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Single Idea 15940

[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite ]

Full Idea

The intuitionist endorse the actual finite, but only the potential infinite.

Gist of Idea

The intuitionist endorses only the potential infinite

Source

Shaughan Lavine (Understanding the Infinite [1994], VI.2)

Book Ref

Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.176

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]
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]
15922 For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
15929 Set theory will found all of mathematics - except for the notion of proof [Lavine]
15926 Second-order logic presupposes a set of relations already fixed by the first-order domain [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]
15947 The infinite is extrapolation from the experience of indefinitely large size [Lavine]
15949 The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]