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Single Idea 14199
[filed under theme 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / b. Combinatorial sets
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Full Idea
Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack.
Gist of Idea
Cantor's sets were just collections, but Dedekind's were containers
Source
report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro
Book Ref
'Metaphysics (Philosophical Perspectives 20)', ed/tr. Hawthorne,John [Blackwell 2006], p.123
A Reaction
As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets.
The
42 ideas
from 'works'
15901
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Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory
[Cantor, by Lavine]
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13444
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Cantor's Theorem: for any set x, its power set P(x) has more members than x
[Cantor, by Hart,WD]
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18098
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Cantor proved that all sets have more subsets than they have members
[Cantor, by Bostock]
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10865
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The continuum is the powerset of the integers, which moves up a level
[Cantor, by Clegg]
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15505
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If a set is 'a many thought of as one', beginners should protest against singleton sets
[Cantor, by Lewis]
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10701
|
Cantor showed that supposed contradictions in infinity were just a lack of clarity
[Cantor, by Potter]
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13016
|
The Axiom of Union dates from 1899, and seems fairly obvious
[Cantor, by Maddy]
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14199
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Cantor's sets were just collections, but Dedekind's were containers
[Cantor, by Oliver/Smiley]
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10082
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There are infinite sets that are not enumerable
[Cantor, by Smith,P]
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13483
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Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it
[Cantor, by Hart,WD]
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8710
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The powerset of all the cardinal numbers is required to be greater than itself
[Cantor, by Friend]
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15910
|
Cantor named the third realm between the finite and the Absolute the 'transfinite'
[Cantor, by Lavine]
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15905
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Cantor proved the points on a plane are in one-to-one correspondence to the points on a line
[Cantor, by Lavine]
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9983
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Cantor took the ordinal numbers to be primary
[Cantor, by Tait]
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17798
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Cantor presented the totality of natural numbers as finite, not infinite
[Cantor, by Mayberry]
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9971
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Cantor introduced the distinction between cardinals and ordinals
[Cantor, by Tait]
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15893
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Cantor's theory concerns collections which can be counted, using the ordinals
[Cantor, by Lavine]
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9892
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Cantor showed that ordinals are more basic than cardinals
[Cantor, by Dummett]
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14136
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A cardinal is an abstraction, from the nature of a set's elements, and from their order
[Cantor]
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15906
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Cantor tried to prove points on a line matched naturals or reals - but nothing in between
[Cantor, by Lavine]
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11015
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Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1
[Cantor, by Read]
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15903
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A real is associated with an infinite set of infinite Cauchy sequences of rationals
[Cantor, by Lavine]
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18251
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Irrational numbers are the limits of Cauchy sequences of rational numbers
[Cantor, by Lavine]
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15902
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Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties
[Cantor, by Lavine]
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15908
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It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers
[Cantor, by Lavine]
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13464
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Cantor proposes that there won't be a potential infinity if there is no actual infinity
[Cantor, by Hart,WD]
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10112
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The naturals won't map onto the reals, so there are different sizes of infinity
[Cantor, by George/Velleman]
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8733
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The Continuum Hypothesis says there are no sets between the natural numbers and reals
[Cantor, by Shapiro]
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17889
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CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals
[Cantor, by Koellner]
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13447
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Cantor: there is no size between naturals and reals, or between a set and its power set
[Cantor, by Hart,WD]
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10883
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Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers
[Cantor, by Horsten]
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18174
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Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities
[Cantor, by Maddy]
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13528
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Continuum Hypothesis: there are no sets between N and P(N)
[Cantor, by Wolf,RS]
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9555
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Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum
[Cantor, by Chihara]
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18173
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Cardinality strictly concerns one-one correspondence, to test infinite sameness of size
[Cantor, by Maddy]
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10232
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Property extensions outstrip objects, so shortage of objects caused the Caesar problem
[Cantor, by Shapiro]
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8631
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Cantor says that maths originates only by abstraction from objects
[Cantor, by Frege]
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8715
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Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability
[Cantor, by Friend]
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13454
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Cantor says (vaguely) that we abstract numbers from equal sized sets
[Hart,WD on Cantor]
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13465
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Only God is absolutely infinite
[Cantor, by Hart,WD]
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10863
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Cantor proved that three dimensions have the same number of points as one dimension
[Cantor, by Clegg]
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18176
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Pure mathematics is pure set theory
[Cantor]
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