Questions tagged [goodstein-sequences]
For questions about Goodstein sequences and Goodstein's Theorem.
6 questions
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Does Peano's axioms prove $\alpha$-induction for primitive recursive sequences for every concrete $\alpha < \varepsilon_0$?
It is well-known that Peano's axioms (PA) cannot prove $\varepsilon_0$-induction for primitive recursive sequences (PRWO($\varepsilon_0$)), because PA + PRWO($\varepsilon_0$) proves the consistency of ...
9
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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 ...
54
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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 ...
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0
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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 ...
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4
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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 ...
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2
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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:
...
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