Search Results

Here are a few areas of overlap for those research topics. Computable model theory is a nice overlap of computability theory and algebra, since one is looking at the nature of computably effective pr …

Presburger arithmetic admits elimination of quantifiers, if one expands the language to include truncated minus and the unary relations for divisibility-by-2, divisibility-by-3 and so on, which are de …

Here is an easy way to see it. Let $A$ assert that if PA is inconsistent, the smallest $k$ for which $\Sigma_k$ induction is inconsistent is a multiple of $3$. Let $B$ make the similar assertion tha …

Allow me to reinterpret your question as the inquiry How can abstract infinitary constructions inform us about the finite? To my mind, this is the troubling or at least surprising possibility at the …

To my way of thinking, the two notions of undecidability are closely related, and the associated undecidability phenomenon and independence phenomenon, which are both pervasive in mathematics, are dee …

You are right to view the Goldbach conjecture as having a particularly simple logical form. Such statements of the form "for every $n$, property $P(n)$ holds", where $P$ is a particularly simple state …

I believe that there are many instances of this phenomenon in set theory, where an elaborate theory is developed over a period of years by many people, even though the theory is not viewed ultimately …

I have heard that Jack Silver's discovery of zero sharp ($0^\#$) was part of his attempt to show measurable cardinals inconsistent. Instead of finding the long-sought-after contradiction, however, he …

The integers can indeed be defined in the rational field, but not in the real field. $\newcommand\Q{\mathbb{Q}}\newcommand\Z{\mathbb{Z}}\newcommand\R{\mathbb{R}}$ The question can be made precise by i …

The two notions are incomparable. To see that the first notion does not imply the second, let's construct a set $S$ with asymptotic density $0$, but with infinite chromatic number. We place infinitely …

Consider the arithmetic progression game, a two-player game of perfect information, in which the players take turns playing natural numbers, or finite sets of natural numbers, all distinct, and the fi …

To start things off, here is a simple observation: the set $S$ is contained in the rational interval $\mathbb{Q}\cap[\frac 12,1]$, the rational numbers $\frac ab$ where $0<a\leq b\leq 2a$. The reaso …

Picking up on the phrase "arithmetic information" in your question, let me give a brief answer coming from logic, although I recognize that this is likely not the answer for which you are looking. L …

Nice question, Erin. Here is one quick easy thing to say. If $\pi$ and $e$ disagree in infinitely many digits, then there are continuum many choices of the particular subset of those digits to swap, …

Goedel did indeed use the Chinese remainder theorem in his proof of the Incompleteness theorem. This was used in what you describe as the `boring' part of the proof, the arithmetization of syntax. Con …