Absolute Basics

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Section 1.1 Absolute Basics

There are some things we will not expand on at all.

We will take most normal mathematical and logical terminology from Math 212 ¹ for granted.

We assume that sets exist. We will take the sets of counting (\(\N\)), integer (\(\Z\)), \(rational\) (\(\Q\)), and real (\(\R\)) numbers as known. In this course \(\N\) starts with 1, not 0. We will denote the irrational numbers by \(\Q^c\text{,}\) a notation for the complement of the rationals with respect to the set \(\R\text{.}\)

In these sets we will assume the most basic properties of addition, subtraction, and multiplication, such as commutativity and distributivity. Likewise, we will later assume without proof basic facts about ordering numbers. At this time we explicitly assume that \(0 \lt 1\) and that there are no integers between them. You will be able to use these facts to prove that is true for other consecutive pairs of integers. (The fact \(0 \lt 1\) does require proof! Typically one uses a Well-Ordering ² type axiom.)

Although there is a lot of fun to be had with infinity, I do not foresee doing much with it other than using generic and specific infinite sets.

The rest of this preliminary chapter is intended to remind us of things for which I do think review might be useful.

prof.mkjanssen.org/ds/notes

ringswithinquiry.org/SubSec-WellOrdering.html#axiom_wellordering

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