a subset of Using proof by contradiction, assume that a subset of is equinumerous with . Then there is a one-to-one correspondence from to Define from to if then if then Since maps onto maps onto contradicting the theorem above stating that a function from to is not onto. Therefore, is not equinumerous with a subset of Besides the axioms of infinity and power set, the axioms of separation, extensionality, and pairing were used in the modern argument. For example, the axiom of separation was used to define the diagonal subset the axiom of extensionality was used to prove and the axiom of pairing was used in the definition of the subset Reception of the argument Initially, Cantor's theory was controversial among mathematicians and (later) philosophers. As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there." Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics. Logician has commented on the energy devoted to refuting this "harmless little argument" (i.e. Cantor's diagonal argument) asking, "what had it done to anyone to make them angry with it?" Mathematician Solomon Feferman has referred to Cantor's theories as “simply not relevant to everyday mathematics.” Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world; for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence. "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already". Carl Friedrich Gauss's views on the subject can be paraphrased as: "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics." In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets. Cantor's ideas ultimately were largely accepted, strongly supported by David Hilbert, amongst others. Hilbert predicted: "No one will drive us from the paradise which Cantor created for us." To which Wittgenstein replied