  Wittgenstein argued that to say that two things are the same, x = y, is nonsense, while x = x is empty and showed how to get rid of identity with an ‘exclusive’ reading of the variables (Hintikka 1956), (Wehmeier 2008), (Wehmeier 2012), with the exception of some propositions such as the axiom of infinity that cannot be so read, and are thus to be rejected (5.535). To save the latter (and the axiom of choice, which was then called the ‘Multiplicative Axiom’), Ramsey introduced ‘functions in extension’ (including arbitrary pairings of arguments with values, and indicated here with the subscript \(e\)), replacing the above with: