6
votes
1answer
117 views

Defining $\Bbb{Q}$ without the axiom of infinity

(TL;DR version: I want a meaningful definition of $\Bbb{Q}$ without $\sf{Inf}$.) In the "conventional" construction of the rationals, we define $\Bbb{Q}$ as follows: $\omega$ is the first limit ...
2
votes
2answers
226 views

Constructing the reals from the rationals

Dr. H. Jerome Keisler, in his book Elementary Calculus: An Infinitesimal Approach, states on page 24: Just as the real numbers can be constructed from the rational numbers, the hyperreal numbers ...
1
vote
1answer
57 views

How many sieves are there on a given rational number $q$?

Consider the poset category $\mathbb{Q}^{op}$, i.e. where $p \rightarrow q$ iff $p \geq q$. Take any $q \in \mathbb{Q}$. Then how many sieves are there for $q$ in $\mathbb{Q}^{op}$? Supposedly, the ...