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)
“I let my neighbor know beyond the hill;
And on a day we meet to walk the line
And set the wall between us once again.
We keep the wall between us as we go.”
—Robert Frost (1874–1963)
“There is in him, hidden deep-down, a great instinctive artist, and hence the makings of an aristocrat. In his muddled way, held back by the manacles of his race and time, and his steps made uncertain by a guiding theory which too often eludes his own comprehension, he yet manages to produce works of unquestionable beauty and authority, and to interpret life in a manner that is poignant and illuminating.”
—H.L. (Henry Lewis)
“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)
“In mathematics he was greater
Than Tycho Brahe, or Erra Pater:
For he, by geometric scale,
Could take the size of pots of ale;
Resolve, by sines and tangents straight,
If bread and butter wanted weight;
And wisely tell what hour o’ th’ day
The clock doth strike, by algebra.”
—Samuel Butler (1612–1680)