Re: Goodstein's Theorem

matthias@xxxxxxxxxxx wrote:

M>> To be more precise: Goodstein's theorem is equivalent (over, say,
PA^- plus Sigma_1 induction) to arithmetical transfinite induction on
all ordinals below epsilon_0. << M

Matthias
=======
The assumption that "PA^- plus Sigma_1 induction" is consistent may
need careful attention.

I have argued that, in the absence of a constructive, and
intuitionistically unobjectionable, proof, or meta-proof, that any
given Goodstein sequence is bounded in Peano Arithmetic, the standard
interpretation of Goodstein's Theorem as a number-theoretic assertion
that is 'consistent' with any formal system of Peano Arithmetic (in the
sense that the range over which the assertion holds can be added as a
constant to the Arithmetic, along with its definitions as axioms), such
as standard PA, ought not to be accepted as definitive.

http://alixcomsi.com/Goodstein_argument.pdf
http://alixcomsi.com/Goodstein_argument.htm

More specifically, I have argued that Goodstein's argument implicitly
assumes that the recursively defined ordinal sequence, {W[n,
m<n+1>|w]}, as defined in the above, is a formal mathematical object in
Cantor Arithmetic.

In other words, the argument implicitly assumes the existence of a
well-defined set of transfinite ordinals, in Cantor Arithmetic, which
has properties corresponding to the properties required of the
number-theoretic sequence L(k) that is defined in the third Goodstein
sequence theorem in the above paper.

Since I have also argued, elsewhere, that such an assumption need not
necessarily hold, Goodstein's Theorem can, reasonably, be viewed as a
number-theoretic proposition whose truth in the standard interpretation
of any formal system of Peano Arithmetic has simply been asserted as a
non-verifiable consequence of a non-constructive argument.

I conclude that, in the absence of a constructive, and
intuitionistically unobjectionable, proof, or meta-proof, that any
given Goodstein sequence is bounded in Peano Arithmetic, the standard
interpretation of Goodstein's Theorem as a number-theoretic assertion
that is consistent with any formal system of Peano Arithmetic, such as
standard PA, ought not to be accepted as definitive.

Regards,

Bhup

.

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