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 | Incompleteness | Tagged: Goodstein's function, Undecidability and incompleteness | Permalink
Posted by andrescaicedo

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Mathoverflow
- Answer by Andrés E. Caicedo for Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$ January 16, 2019
  Perhaps the following may clarify the comments: for any ordinal $\delta$, there is a Boolean-valued extension of the universe of sets where $2^{\aleph_0}>\aleph_\delta$ holds. If you rather talk of models than Boolean-valued extensions, what this says is that we can force while preserving all ordinals, and in fact all initial ordinals, and make the contin […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Arguments against large cardinals October 29, 2010
  I do not know of any active set theorists who think large cardinals are inconsistent. At least, within the realm of cardinals we have seriously studied. [Reinhardt suggested an ultimate axiom of the form "there is a non-trivial elementary embedding $j:V\to V$". Though some serious set theorists found it of possible interest immediately following it […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Nonessential use of large cardinals December 10, 2010
  There is a fantastic (and not too well-known) result of Shelah stating that $L({\mathcal P}(\lambda))$ is a model of choice whenever $\lambda$ is a singular strong limit of uncountable cofinality. This is a consequence of a more general theorem that can be found in 4.6/6.7 of "Set Theory without choice: not everything on cofinality is possible", Ar […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for What definitions were crucial to further understanding? December 3, 2018
  In set theory, definitely the notion of a Woodin cardinal. First, it is not an entirely straightforward notion to guess. Significant large cardinals were up to that point defined as critical points of certain elementary embeddings. This is not the case here: Woodin cardinals need not be measurable. If $\kappa$ is Woodin, then $V_\kappa$ is a model of set the […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Examples of Mathematical Papers that Contain a Kind of Research Report November 21, 2018
  The first example that came to mind was MR0270881 (42 #5764) van der Waerden, B. L. How the proof of Baudet's conjecture was found. 1971 Studies in Pure Mathematics (Presented to Richard Rado) pp. 251–260 Academic Press, London. There, van der Waerden describes some of the history as well as his proof of his well-known theorem. Another example: MR224589 […]
  Andrés E. Caicedo
Math.StackExchange
- Answer by Andrés E. Caicedo for Can Tarski's circle squaring problem be solved with measurable sets and/or without the Axiom of Choice? November 14, 2013
  Update: The problem has been solved. See below for the original answer, with the state of the art in 2013. In 2017, Ł. Grabowski, A. Máthé and O. Pikhurko showed in Measurable circle squaring, Ann. of Math. (2) 185 (2017), no. 2, 671–710, MR3612006, that Tarski's problem can be solved using pieces that are both Lebesgue and Baire measurable. Their proof […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Particular case of Ramsey's theorem $\mu \rightarrow (\mu)_{\kappa}^{2}$ March 10, 2019
  The condition $\kappa\to(\kappa)^2_2$ is quite significant: Any such $\kappa$ larger than $\aleph_0$ is a weakly compact cardinal. These are strongly inaccessible cardinals that are a limit of strongly inaccessible cardinals. In fact, they are a Mahlo limit of Mahlo cardinals, and much more. Weak compactness is a natural rung in the large cardinal ladder. As […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Can there be an infinite $\kappa$ with $\kappa\to(\omega)^\omega_2$ in $\mathsf{ZF}$? February 10, 2019
  Yes, it is consistent to have such cardinals. In fact, it is consistent relative to an inaccessible cardinal that $\omega\to(\omega)^\omega_2$. This is a famous result of Mathias, in MR0491197 (58 #10462). Mathias, A. R. D. Happy families. Ann. Math. Logic 12 (1977), no. 1, 59–111. (It is still open whether the inaccessible cardinal is required.) The result […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Unclear manipulation with forcing conditions January 25, 2019
  The inductive definition of forcing (by complexity of formulas) gives in particular that $p$ forces $\lnot\psi$ if and only if no extension of $p$ forces $\psi$. That is exactly what is being claimed. As for why this general fact about forcing of negations holds, it is either immediate from the fact that a statement holds in a generic extension if and only i […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for What is the $\text{♢-Axiom}$? (now known to be the 'Diamond Principle / Axiom') January 14, 2019
  The principle $\lozenge$ (diamond) is in a sense the right set-theoretic version of the continuum hypothesis, as it presents it instead as a reflection principle. Formally, it asserts that there is a diamond sequence, that is, a sequence $(A_\alpha:\alpha
  Andrés E. Caicedo
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