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MathML Test : Goodstein's sequences

1. Introduction

The sequences we are going to study grow very fastly and get values so high that they are hardly computable by a machine. Using Zermelo-Frankel Theory, Goodstein established that these sequences finally fall down to zero. Moreover, Kirby and Paris showed later that this result is undecidable in Peano arithmetic.

1.1. Hereditary base-b notation

Goodstein's sequences are defined using the concept of hereditary base-b notation iterated. It is well known that for all $n\in \mathbb{N}$ and a base $b\ge 2$ you can find a base-b decomposition of $n$:

$n=\sum _{i=0}^k{a}_i{b}^i$

This notation can be made unique if you add conditions on $k$ (for instance $k$ the least possible). Now the hereditary base-b notation of $n$ is defined in the same way, but you recursively apply this notation to the exponents $i$. Because the numbers are strictly decreasing at each level, there is only a finite number of steps to decompose $n$. For instance, the base-3 notation and the heriditary base-3 notation of 123456789 are:

$123456789=2\times 3^{16}+2\times 3^{15}+1\times 3^{14}+2\times 3^{13}+1\times 3^{12}+2\times 3^{10}+2\times 3^9+2\times 3^7+2\times 3^5+1\times 3^4+2\times 3^3+2\times 3^2$

$123456789=2.3^{3^2+2.3^1+1}+2.3^{3^2+2.3}+3^{3^2+3^1+2}+2.3^{3^2+3^1+1}+3^{3^2+3^1}+2.3^{3^2+1}+2.3^{3^2}+2.3^{{2.3}^1+1}+2.3^{3^1+2}+3^{3^1+1}+2.3^{3^1}+2.3^2$

Definition 1.2 (Goodstein's sequences)

For all bases $b_1,b_2\ge 2$, we define $R_{b_1,b_2}:\mathbb{N}\to \mathbb{N}$ the function such that $R_{b_1,b_2}\left(n\right)$ is the number you get when you replace each occurence of $b_1$ by $b_2$ in the hereditary base-$b_1$ notation of $n$. For the sake of completeness, we take $R_{b_1,b_2}\left(0\right)=0$.

For all $a\in \mathbb{N}$ and $n_0\ge 2$ we define the sequence ${\left({g^{a,n_0}}_n\right)}_{n\ge n_0}$ by:

• ${g^{a,n_0}}_{n_0}=a$
• $\forall n\ge n_0,{g^{a,n_0}}_{n+1}=\{\begin{array}{c}{R}_{n,n+1}\left({g^{a,n_0}}_n\right)-1 \text{if} {g^{a,n_0}}_n\ne 0\\ 0\text{otherwise}\end{array}$

For $n_0=2$, we write${\left({g^a}_n\right)}_{n\ge 2}$ and call them Goodstein's sequences.

1.3. Trivial cases

Goodstein's Theorem is trivial for $a=\mathrm{0,1,2,3}$:

2. Study of Goodstein's sequences

2.1. Examples

For $a\ge 4$, the first values of the Goostein's sequence are not enough to see whether they reach zero:

Proposition 2.2

For all $\left(a,b,c\right)\in {\mathbb{N}}^3$ such that $b\ge 2\wedge a<b\wedge c<b$, $f\left(a,b,c\right)=min\left\{n\in \mathbb{N}|{g^{a{b}^c,b}}_n=0\right\}-b$ (i.e. the number of step to reach zero if you apply the Goodstein's algorithm to $a{b}^c$) is well-defined.

Proof:

We show the result by induction, using the lexicographical order on $\left(c,a\right)$. For $a=0$, we have $a{b}^c=0$, hence $f\left(a,b,c\right)=0$ is defined. Otherwise, the element is non-minimal and we have:

One can check that the $f\left(a\text{'},b\text{'},c\text{'}\right)$ that appear in this table are for $a\text{'},c\text{'}$ such that $\left(c\text{'},a\text{'}\right)<\left(c,a\right)$ and with the condition $b\text{'}\ge 2\wedge a\text{'}<b\text{'}\wedge c\text{'}<b\text{'}$□

From the previous proof, we get (see [4]):

Corollary 2.3

$f$ is defined by induction:

• $f\left(0,b,c\right)=0$
• $f\left(a,b,c\right)=1+\sum _{k=1}^c{u}_k+f\left(a-1,\sum _{k=0}^c{u}_k,c\right)$ where
  • $u_0=b+1$
  • $\forall n\in [0,c-1]u_{n+1}=f\left(b,\sum _{k=0}^n{u}_k,c\right)$

Corollary 2.4

If a Goodstein's sequence reach a value ${g^a}_n$ where all the exponents in the base-$n$ decomposition are strictly less than $n$, then this sequence finally reach zero.

