Objections To Set Theory As A Foundation For Mathematics
From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. Ludwig Wittgenstein questioned the way Zermelo–Fraenkel set theory handled infinities. Wittgenstein's views about the foundations of mathematics were later criticised by Georg Kreisel and Paul Bernays, and investigated by Crispin Wright, among others.
Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory.
Read more about this topic: Set Theory
Famous quotes containing the words objections, set, theory, foundation and/or mathematics:
“Miss Western: Tell me, child, what objections can you have to the young gentleman?
Sophie: A very solid objection, in my opinion. I hate him.
Miss Western: Well, I have known many couples who have entirely disliked each other, lead very comfortable, genteel lives.”
—John Osborne (1929–1994)
“It is not unkind to say, from the standpoint of scenery alone, that if many, and indeed most, of our American national parks were to be set down on the continent of Europe thousands of Americans would journey all the way across the ocean in order to see their beauties.”
—Franklin D. Roosevelt (1882–1945)
“There could be no fairer destiny for any physical theory than that it should point the way to a more comprehensive theory in which it lives on as a limiting case.”
—Albert Einstein (1879–1955)
“The foundation of humility is truth. The humble man sees himself as he is. If his depreciation of himself were untrue,... it would not be praiseworthy, and would be a form of hypocrisy, which is one of the evils of Pride. The man who is falsely humble, we know from our own experience, is one who is falsely proud.”
—Henry Fairlie (1924–1990)
“... though mathematics may teach a man how to build a bridge, it is what the Scotch Universities call the humanities, that teach him to be civil and sweet-tempered.”
—Amelia E. Barr (1831–1919)