Jan Burse gives support that godels use of impredicative statments invalidates his theorem


Jan Burse points out

The axiom of reducibility draws attention to the
problematic status of impredicative definitions. To quote Weyl 1946,

as it is another example colin leslie dean cliams makes godel invalid as
godel state himself that he constructs impredicative statements- which
text books on logic say are invalid and your quote shows are a problem and
make his theorem invalid

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

â?? 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,â?¦ ((K
Godel , On undecidable propositions of formal mathematical systems in The
undecidable , M, Davis, Raven Press, 1965, p.63 of this work Dvis notes,
â??it covers ground quite similar to that covered in Godels orgiinal 1931
paper on undecidability,â?? 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



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