9603 | An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR] |
15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
9886 | Cardinals say how many, and reals give measurements compared to a unit quantity [Frege] |
14438 | New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell] |
18254 | Russell's approach had to treat real 5/8 as different from rational 5/8 [Russell, by Dummett] |
14144 | Ordinals result from likeness among relations, as cardinals from similarity among classes [Russell] |
9896 | A prime number is one which is measured by a unit alone [Dummett] |
15273 | Points can be 'dense' by unending division, but must meet a tougher criterion to be 'continuous' [Harré/Madden] |
12333 | Each type of number has its own characteristic procedure of introduction [Badiou] |
12322 | Must we accept numbers as existing when they no longer consist of units? [Badiou] |
18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
3330 | Negatives, rationals, irrationals and imaginaries are all postulated to solve baffling equations [Benardete,JA] |
3337 | Natural numbers are seen in terms of either their ordinality (Peano), or cardinality (set theory) [Benardete,JA] |
10573 | Dedekind cuts lead to the bizarre idea that there are many different number 1's [Fine,K] |
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
8454 | The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc. [Orenstein] |
15712 | 1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
8668 | The 'rational' numbers are those representable as fractions [Friend] |
8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |