Exploring Ontology

It's all about the deep questions.

A Problem for Platonism

A familiar problem for the philosophy of mathematics is the ontology of mathematics. One view, platonism, maintains that the things that mathematicians talk about, ultimately sets, actually exist as mind-independent abstract objects. Here’s one problem for the view that I’ve come up with, although being no expert in the field, it could have some conceptual mistake.

1. If platonism is true, there is a fact of the matter as to what cardinalities sets can be.

2. If there is a fact of the matter as to what cardinalities sets can be, we can take the union of one set for every cardinality that there is.

3. The resulting set A must have the same cardinality as one of its subsets A’, since by hypothesis there exists subsets of A with every cardinality.

4. The resulting set A must have at least the same cardinality as the powerset of A’, since the powerset of A’ is a distinct cardinality and there must be a subset of A with the cardinality of the powerset of A’.

5. Therefore, the resulting set A must have a cardinality both equal to and not equal to (greater than) A’.

6. Therefore, platonism is false.

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