- This is our last section from Chapter 1.
- Keep up with the homework! Your second assignment is due today
at midnight. Email with questions, or post to the discussion.
Please save your work as a single pdf (lots easier for me to grade), and upload that to Canvas.
- You have a new assignment, which will be due a week from today.
- I'm currently grading the first assignment, and the problems
11(de), 17(bc), 41(ac), and 48 have been randomly selected for
grading.
Just a reminder that I won't be grading every problem, but a selection. Over the course of the semester and the many assignments, this random process will represent a good sample of your work.
Sometimes you're the windshield, sometimes you're the bug -- but statistics assures that on average, you're some windshield and some bug...:) You'd like to be more windshield than bug, however. So do your best on your homework, and, as I say, keep up.
I will be dropping several of your lowest scores, however. That's some protection, too.
- I have a comment in the link below about the choice of a
domain. Our author does something that I don't care for, and that's to
specify a domain as the "whole world". What does this mean? Does it
include our thoughts? Does she mean everything in the universe? I like
to be a little more specific in domains -- e.g. all living things on
Earth, or the like.
A domain is a set. We'll soon be discussing sets in more detail.
There are certain ways that one can define sets, consistently, and there are things that one can't do. Can you define a set containing all sets, for example? Can it therefore contain itself? This was an issue with which the famous logician, philosopher, and mathematician Bertrand Russell wrestled....
- What does it mean when one says that "The domain is the whole world."? (e.g. p. 51, #13).
On a set containing all sets: Russell's paradox.
"Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.
"Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox has prompted much work in logic, set theory and the philosophy and foundations of mathematics."