Corollary 2.5

1. $f\left(a,b,0\right)=a$
2. $f\left(a,b,1\right)=\left(b+1\right)(2^a-1)$
3. $f\left(1,b,2\right)=(2^{b+1}-1)\left(b+1\right)$
4. $f\left(a,b,2\right)=2^{u_a+1}(u_a+1)-\left(b+1\right)$ where
   • $u_1=b$
   • $\forall k\in [1,a-1]u_{k+1}=2^{u_k+1}(u_k+1)-1$

2.6. Case $a=4$

Using the corollary 2.5, we can find $n_0$ :

• $u_1=23$
• $u_2=(2^{u_1+1}-1)(u_1+1)=3.2^{27}-1$
• $n_0=23+2^{u_2+1}(u_2+1)-\left(23+1\right)=2^{3.2^{27}}\left(3.2^{27}\right)-1=3.2^{3.2^{27}+27}-1=3.2^{402653211}-1$

2.7. Case $a=5$

Once again, we can try to compute the value of the $n_i$, using the corollary 2.5:

• $n_5=2^{2^{2^{2^6+6}+2^6+6}+2^{2^6+6}+2^6+6}-1$
• $n_6=2^{2^{2^{2^6+6}+2^6+6}+2^{2^6+6}+2^6+6}$
• $n_7=2^{2^{2^{2^6+6}+2^6+6}+2^{2^6+6}+2^6+7}-1$
• $n_8=2^{2^{2^{2^{2^6+6}+2^6+6}+2^{2^6+6}+2^6+6}+2^{2^{2^6+6}+2^6+6}+2^{2^6+6}+2^6+6}-1$

It is really hard to express $n_9$ as you need to compute a sequence with about $n_5=2^{2^{2^{2^6+6}+2^6+6}+2^{2^6+6}+2^6+6}-1$ steps. Moreover, if I want to write its value with the same notation as before, you also need about $n_5$ levels of exponents! The rank $n_{17}$ where the sequence reaches zero is even more difficult to express!

3. Proof of Goodstein Theorem

We work in the Zermerlo-Frankel set theory (ZF) and follow the proof of [2]. We use the traditional notation $\omega$ for the least infinite ordinal. $R_{b,\omega }\left(n\right)$ is an ordinal defined in the same way as above: you replace each occurence of $b$ by $\omega$ in the hereditary base-$b$ notation of $n$. Recall the ordinal addition and product are not commutative. Hence, when we work with infinite ordinals, the base-$b$ decompositon must be written with the "$c_i$" on the left, and $\sum _{i=0}^n{b}^i{c}_i$ means the sum $b^n{c}_n+...c_0$, not $c_0+...b^n{c}_n$.

Lemma 3.1

For all $b\ge 2$ and $n\in \mathbb{N}$ we have $R_{b,\omega }\left(n\right)<R_{b,\omega }\left(n+1\right)$

Proof: by induction on $n$. The result is obvious for $n=0$. Now let $n=\sum _{i=0}^{+\infty }{c}_i{b}^i$ be the base-$b$ decomposition of $n$. Because $b^n>n$, we can actually write $n=\sum _{i=0}^{n-1}{c}_i{b}^i$ and the induction hypothesis can be applied to the exponents. Consider the least $m$ such that $c_m<b-1$. Then $n=\sum _{i=m}^{n-1}{c}_i{b}^i+\sum _{i=0}^{m-1}{c}_i{b}^i$ and $n+1=\sum _{i=m}^{n-1}{c}_i{b}^i+b^m$. Hence you simply have to check that $\sum _{i=0}^{m-1}{\omega }^{R_{b,\omega }\left(i\right)}{c}_i<{\omega }^{R_{b,\omega }\left(m\right)}$, using classical rules of ordinal arithmetic and the induction hypothesis.□

Theorem 3.2

For all $a\in \mathbb{N}$, $\exists n\in \mathbb{N},{g^a}_n=0$

Proof:

For all $n\ge 2$, let $h_n\left(a\right)=R_{n,\omega }\left(g_n\left(a\right)\right)$. Suppose that $\forall n,{g^a}_n\ne 0$, then: $h_{n+1}\left(a\right)=R_{n+1,\omega }\left(g_{n+1}\left(a\right)\right)=R_{n+1,\omega }(R_{n,n+1}\left(g_n\left(a\right)\right)-1)$

By lemma 3.1, $h_{n+1}\left(a\right)<R_{n+1,\omega }\left(R_{n,n+1}\left(g_n\left(a\right)\right)\right)=R_{n,\omega }\left(g_n\left(a\right)\right)=h_n\left(a\right)$

Finally, ${\left(h_n\left(a\right)\right)}_{n\ge 2}$ is a strictly decreasing sequence of ordinals. A contradiction.□

4. References

1. Wikipedia.en - Goodstein's theorem
2. Dehornoy (http://www.math.unicaen.fr/~dehornoy/) : Logique et théorie des ensembles; Notes de cours, FIMFA ENS - Chapitre 2 : Les ordinaux.
3. Miller (http://www.u.arizona.edu/~miller/thesis/) On the Independence of Goodstein's Theorem
4. Wang, Delisle, Freton - Démonstrations et algorithmes : infiniment longs ou impossibles ?