Quotes About Set theory

I see the beauty of God's archetypal infinity reflected in the towers of infinities in set theory

T]he enigma of the continuum arises because language misleads us into applying to it a picture that doesn't fit. Set theory preserves the inappropriate picture of something discontinuous, but makes statements about it that contradict the picture, under the impression that it is breaking with prejudices; whereas what should really have been done is to point out that the picture just doesn't fit…

There is, as we shall see, a real and precise difference between the number zero and the concept of a set that posesses no members - the null, or empty set. Indeed, the second idea, pointless as it sounds, turns out to be by far the most fruitful of the two. From it, all of the rest of mathematics can be created step by step.

When I dealt with set theory, I could never make it be the music that I wanted.

Minimalism now is a reaction to what came before. It's absolutely of its time. Music moved into the set theory thing, and moved out of it.

The ocean—a source of electrical, magnetic, and gravitational impulses—spoke as it were in the language of mathematics; certain sequences of its electrical discharges could be classified by drawing on the most abstract branches of terrestrial analysis and of set theory; they contained homologues of structures known from the area of physics that is concerned with the mutual relationship between energy and matter, finite and infinite magnitude, particles and fields.

The number 0 is the number of terms in a class which has no members, i.e. in the class which is called the null-class.

It remains to define successor. Given any number n, let ? be a class which has n members, and let x be a term which is not a member of ?. Then the class consisting of ? with x added on will have n+1 members.

But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.

Only someone who (like the Intuitionist) denies that the concepts and axioms of classical set theory have any meaning could be satisfied with such a solution, not someone who believes them to describe some well-determined reality. For in reality Cantor's conjecture must be either true or false, and its undecidability from the axioms as known today can only mean that these axioms do not contain a complete description of reality.

In the battle between Kronecker and Cantor, Cantor would ultimately prevail. Cantor's theory would show that Kronecker's precious integers-and even the rational numbers-were nothing at all. They were an infinite zero.

Cantor's discovery that lines, planes, cubes, and polytopes were all equivalent as sets of points goes a long way toward explaining why set theory was such a revolutionary development for math-revolutionary in theory and practice both.

To make a long story short, Cantorian set theory helps unify and clarify math in the sense that all mathematical entities can now be understood as fundamentally the same kind of thing-a set.

But what G. Cantor posits as the defining formal property of an infinite set is that such a set can be put in a 1-1C with at least one of its proper subsets. Which is to say that an infinite set can have the same cardinal number as its proper subset, as in Galileo's infinite set of all positive integers and that set's proper subset of all perfect squares, which latter is itself an infinite set.

In other words, Cantor is able to show that real numbers themselves can serve as the limits of fundamental sequences of reals, meaning his system of definitions is self-enclosed and VIR-proof.

No one shall expel us from the paradise which Cantor has created for us. { Expressing the importance of Georg Cantor 's set theory in the development of mathematics .}

BOTH GÖDEL'S AND COHEN'S arguments proceed by constructing a model of set theory, though I will not explain them in detail.

We met the list comprehension briefly in Chapter 4. Syntactically, its syntax is derived from a construct in set theory notation that applies an operation to each item in a set, but you don't have to know set theory to use this tool.