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Single Idea 15919

[filed under theme 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets ]

Full Idea

The Peano-Russell notion of class is the 'logical' notion, where each collection is associated with some kind of definition or rule that characterises the members of the collection.

Gist of Idea

The 'logical' notion of class has some kind of definition or rule to characterise the class

Source

Shaughan Lavine (Understanding the Infinite [1994], IV.1)

Book Ref

Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.63

The 33 ideas from 'Understanding the Infinite'

18250 Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
15898 The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine]
15899 Replacement was immediately accepted, despite having very few implications [Lavine]
15900 The iterative conception of set wasn't suggested until 1947 [Lavine]
15904 The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
15907 Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]
15909 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]
15915 Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
15912 Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
15913 A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]
15914 An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine]
15917 Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
15918 Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]
15919 The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine]
15922 For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
15921 Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine]
15920 Pure collections of things obey Choice, but collections defined by a rule may not [Lavine]
15926 Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
15929 Set theory will found all of mathematics - except for the notion of proof [Lavine]
15928 Intuitionism rejects set-theory to found mathematics [Lavine]
15930 Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine]
15931 The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]
15932 The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine]
15933 Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]
15934 Mathematical proof by contradiction needs the law of excluded middle [Lavine]
15936 The Power Set is just the collection of functions from one collection to another [Lavine]
15935 Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine]
15940 The intuitionist endorses only the potential infinite [Lavine]
15937 Those who reject infinite collections also want to reject the Axiom of Choice [Lavine]
15942 Every rational number, unlike every natural number, is divisible by some other number [Lavine]
15945 Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]
15949 The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
15947 The infinite is extrapolation from the experience of indefinitely large size [Lavine]