TO GIVE DETAIL
Godel states that he is going to use the system of PM
" before we go into details lets us first sketch the main ideas of
the proof … the formulas of a formal system (we limit ourselves
here to the system PM) …" ((K Godel , On formally undecidable
propositions of principia mathematica and related systems in The
undecidable , M, Davis, Raven Press, 1965,pp.-6)
Godel uses the axiom of reducibility and axiom of choice from the PM
Quote
mrob.com/pub/math/goedel.htm
“A. Whitehead and B. Russell, Principia Mathematica, 2nd edition,
Cambridge 1925. In particular, we also reckon among the axioms of PM
the axiom of infinity (in the form: there exist denumerably many
individuals), and the axioms of reducibility and of choice (for all
types)” ((K Godel , On formally undecidable propositions of principia
mathematica and related systems in The undecidable , M, Davis, Raven
Press, 1965, p.5)
AXIOM OF REDUCIBILITY
(1) Godel uses the axiom of reducibility axiom 1V of his system is the
axiom of reducibility "As Godel says “this axiom represents the axiom
of reducibility (comprehension axiom of set theory)” (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 "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
" [40. R-Ax(x) ≡ (∃u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u
&
u Fr y & Form(y) & x = u ∃x {v Gen [[R(u)*E(R(v))] Aeq y]}]
x is a formula derived from the axiom-schema IV, 1 by substitution "
mrob.com/pub/math/goedel.html
( 2) “As a corollary, the axiom of reducibility was banished as
irrelevant to mathematics … The axiom has been regarded as
re-instating the semantic paradoxes” -
mind.oxfordjournals.org/cgi/repr … 28/823.pdf
2)“does this mean the paradoxes are reinstated. The answer seems to
be yes and no” - fds.oup.com/www.oup.co.uk/pdf/0-19-825075-4.pdf )
-
It has been repeatedly pointed out this Axiom obliterates the
distinction according to levels and compromises the vicious-circle
principle in the very specific form stated by Russell. But The
philosopher and logician FrankRamsey (1903-1930) was the first to
notice that the axiom of reducibility in effect collapses the
hierarchy of levels, so that the hierarchy is entirely superfluous in
presence of the axiom.
(helsinki.fi/filosofia/gts/ramsay.pdf)
-
Russell Ramsey and Witgenstein regarded it as illegitimate Russell
abandoned this axiom and many believe it is illegitimate and must be
not used in mathematics
Ramsey says
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
AXIOM OF CHOICE
Godel states he uses the axiom of choice “this allows us to deduce
that even with the aid of the axiom of choice (for all types) … not
all sentences are decidable…” (K Godel , On formally undecidable
propositions of principia mathematica and related systems in The
undecidable , M, Davis, Raven Press, 1965. p.28.) Quite clearly the
axiom of choice is part of the meta-theory used in the deduction
("The Axiom of Choice (AC) was formulated about a century ago, and it
was controversial for a few of decades after that; it may be
considered the last great controversy of mathematics…. A few pure
mathematicians and many applied mathematicians (including, e.g., some
mathematical physicists) are uncomfortable with the Axiom of Choice.
Although AC simplifies some parts of mathematics, it also yields some
results that are unrelated to, or perhaps even contrary to, everyday
“ordinary” experience; it implies the existence of some rather
bizarre, counterintuitive objects. Perhaps the most bizarre is the
Banach-Tarski Paradox "–
math.vanderbilt.edu/~schecte … hoice.html)
IMPREDICATIVE DEFINITIONS
Godel used impredicative definitions
Ponicare Russell and philosophers argue these types of definitions are
invalid Ponicare Russell point out that they lead to contradictions in
mathematics
Quote from Godel
" The solution suggested by Whitehead and Russell, that a proposition
cannot say something about itself , is to drastic… We saw that we
can construct propositions which make statements about themselves,…
((K Godel , On undecidable propositions of formal mathematical
systems in The undecidable , M, Davis, Raven Press, 1965, p.63 of this
work Dvis notes, “it covers ground quite similar to that covered in
Godels orgiinal 1931 paper on undecidability,” p.39.)
What Godel understood by “propositions which make statements about
themselves”
is the sense Russell defined them to be
‘Whatever involves all of a collection must not be one of the collection.’
Put otherwise, if to define a collection of objects one must use the total
collection itself, then the definition is meaningless. This explanation
given by Russell in 1905 was accepted by Poincare’ in 1906, who coined the
term impredicative definition, (Kline’s “Mathematics: The Loss of
Certainty”
Note Ponicare called these self referencing statements impredicative
definitions
texts books on logic tell us self referencing ,statements (petitio
principii) are invalid
Godels has argued that impredicative definitions destroy mathematics and
make it false
friesian.com/goedel/chap-1.htm
Gödel has offered a rather complex analysis of the vicious circle
principle and its devastating effects on classical mathematics
culminating in the conclusion that because it “destroys the derivation
of mathematics from logic, effected by Dedekind and Frege, and a good
deal of modern mathematics itself” he would “consider this rather as a
proof that the vicious circle principle is false than that classical
mathematics is false”