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The Insolubility of the So-called Gödel's First Incompleteness Theorem. Gödelian and Hilbertian Mathematics.

The Insolubility of the So-called Gödel's First Incompleteness Theorem. Gödelian and Hilbertian Mathematics.

Authors :
Penchev, Vasil
Source :
Philosophical Alternatives Journal / Filosofski Alternativi. 2010, Vol. 19 Issue 5, p104-119. 16p.
Publication Year :
2010

Abstract

Can the so-called first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected with some kind of self-reference. Gödel built his proof on the ground of self-reference: a statement which claims its unprovability. So, he demonstrated that undecidable propositions exist in any enough rich axiomatics (i.e. such one which contains Peano arithmetic in some sense). What about the decidability of the very first incompleteness theorem? We can display that it fulfills its conditions. That's why it can be applied to itself, proving that it is an undecidable statement. It seems to be a too strange kind of proposition: its validity implies its undecidability. If the validity of a statement implies its untruth, then it is either untruth (reductio ad absurdum) or an antinomy (if also its negation implies its validity). A theory that contains a contradiction implies any statement. Appearing of a proposition, whose validity implies its undecidability, is due to the statement that claims its unprovability. Obviously, it is a proposition of self-referential type. By Gödel's words, it is correlative with Richard's or liar paradox, or even with any other semantic or mathematical one. What is the cost, if a proposition of that special kind is used in a proof? In our opinion, the price is analogous to «applying» of a contradictory in a theory: any statement turns out to be undecidable. If the first incompleteness theorem is an undecidable theorem, then it is impossible to prove that the very completeness of Peano arithmetic is also an undecidable statement (the second incompleteness theorem). Hilbert's program for an arithmetical self-foundation of mathematics is partly rehabilitated: only partly, because it is not decidable and true, but undecidable; that's why both it and its negation may be accepted as true, however not simultaneously true. The first incompleteness theorem gains the statute of axiom of a very special, semi-philosophical kind: it divides mathematics as whole into two parts: either Gödel mathematics or Hilbert mathematics. Hilbert's program of self-foundation of mathematic is valid only as to the latter. [ABSTRACT FROM AUTHOR]

Details

Language :
Russian
ISSN :
08617899
Volume :
19
Issue :
5
Database :
Academic Search Index
Journal :
Philosophical Alternatives Journal / Filosofski Alternativi
Publication Type :
Academic Journal
Accession number :
60389682