A cardinal fraction might be defined as a pair of bijections.

So (2,3) would be a bijection with a set of cardinality 2 and a bijection with a set of cardinality 3.

The ratio idea could also emerge by a way to generate the pair of bijections for (4,6) from (2,3).

==Following seems flawed was earlier post.

We can use cardinal fractions to do comparisons among sets. Once we have the ordinal fractions, we can then use them in 1 to 1 comparisons of sets. This gives us a way to talk about cardinal fractions.

Of course, we can always think of this as coming back to atomic numbers by labeling or rearranging. This is true. The machinery of fractions however implemented is a way to reuse the machinery of whole numbers to represent a fractional relationship. So every fraction method comes back to being about whole numbers and relating whole numbers.

The Peano Axioms tell us the properties of whole numbers, all we know and all we need to know. The Peano Bin Construction we can label the parallel Peano Axiom lines of varying bin sizes, where we look in the bins, and count by bin members along a bin line.

Bin lines generalize number lines. A set of bin lines, generalizes a single number line. The construction though relates back to itself. We can think of the bottom line of wholes as being mapped to each bin line so that a bin of 2 or bin of 3 simultaneously match to the bin of 1 on the bottom line.

We can then label the numbers on each bin line, and then put them on a single fraction line. If we think of ordinal comparisons by set membership along that line we can call these ordinal fractions. If we think of bijections of these sets, we can think of cardinality.

(0,1)(1,6)(1,3)(1,2)(2,3)(5,6)(1,1)

Here we have marked the numbers along each bin line with their bin size and then put them together to form a fraction line or fraction set. These vary with the bin lines we combine.

We can extend this set to go up to (2,1). We can then take a bijection between the set from 0 to 1 and from 1 to 2. This gives us a cardinality comparison of these numbers.

Ordinality is a comparison based on order and can be done constructively with set membership. Cardinality is a comparison between sets using 1 to 1 correspondences. So we can use sets of fractions to do cardinal comparisons and then can call these cardinal fractions. What is presented here is just the start of the idea really.

=

The problem with a single bijection to an ordinal set is that for a finite ordinal set, we simply get a finite number as its cardinality, not any sort of fraction.

A relation between whole cardinals is what we need to seek. This can be represented by pairs of cardinal bijections is a better way to go it seems. An equivalence class of such bijections is an infinite object which also gives us more room to play with. The infinite nature of a rational as an equivalence class may help us with defining rational cardinals.