July | 2007 | A kind of library

Goodstein sequences

July 27, 2007

Will Sladek, a student at Caltech, wrote an excellent introductory paper on incompleteness in $\mathsf{PA}$ , The termite and the tower. While Will was working on his paper, I wrote a short note, Goodstein’s function, on how to compute Goodstein’s function. Please let me know of any comments of corrections to either article.

Update: My paper has been published.

Goodstein’s function. Revista Colombiana de Matemáticas 41 (2)(2007), 381–391. MR2585906 (2011c:03139). Zbl 1156.03053.

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2 Comments | 116b: Computability and incompleteness | Tagged: Goodstein's function, Undecidability and incompleteness | Permalink
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Mathoverflow
- Answer by Andrés E. Caicedo for Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$ May 10, 2018
  The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Does constructing non-measurable sets require the axiom of choice? October 14, 2010
  As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]
  Andrés E. Caicedo
- Relative Densities September 12, 2013
  Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Solutions to the Continuum Hypothesis May 18, 2010
  (1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Are there any non-linear solutions of Cauchy's equation ($f(x+y)=f(x)+f(y)$) without assuming the Axiom of Choice? March 6, 2011
  It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]
  Andrés E. Caicedo
Math.StackExchange
- Answer by Andrés E. Caicedo for Collection of measurable sets closed under countable unions implies existence of a set with maximal measure July 12, 2018
  Let $s$ be the supremum of the $\mu$-measures of members of $\mathcal G$. By definition of supremum, for each $n$, there is $G_n\in\mathcal G$ with $\mu(G_n)>s-1/n$. Letting $G=\bigcup_n G_n$, then $G\in \mathcal G$ since $\mathcal G$ is closed under countable unions, and $\mu(G)=s$, since it is at least $\sup_n\mu(G_n)$ but it is at most $s$ (by definiti […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Set theory ordinals problem Prove that if $b$ is a limit ordinal and $a June 27, 2018
  The result you are trying to prove is false. For example, if $a=\omega+1$ and $b=\omega+\omega$, then $a+b=\omega\cdot 3>b$. Here is what is true: first, the key result you should establish (by induction) is that An ordinal $\alpha>0$ has the property that for all $\beta
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Are there any new axioms being developed? June 5, 2018
  Very briefly: Yes, there are several programs being developed that can be understood as pursuing new axioms for set theory. For the question itself of whether pursuing new axioms is a reasonably line of inquiry, see the following (in particular, the paper by John Steel): MR1814122 (2002a:03007). Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Are uncountable dense subsets of $\mathbb{R}$ with the same cardinality homeomorphic? June 2, 2018
  This is a very interesting question and the subject of current research in set theory. There are, however, some caveats. Say that a set of reals is $\aleph_1$-dense if and only if it meets each interval in exactly $\aleph_1$-many points. It is easy to see that such sets exist, have size $\aleph_1$, and in fact, if $A$ is $\aleph_1$-dense, then between any tw […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Does every cover of a set have a minimal subcover? June 2, 2018
  Consider $S=\aleph_\omega $ and $C=\{\aleph_n:n
  Andrés E. Caicedo
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