Math From Scratch 39: Losing the Axiom of Inequality

The latest chapter of Math From Scratch is ready here, and details the exact justifications for removing the Axiom of Inequality, also known as the well-ordering principle, and then introduce the new axiom that replaces it.

4 replies on “Math From Scratch 39: Losing the Axiom of Inequality”

1. pythor says:

The axioms section is very confusing to me. I can understand axiom 3, but several others are confusing because of missing commas between the conditions and the results (my words, hopefully you can understand), and/or some separator between the name/description of the axiom and its definition. Others have a comma separator used both to separate variable names and to separate the condition from the result. I’m guessing this is a standard notation, but having some other separator to differentiate would make things easier for me.

Maybe an example is in order…

Axiom 1 is missing a comma between condition and result, so I naturally interpret the first comma for that purpose, which then makes it read unintelligibly: “(For All x) (y is an element of Real x+y is an element of Real)” is my first reading. Axiom 2 is essentially the same.

Axiom 3 is better, and in fact, I had to read 3 before I could figure out what 1 and 2 meant at all. 3 to 6 are actually all fairly readable, but again, changing one separator to something other than comma (semicolon would work for me) would have made them easier.

7 to 11 are each missing a separator, and 12-13 are pretty much written in English, so no problem there.

• Standard mathematical notation doesn’t use the separators you are asking for, and part of my goal is to use the standard stuff. Commas, semicolons and colons all have specific meanings which do not apply in the cases you are suggesting. This is why they all had full English descriptions when first introduced. It is only in the review lists that these are summarized using the standard notation only.

• pythor says:

I kinda thought that was the case, but then I’m confused why some of the list had them while others didn’t.

• If some have them, those are my errors. I wrote most of this while sick with a fever, and may not effectively have overruled the English grammar typing muscles with math-oriented typing muscles.