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Single Idea 9603
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
]
Full Idea
Assume a largest prime, then multiply the primes together and add one. The new number isn't prime, because we assumed a largest prime; but it can't be divided by a prime, because the remainder is one. So only a larger prime could divide it. Contradiction.
Gist of Idea
An assumption that there is a largest prime leads to a contradiction
Source
report of Euclid (Elements of Geometry [c.290 BCE]) by James Robert Brown - Philosophy of Mathematics Ch.1
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.1
A Reaction
Not only a very elegant mathematical argument, but a model for how much modern logic proceeds, by assuming that the proposition is false, and then deducing a contradiction from it.
The
25 ideas
with the same theme
[the various families of numbers]:
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]
|
10106
|
Rational numbers give answers to division problems with integers
[George/Velleman]
|
10102
|
The integers are answers to subtraction problems involving natural numbers
[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]
|