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 […]

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Not necessarily. That $\mathfrak m$ is consistently singular is proved in MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first fails. J. Symbolic Logic 53(2), (1988), 429–433. There, Ken shows that $\mathfrak{m}$ can be singular of cofinality $\omega_1$. (Both links above are behind paywalls.)

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. […]

The ordinal $\omega_1$ can (consistently) be a countable union of countable ordinals, i.e., the supremum of a countable set of countable ordinals. This consistency result was one of the first found with forcing. It was announced in S. Feferman-A. Levy, "Independence results in set theory by Cohen's method II", Notices Amer Math Soc., 10, (1963 […]

Supercompactness is much stronger than measurability: Supercompactness of $\kappa$ states that for any $\mu$, we can find embeddings $j:V\to M$ of the universe of sets with $M$ transitive, critical point $\kappa$ and $M$ closed under $\mu$ sequences. This implies in particular that for any set $x\in V$, we can find one such embedding $j$ with $x\in M$ (that […]

Since successor cardinals are regular, if $\kappa^+>\lambda$ then any function from $\lambda$ to $\kappa^+$ is bounded, thus ${}^\lambda\kappa^+=\bigcup_{\alpha\tau$, then $\kappa>\aleph_2$ since ${\aleph_2}^{\aleph_2}\le{\aleph_3}^{\aleph_3}=2^{\aleph_3}$.

(There is a huge amount of leeway, that can be used for instance to arrange that the permutation codes some additional information.) Here is an example. Given $A\subseteq \mathbb N$ define $\pi\in S$ by: If $n\in A$, $\pi(4n+i)=4n+i+1$ for $i

There is a general argument without choice: Suppose ${\mathfrak m}+{\mathfrak m}={\mathfrak m}$, and ${\mathfrak m}+{\mathfrak n}=2^{\mathfrak m}$. Then ${\mathfrak n}=2^{\mathfrak m}.\,$ This gives the result. The argument is part of a nice result of Specker showing that if CH holds for both a cardinal ${\mathfrak m}$ and its power set $2^{\mathfrak m}$, th […]

[…] 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 […]