Language, Teaching and Set theory

0.0.1 The Problem

“sets are sorta like bags but only loosely speaking”
“sets can contain things”

This is the typical introduction to sets given in math classes.
I believe this introduction misleads students into seeing sets as bags or containers that can physically hold elements.
This conceptualization results in the belief that elements could be taken or placed into sets at whim which can inhibit learning proofs.
So without futher ado.
what are sets?

A set specifies a predicate.
An element MUST be in a set if it satisify such predicate.

If we really wanted to use the bag and container analogy, then we would have to believe this bag had some magical vortex that sucked every element that the predicate satisfies inside the bag.(Hard to imagine as a first intro so lets drop this possibility)

Moving on, let’s get to the main issue.
Most teachers would introduce/intialize a set by saying “Here we have a set that contains all even numbers”

The problematic word “Contains” rears it’s head.
We must emphasize that sets are not objects like bags but specifications.

0.0.2 Sets are Specifications

First we need to:
STOP INITIALIZING sets by saying “Let S be the set that contains …”

example:
1. DO NOT: “Let S be the set that contains apple pies that are tasty”
2. DO:“Let S be the set that specifies ALL red apple pies that are tasty”

Now instead of focusing on the set S as it’s own object, I am now focusing on the elements that satisfies the specification which is really what sets were meant for.

0.0.3 Stop using Contains

After introducing the set, we prod the set with elementhood \(\in\) and some element \(x\).
We may ask “is \(x\) in the set?” or assert “\(x\) is in the set”.

“apple pie \(x\) is in set S”
\[ x \in S \]

Isnt “apple pie \(x\) is in set S” and
“set S contains apple pie \(x\)” the same thing?

Yes, semantically overall, they are equivalent but the first statement is closer to the mathematical notation in terms of discrete syntax and semantic ordering (as shown in the chart above).

Again, there is the issue that “contains” implies that sets have some physical manifestation.

Therefore, it is preferable to say “apple pie \(x\) is in set S” over
“set S contains apple pie \(x\)”.

example:
DO NOT: “\(\mathbb{N}\) contains 1,2,3”
DO: “1,2,3 are in \(\mathbb{N}\)”