The ideals in which the whole is prior to the part, are numbers, space, and time. As regards numbers, it is evident that unity, and even the other integers are prior to fractions(pl112)

1 is the class of all classes which are not null and are such that, if x and y belong to the class, then x and y are identical(pm128)

Collections do not presuppose numbers, since they result simply from the terms together with and(pm134)

Euclid, as is evident from his definitions of ratio and proportion, and indeed his whole procedure, was not persuaded of the applicability of numbers to spatial magnitudes(pm157)

Zero is understood as the negation of magnitude(pm168)

The natural numbers have a beginning but no end. The positive and negative integers together have neither(pm201)

What Dedekind presents to us is not the numbers, but any progression, what he says is true of all progressions alike, and his demonstrations nowhere- not even where he comes to cardinals- involve any property distinguishing numbers from other progressions(pm249)

The only solution I(Bertrand Russell) can suggest is, to accept the conclusion that there is no greatest number and doctrine of types, and to deny that there are any true propositions concerning all objects or all propositions. Yet the latter, at least, seems plainly false, since all propositions are at any rate true or false, even if they had no other common properties. In this unsatisfactory state, I reluctantly leave the problem to the ingenuity of the reader(pm368)

According to Gottlob Frege we can define a natural number as anything that belongs to every class y such that y contains 0 and contains x + 1 whenever it contains x(flpov98)

The construction illustrated in the definition of ancestor was introduced by Gottlob Frege in 1879 for application to number. It was rediscovered independently a few years later by Peirce, and again by Dedekind, who propounded it in 1887 under the name of the method of chains . To be a number is to belong to every class to which 0 belongs and the successor of each member belongs. This definition supports mathematical induction, a proof procedure central to number theory(mol294)

The first of the numbers, 0, is then (A]: the class whose sole member is the empty class. 1 is then definable as the successor of O, and 2 as the successor of l, and so on, once we define successor. How to do this is evident from (1): the successor of z, call it Sz, is the class of all those classes which, when deprived of a member, become members of z. An elegant alternative due to von Neumann, and more widely followed nowadays, takes 0 simply as A and Sz as z v {z}. Each number thus becomes the class of all earlier ones. Addition of numbers is easily defined in view of this circumstance: a class has y + z members if and only if it can be broken into two parts having y and z members. Multiplication is definable in view of a similar circumstance: a class has y * z members if it can be broken into y parts having z members each(mol296)

Numbers are known only by their laws, the laws of arithmetic, so that any constructs obeying those laws- certain sets, for instance- are eligible in turn as explications of number(or44)

A real number is explained, substantially, as any class of ratios that (1) does not contain all the ratios, but (2) contains any given ratio if and only if it contains also a higher(slp34)

John von Neumann's version of numbers is looked upon as more natural than Zermelo's because of its closer relation to counting(stl82)

A class of real numbers may be at once bounded and Infinite(stl120)

Reals are on a par with ratios, they are not classes of ratios but unions of ratios, they are classes of the things that ratios are classes of(stl129)

Georg Cantor established that under his criterion there are more real numbers than natural numbers. One is perhaps not surprised that N < Real, since there are infinitely many reals between each integer and the next. But this is a bad reason for not being surprised; for the same reason would wrongly lead us to suppose that N < Rat(stl200)

The fundamental use of natural numbers is in measuring classes(tt15)

When we feel the need of ratios and irrationals, we can simply reach for appropriate subclasses of one of the progressions of classes. We need never talk of numbers, though in practice it is convenient to carry over the numerical jargon(tt16)

The reason for admitting numbers as objects is precisely their efficacy in organizing and expediting the sciences. The reason for admitting classes is much the same. The access of power that comes with classes…(wo237)

The philosophical question 'What is a number?' is on a par with the corresponding question about ordered pairs(wo262)

Pythagoras thought of numbers as shapes and came up with the idea of squares and cubes of numbers, terms that stay with us today(gp66)