In mathematical logic, the theory of infinite sets was first developed by Georg Cantor.
Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers.
Because Leopold Kronecker did not accept these constructions, Cantor was motivated to develop a new proof.
[4] Each of these functions corresponds to a subset of A, so his generalized argument implies the theorem: The power set P(A) has greater cardinality than A.
By presenting a modern argument, it is possible to see which assumptions of axiomatic set theory are used.
Because Cantor implicitly assumed that cardinalities are linearly ordered, this case cannot occur.
[8] After using his 1878 definition, Cantor stated that in an 1883 article he proved that cardinalities are well-ordered, which implies they are linearly ordered.
[12] In 1895, Cantor also gave a new definition of "greater than" that correctly defines this concept without the aid of his well-ordering principle.
"[14] Mathematician Solomon Feferman has referred to Cantor's theories as “simply not relevant to everyday mathematics.”[15] 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.
What we call infinite is only the endless possibility of creating new objects no matter how many exist already".
[17] 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.
Cantor's ideas ultimately were largely accepted, strongly supported by David Hilbert, amongst others.
"[20] The rejection of Cantor's infinitary ideas influenced the development of schools of mathematics such as constructivism and intuitionism.
[citation needed] Wittgenstein did not object to mathematical formalism wholesale, but had a finitist view on what Cantor's proof meant.
Mayberry has noted that "the set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees.
Hermann Weyl wrote: ... classical logic was abstracted from the mathematics of finite sets and their subsets ….