structure for 'Mathematics'    |     alphabetical list of themes    |     expand these ideas

6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number

[the various families of numbers]

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