People normally believe
that Arithmetic is not complete because GÖdel launched this idea a long time ago,
and it looks as if nobody has presented sound evidence on the contrary. We here
intend to do that perhaps for the first time in history. We prove that what Stanford
Encyclopedia has referred to as Theorem 3 cannot be true, and, therefore, if nothing
else is presented in favour of GÖdel’s thesis, we actually do not have evidence
on the incompleteness of Arithmetic: All available evidence seems to point at the
extremely opposite direction.
References
[1]
Kennedy, J. (2015) Kurt GÖdel. http://plato.stanford.edu/entries/goedel/#BioSke
[2]
GÖdel, K. (1929) “I”. University of Vienna, Vienna.
[3]
Hilbert, D. and Ackermann, W. (1928) Grundzüge der theoretischen Logik. Springer-Verlag, Berlin.
[4]
van Heijenoort, J., Ed. (1967) From Frege to GÖdel: A Sourcebook in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge.
[5]
Zach, R. (1999) Completeness before Post: Bernays, Hilbert, and the Development of Propositional Logic. Bulletin of Symbolic Logic, 5, 331-366. http://dx.doi.org/10.2307/421184
[6]
Post, E.L. (1921). Introduction to a General Theory of Elementary Propositions. American Journal of Mathematics, 43, 163-185. http://dx.doi.org/10.2307/2370324
[7]
Bernays, P. (1926) Axiomatische Untersuchung des Aussagen-Kalkuls der “Principia Mathematica.” Mathematisches Zeitschrift, 25, 305-320. http://dx.doi.org/10.1007/BF01283841
[8]
Godel, K. (1930) Die Vollständigkeit der Axiome des logischen Functionenkalküls. Monatshefte Für Mathematik Und Physik, 37, 349-360. http://dx.doi.org/10.1007/BF01696781
[9]
Mal’cev, A.I. (1971) The Metamathematics of Algebraic Systems. In: Wells III, B.F., Ed., North-Holland Publishing Co., Amsterdam.
[10]
Pinheiro, I.M.R. (2012) Concerning the Solution to the Liar Paradox. E-Logos, 21, 15. http://www.academia.edu/9787988/Solution_to_the_Liar_Paradox
[11]
Pinheiro, M.R. (2015) Words for Science. Indian Journal of Applied Research, 5, 19-22. https://www.worldwidejournals.com/ijar/articles.php?val=NjQ0MQ==&b1 =853&k=214
Schechter, E. (2005) Classical & Nonclassical Logics an Introduction to the Mathematics of Propositions. http://www.math.vanderbilt.edu/~schectex/logics/
[14]
Meyer, J.R. (2009) The Fundamental Flaw in GÖdel’s Proof of His Incompleteness Theorem “on Formally Undecidable Propositions of Principia Mathematica and Related Systems”. https://www.academia.edu/4706759/The_Fundamental_Flaw_in_G%C3%B6del _s_Proof_of_the_Incompleteness_Theorem
[15]
Pinheiro, M.R. (2008) On the Consistency and the Completeness of Arithmetic. Research Report, Washington Public Library, Washington, 32.
[16]
MathsIsFun.com (2014) Definition of Power. https://www.mathsisfun.com/definitions/power.html
[17]
McCall, S. (2014) Oxford University Pr. (The Consistency of Arithmetic: And Other Essays, Ed.). Oxford University Press. https://books.google.com.au/books?id=7d9_AwAAQBAJ&pg=PR7&lpg=PR7&dq=mccall +2008+the+consistency+of+arithmetic&source=bl&ots=Nk4rx6vGc6&sig =FQojeCga4E50uwRi5i3yU4lY2PM&hl=en&sa=X&ved= 0ahUKEwjnov2gioPNAhWhg6YKHcakAVsQ6AEIGzAA#v =onepage&q&f=false
[18]
Pinheiro, M.R. (2016) GÖdel and the Incompleteness of Arithmetic: He Had a Dream. http://itshouldallbeaboutlogic.blogspot.com.au/2016/05/godel-and-incompleteness -of-arithmetic.html
