Questions tagged [goodstein-sequences]

For questions about Goodstein sequences and Goodstein's Theorem.

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54 votes
1 answer
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In the two-person Killing the Hydra game, what is the winning strategy?

My question is which player has a winning strategy in the two-player version of the Killing the Hydra game? In their amazing paper, Kirby, Laurie; Paris, Jeff, Accessible independence results for ...
Joel David Hamkins's user avatar
9 votes
0 answers
187 views

primes concatenation sequence

Let us take a natural number x > 1. Then define a sequence $x_n$ as follows: $x_0=x$; if $x_n = p_1\cdots p_s$, where $p_1\leqslant\dots\leqslant p_s$ are prime numbers, then $x_{n+1}$ is the ...
Nikolay Kazimirov's user avatar
2 votes
4 answers
1k views

Can transfinite induction be defined as axiom scheme in FOL on bin-tree structures?

Transfinite induction requires a second order induction hypothesis. So, that can not be defined as axiom scheme in FOL. However, if I look to Goodstein's theorem en the Hydra games, then they have to ...
Lucas K.'s user avatar
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1 vote
2 answers
1k views

Can Goodstein's theorem been proven with first order PA + Constructive Omega Rule?

I am trying to understand transfinite induction and Gentzen's theories. But I was wondering, if there is any connection with the Constructive Omega Rule (COR). With COR I mean that if you can proof: ...
Lucas K.'s user avatar
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0 votes
1 answer
285 views

Non-standard naturals and goodstein sequences [closed]

By the Kirby–Paris theorem, Goodstein's theorem is independent of Peano arithmetic (PA). Therefore there are non-standard models in which every Goodstein sequence terminates. However, Tennenbaum's ...
g5375637's user avatar