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. […]

$\mathrm{HOD}$ always contains $L$ because any inner model contains $L$, by absoluteness. How easy it is to exhibit a difference really depends on your background. For instance, $0^\sharp$, if it exists, is a real that always belongs to $\mathrm{HOD}$ but is not in $L$. If you are not too comfortable with large cardinals, but know forcing, you may enjoy prov […]

The classical definition of $0^\sharp$ is as (the set of Gödel numbers of) a theory, namely, the unique Ehrenfeucht-Mostowski blueprint satisfying certain properties (coding indiscernibility). This is a perfectly good definition formalizable in $\mathsf{ZFC}$, but $\mathsf{ZFC}$ or even mild extensions of $\mathsf{ZFC}$ are not enough to prove that there are […]

This is the descriptor operator. $(\iota x)\varphi x $ is the unique $x $ with the property specified by $\varphi $ (should it be the case that, indeed, there is precisely one such $x $). The Wikipedia entry on Principia has a very decent explanation of their notation.

The result is proved starting on page 90 ($\S4$ of Chapter III) of MR0953784 (90a:42008). Alexander S. Kechris and Alain Louveau. Descriptive set theory and the structure of sets of uniqueness. London Mathematical Society Lecture Note Series, 128. Cambridge University Press, Cambridge, 1987. viii+367 pp. ISBN: 0-521-35811-6. (The link is to the review at Mat […]

We are given an uncountable set $A$ of reals, and want to define $f:A\to A$ regressive with the property that $f$ is not constant on any uncountable set. (We need some convention on how to define $f(a)$ if $a$ is the minimum of $A$; this is not really important and I'll leave it up to you what to do here.) Let $\kappa=|A|$. We proceed by induction on $\ […]

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@FedericoArdila "In Flatland, Abbott intends to point out the discriminatory role of women... Abbott believed that the conventional roles 2 days ago

[…] 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 […]