Friday, January 26, 2024

Counting with plural quantification

I’ve been playing with the question of what if anything we can say with plural quantification that we can’t say with, say, sets and classes.

Here’s an example. Plural quantification may let us make sense of cardinality comparisons that go further than standard methods. For instance, if our mathematical ontology consists only of sets, we can still define cardinality comparisons for pluralities of sets:

  1. Suppose the xx and the yy are pluralities of sets. Then |xx| ≤ |yy| iff there are zz that are an injective function from the xx to the yy.

What is an injective function from the xx to the yy? It is a plurality, the zz, such that each of the zz is an ordered pair of classes, and such that for any a among the xx there is unique b among the yy such that (a,b) is among the zz and for any b among the yy there is at most one a among the xx such that [a,b] is among the zz.

This lets us say stuff like:

  1. There are more sets than members of any set.

Or if our mathematical ontology includes sets and classes, we can compare the cardinalities of pluralities of classes using (1), as long as we can define an ordered pair of classes—which we can, e.g., by identifying the ordered pair of a and b with the class of all ordered pairs (i,x) where i = 0 and x ∈ a or where i = 1 and x ∈ b.

This would then let us say (and prove using a variant of Cantor’s diagonal argument, assuming Comprehension for pluralities):

  1. There are more classes than sets.

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