Will Sladek, a student at Caltech, wrote an excellent introductory paper on incompleteness in 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.

[…] a nice introduction to incompleteness and Goodstein’s theorem, see Will Sladek’s paper here. Possibly related posts: (automatically generated)Partitioning numbersUS concerns of decline in […]

[…] (an undergraduate student of mine at Caltech wrote a nice paper on this a few years ago, “The termite and the tower.“). There are others. A nice one is about a game, Hercules and the […]

Ignas: 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. […]

MR2449474 (2009j:03067) Woodin, W. Hugh. A tt version of the Posner-Robinson theorem. Computational prospects of infinity. Part II. Presented talks, 355–392, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 15, World Sci. Publ., Hackensack, NJ, 2008. The proof is nice, invoking both recursion-theoretic and set-theoretic tools. Hugh uses a Prikry-like f […]

The argument you are looking for is given in Kanamori's book, see Theorem 28.15. For the more nuanced version of the lemma, see section 7D in Moschovakis's descriptive set theory book (particularly 7.D.5-8), or section 3.1 in the Koellner-Woodin chapter of the Handbook.

This problem is very much open. Cheng Yong calls Harrington's $\star$ the assumption that there is a real $x$ such that all $x$-admissible ordinals are $L$-cardinals. From the work of Yong we know that Second- and even Third-order arithmetic do not suffice to prove that Harrington's $\star$ implies the existence of $0^\sharp$. Whether this was poss […]

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

Through this question, I was made aware of Ádám Besenyei. Peano's unnoticed proof of Borel's theorem, Amer. Math. Monthly 121 (2014), no. 1, 69–72. In this short note, Besenyei presents a proof due to Peano of the theorem usually attributed to Borel. Peano's result first appeared in Angelo Genocchi , Giuseppe Peano. Calculo differenziale e pri […]

${}$ Hi Ramiro! I looked at very similar questions in my undergraduate thesis (see here). Your question is related to a conjecture of Tarski, in A. Tarski, Quelques théorèmes sur les alephs, Fund. Math. 7 (1925), 1-14. In that paper, he proves that $$ \prod_{\alpha

There is a fairly extense literature detailing uses of determinacy in a variety of situations. A good place to start is Akihiro Kanamori's The higher infinite. The last part of the book is devoted to determinacy. Eventually, Aki concentrates on the question of the consistency of determinacy from large cardinals, but before getting there, he provides man […]

I. Some of the answers reveal a confusion, so let me start with the definition. If $I$ is an interval, and $f:I\to\mathbb R$, we say that $f$ has the intermediate value property iff whenever $a

My favorite family of examples come from the partition calculus. The original proof of the Baumgartner-Hajnal theorem $\omega_1\to (\alpha)^2_n$ for all finite $ n $ and all countable $\alpha $ appealed to the absoluteness of well-foundedness. The homogeneous set was found in an extension where $\mathsf{MA} $ holds, and that means that a certain ground-model […]

[…] a nice introduction to incompleteness and Goodstein’s theorem, see Will Sladek’s paper here. Possibly related posts: (automatically generated)Partitioning numbersUS concerns of decline in […]

[…] (an undergraduate student of mine at Caltech wrote a nice paper on this a few years ago, “The termite and the tower.“). There are others. A nice one is about a game, Hercules and the […]