Crisis in mathematics

Mathematics is meant to be a rigorious deductive discipline based upon
sound principles

but
colin leslie dean showing that godels theorem - what godel did- is invalid
as it is based on invalid axioms
throws maths into crisis
because it now turns out that maths is not based upon sound principles

and ad hoc principles can be used if they apparently give the right
result

take the axiom of reducibility used by godel
it is ad hoc and unjustifiable as The Stanford Dictionary of Philosophy
states

The Stanford Dictionary of Philosophy states that ", many critics
concluded that the axiom of reducibility was simply too ad hoc to be
justified philosophically."


with this admission and the fact that godel used an ad hoc principle
the foundations of maths have been destroyed for any one can now use any
ad hoc principle to prove anything
take Fermats last theorem
any one can now create an ad hoc principle which will prove the theorem

colin leslie dean has thrown mathematics into crisis by shattering its
logical foundations
and by showing that truth can be arrived at by any ad hoc avenue
thus showing the myth of mathematics as a rigorous deductive discipline
based upon sound principles


to reiterate Godel does use the axiom of reducibility in his proof of HIS
incompleteness theorem

it is is his axiom 1v
and he uses it in his formular 40


Godel uses the axiom of reducibility axiom 1V of his system is the axiom
of reducibility â??As Godel says â??this axiom represents the axiom of
reducibility (comprehension axiom of set theory)â?? (K Godel , On formally
undecidable propositions of principia mathematica and related systems in
The undecidable , M, Davis, Raven Press, 1965,p.12-13)
..
Godel uses axiom 1V the axiom of reducibility in his formula 40 where he
states â??x is a formula arising from the axiom schema 1V.1 ((K Godel , On
formally undecidable propositions of principia mathematica and related
systems in The undecidable , M, Davis, Raven Press, 1965,p.21

â?? [40. R-Ax(x) â?¡ (â??u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u
& u Fr y & Form(y) & x = u â??x {v Gen [[R(u)*E(R(v))] Aeq y]}]

x is a formula derived from the axiom-schema IV, 1 by substitution â??


ramsey says of the axiom

Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

The Stanford Dictionary of Philosophy states that ", many critics
concluded that the axiom of reducibility was simply too ad hoc to be
justified philosophically."

--
Message posted using http://www.talkaboutscience.com/group/sci.math/
More information at http://www.talkaboutscience.com/faq.html

.

Relevant Pages

  • Godels incompleteness theorm proven wrong
    ... either lead to paradox or end in paradox. ... axiom of reducibility but this axiom was rejected as being invalid by ... Thus just on this point Godel is invalid as by using an axiom most people ... Godel proved that mathematics was inconsistent ...
    (sci.math)
  • Godels incompleteness theorem proven invalid
    ... axiom 1V - axiom of reducibility- in his proof. ... either lead to paradox or end in paradox. ... Thus just on this point Godel is invalid as by using an axiom most people ... Godel proved that mathematics was inconsistent ...
    (sci.math)
  • Re: Godels incompleteness theorem proven invalid
    ... Godel tells us he uses his system P to prove his incompleteness theorem ... 178] wrote of his comprehension axiom IV, ... Such an axiom has no place in mathematics, ... construct propositions which make statements about themselves,… ...
    (sci.logic)
  • Crisis in mathematics
    ... Mathematics is meant to be a rigorious deductive discipline based upon ... colin leslie dean showing that godels theorem - what godel did- is invalid ... because it now turns out that maths is not based upon sound principles ... concluded that the axiom of reducibility was simply too ad hoc to be ...
    (sci.logic)
  • Re: Godels incompleteness theorem was not just about a version o
    ... ad hoc axiom ie the axiom of reducibility and that maths is not based upon ... colin leslie dean showing that godels theorem - what godel did- is ... because it now turns out that maths is not based upon sound principles ... colin leslie dean has thrown mathematics into crisis by shattering its ...
    (sci.logic)