Paradox without Basic Law V: a problem with Frege’s ontology
1893, 1903). Suppose there are K objects. Then there are 2'> 'concepts. But, by Basic Law V, for every concept there is an object, its value-range, assigned in such a way that distinct concepts have distinct value-ranges. Hence there are at least as many objects as concepts contradiction. If thoughts are objects, something very similar happens. Suppose as before that there are K objects and hence 2'> Kconcepts. Pick an arbitrary object, say Ben Lomond. For each concept, there is a thought (a Ben Lomond thought) that Ben Lomond falls under that concept.1 Clearly, distinct concepts give rise to distinct Ben Lomond thoughts. Hence there are at least as many thoughts, and so at least as many objects, as concepts contradiction. This Cantor-style paradox may seem to throw suspicion on the (fishy) notion of the cardinality of the collection of objects. It is, however, possible to construct a Russell-style version which does not mention cardinality at all.