Suppose we have defined finite sets. In the prior post these are sets with a bijection to a subset of a head segment of the natural numbers.
For finite sets, can we prove trichotomy of cardinality comparisons?
For finite sets,
we define A is less than B in cardinality if there is a 1 to 1 correspondence from A to a proper subset of B.
If there is a one to one correspondence of A to B, we say they have equal cardinality.
If there is a 1 to 1 correspondence from a proper subset of A to B, then B has less cardinality than A.
Can we guarantee in advance that for finite sets A and B we can always make this comparison?
How much machinery do we have to build up first? Can we do this before we do much on head segments?
Can we do it with a very minimal definition of finite sets?
For infinite cardinals, trichotomy is equivalent to the Axiom of Choice.
Axiom of Choice equivalents:
These are basic questions. Where does trichotomy of cardinality for finite sets come from? I.e. what axioms and machinery are needed. Do we have to go far down the path of head segments of natural numbers as in Dedekind or modern treatments to use trichotomy of cardinality of finite sets?
These types of questions are often not raised and if they are, the answers are not organized and are often in obscure places if anywhere. Yet these are basic conceptual issues for understanding counting and natural numbers. They should be explained in a way accessible to grade K teachers. Yet they are not. Nor do math ed college programs even conceive of using such materials if they existed. They would be indifferent at least to start if they existed.
This shows how far the establishment is from supporting the Common Core Standards in Math. Teaching math in grade K is not well supported by the math ed establishment. They are indifferent to building the materials or even their own conceptual understanding of the math behind elementary math.
This comes from inflated self esteem. It is the same as with students who don’t learn because their inflated self esteem tells them they already know it when they don’t. The college math ed establishment has the same issues.