1994 | Understanding the Infinite |
2.5 | p.33 | 18250 | Cauchy gave a necessary condition for the convergence of a sequence |
4.2 | p.92 | 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers |
I | p.3 | 15893 | Cantor's theory concerns collections which can be counted, using the ordinals |
I | p.4 | 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules |
I | p.5 | 15900 | The iterative conception of set wasn't suggested until 1947 |
I | p.5 | 15899 | Replacement was immediately accepted, despite having very few implications |
II.6 | p.38 | 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios |
III.2 | p.47 | 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity |
III.3 | p.50 | 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal |
III.4 | p.53 | 15912 | Counting results in well-ordering, and well-ordering makes counting possible |
III.4 | p.53 | 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified |
III.4 | p.53 | 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one |
III.4 | p.54 | 15915 | Ordinals are basic to Cantor's transfinite, to count the sets |
III.5 | p.61 | 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal |
III.5 | p.62 | 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest |
IV.1 | p.63 | 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class |
IV.2 | p.78 | 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size |
IV.2 | p.78 | 15920 | Pure collections of things obey Choice, but collections defined by a rule may not |
IV.2 | p.92 | 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true |
V.3 | p.123 | 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain |
V.3 | p.133 | 15929 | Set theory will found all of mathematics - except for the notion of proof |
V.3 n33 | p.132 | 15928 | Intuitionism rejects set-theory to found mathematics |
V.4 | p.135 | 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets |
V.5 | p.148 | 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate |
V.5 | p.149 | 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs |
V.5 | p.150 | 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement |
VI.1 | p.155 | 15934 | Mathematical proof by contradiction needs the law of excluded middle |
VI.1 | p.157 | 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' |
VI.1 | p.160 | 15936 | The Power Set is just the collection of functions from one collection to another |
VI.2 | p.164 | 15937 | Those who reject infinite collections also want to reject the Axiom of Choice |
VI.2 | p.176 | 15940 | The intuitionist endorses only the potential infinite |
VI.3 | p.198 | 15942 | Every rational number, unlike every natural number, is divisible by some other number |
VII.1 | p.215 | 15943 | Limitation of Size is not self-evident, and seems too strong |
VII.4 | p.226 | 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions |
VIII.2 | p.248 | 15947 | The infinite is extrapolation from the experience of indefinitely large size |
VIII.2 | p.256 | 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes |