Abstract:
According to Frege logicism is the thesis that the truths of arithmetic can be transformed into logical truths by means of definitions.  Frege is guided by a criterion for the identity of numbers that he calls Hume's Principle:

'The number of Fs = the number of Gs iff there are as many Fs as Gs'.

Frege insists that this principle is not itself a definition of numbers.  Rather, it serves as a test; any proposed identification of the numbers must satisfy it.  Frege then proceeds to identify certain sets as the numbers. Unfortunately, the resulting theory is inconsistent. 

Following Frege's general strategy I take Hume's Principle as the starting point.  Instead of identifying numbers with sets I take numbers to be the properties expressed by numerical quantifiers.  These properties, however, need to be treated as objects.  Numbers as objects are introduced by definitions.  Hume's Principle is satisfied and the basic principles of arithmetic are implied.