of zeroth-order logic containing k variables, restricted to the set {A1, ..., Ak}, with an additional restriction as follows: a sentence may only contain Ai iff it contains every Aj for j=1..i-1 (this gives a total of A000670(k-1) legal permutations for a sentence containing k variables.)
The rules for a well-formed formula (wff) of zeroth-order logic are defined recursively as follows (see M. Bergmann et al.):
1. Every variable is a wff.
2. If P is a wff, then so is ~P.
3. If P and Q are wffs, then so is (PxQ), where 'x' is any binary logical operator.
It is also customary to remove the outermost parentheses of a sentence.

Examples

			a(4) = 25, since the number of sentences of zeroth-order logic containing four characters are as follows: ~~~A, ~AxA, Ax~A, ~AxB, Bx~A, ~BxA, and Ax~B, where 'x' is any of the four binary logical operators.

References

  • Merrie Bergmann, James Moor, and Jack Nelson. The logic book. Vol. 2. New York: McGraw-Hill, 1990, p. 54.

Links

Crossrefs

Cf. A300696, A000670.
Related sequences: A101248, A101273, A308616. - N. J. A. Sloane, Aug 17 2021

Extensions

More terms from Sean A. Irvine, Jul 24 2021

A140861 Decimal Goedelization of Heyting's 11 axioms for intuitionistic propositional logic.

Original entry on oeis.org

1791410, 91420792410, 91720799141109241100, 991720492711007917, 2791720, 91491720072, 1791620, 91620792610, 99171104927110079916207110, 31791720, 99172049173200731
Offset: 1

Views

Author

Jonathan Vos Post, Jul 18 2008

Keywords

Comments

Axioms of Heyting (1930) as explained in Mark van Atten (2008). The same notation as in A101273, including: 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 hard thing, given errors in my related earlier submissions within Richard C. Schroeppel's metatheory, is to list in numerical order the theorems that can be proved from these 11 axioms.

Examples

			axiom 1: A->(A^A).
axiom 2: (A^B)->(B^A).
axiom 3: (A->B)->((A^C)->(B^C)).
axiom 4: ((A->B)^(B->C))->(A->C).
axiom 5: B->(A->B).
axiom 6: (A^(A->B))->B.
axiom 7: A->(AvB).
axiom 8: (AvB)->(BvA).
axiom 9: ((A->C)^(B->C))->((AvB)->C).
axiom 10: -A->(A->B).
axiom 11: ((A->B)^(A->-B))->-A.

References

  • Heyting, A., 1930, Die formalen Regeln der intuitionistischen Logik I, Sitzungsberichte der Preussischen Akademie der Wissenschaften, 42-56. English translation in Mancosu, 1998, pp.311-327.

Links

Crossrefs

Cf. A101273, A100200, A101248, A101273.

A166746 Count of propositional theorems up to 10^n in Richard C. Schroeppel's Goedelization of A101273.

Original entry on oeis.org

0, 0, 4, 12, 72, 262, 1396, 6126, 31186, 144606, 713078, 3384796
Offset: 1

Views

Author

Charles R Greathouse IV, Oct 21 2009

Keywords

Comments

Jonathan Vos Post conjectures that a(n) is approximately D^n for some real number D.

Examples

			The first five theorems in A101273 are 171 ("A->A"), 181 ("A=A"), 272 ("B->B"), 282 ("B=B"), and 1531 ("A xor -A"), so a(1) = 0, a(2) = 0, and a(3) = 4.

Links

Crossrefs

A101273, A100200, A101248
Showing 1-5 of 5 results.

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