Re: Goodstein's Theorem


Bhupinder Singh Anand wrote:
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

What do you mean by "constructive"? If you accept transfinite induction
up to epsilon-null, you should accept Goodstein's theorem. If not, then
you don't have to.

.

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