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Single Idea 10232

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem ]

Full Idea

Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem.

Clarification

How do we know that Julius Caesar is not a number?

Gist of Idea

Property extensions outstrip objects, so shortage of objects caused the Caesar problem

Source

report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6

Book Ref

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.127

A Reaction

So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem?

The 49 ideas from George Cantor

9145 We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor]
15911 Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine]
15946 Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine]
9992 The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor]
17831 Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake]
15896 Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]
9616 A set is a collection into a whole of distinct objects of our intuition or thought [Cantor]
13444 Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD]
18098 Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock]
10865 The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
15505 If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis]
10701 Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
10232 Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro]
8631 Cantor says that maths originates only by abstraction from objects [Cantor, by Frege]
8715 Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend]
13454 Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor]
17889 CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner]
13447 Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD]
10883 Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten]
18174 Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
13528 Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS]
9555 Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara]
18173 Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
13016 The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy]
14199 Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley]
10082 There are infinite sets that are not enumerable [Cantor, by Smith,P]
13483 Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD]
8710 The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend]
15910 Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
15905 Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine]
9983 Cantor took the ordinal numbers to be primary [Cantor, by Tait]
17798 Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry]
9971 Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
15893 Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
9892 Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
14136 A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor]
15906 Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
11015 Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
15903 A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
18251 Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
15902 Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine]
15908 It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine]
13464 Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD]
10112 The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
8733 The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
13465 Only God is absolutely infinite [Cantor, by Hart,WD]
10863 Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg]
15901 Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
18176 Pure mathematics is pure set theory [Cantor]