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
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 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
- Answer by Andrés E. Caicedo for Meaning of "absolute zero" in set theory January 11, 2019
  Unfortunately, Maddy is being imprecise in her use of terminology and the surrounding explanation. The mention of Borel in page 496 is a good hint that the notion she is discussing is that of being strong measure zero, as suggested in the comments. A set of reals is (or has) measure zero if and only if for any $\epsilon>0$ it can be covered by countably m […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Constant functions in set-theory January 8, 2019
  Note that $\alpha\mapsto\|c_\alpha^\lambda\|_S$ is strictly increasing (trivially): After all, $$\{\delta\in S\mid c_\beta^\lambda(\delta)\ge c_\alpha^\lambda(\delta)\}=\{\delta\in S\mid\beta\ge \alpha\}=\emptyset$$ if $\beta
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Clarification for the least upper bound property January 6, 2019
  The property is defined relative to the set $S$, not to any potential superset of $S$. The example you give has $S=(a,b)$ and the subset you chose as $E$, namely $((a+b)/2,b)$ is not bounded above in $S$ (it is bounded above in $\mathbb R$, but this doesn't matter). In fact, you can easily check that $S=(a,b)$ has the least upper bound property, the poi […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for If two functions are equal almost everywhere, the first is continuous a.e., is the second? December 12, 2018
  The claim is not true. An easy counterexample is obtained by letting $f$ be identically 0 and $g$ be the characteristic function of the rationals. We have $f=g$ a.e., $f$ is everywhere continuous and $g$ is nowhere continuous. (Clearly, the restriction of $g$ to the irrationals is continuous (being constant), but this is not enough to ensure that $g$ is cont […]
  Andrés E. Caicedo
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