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15945 | Second-order set theory just adds a version of Replacement that quantifies over functions |
Full Idea: Second-order set theory is just like first-order set-theory, except that we use the version of Replacement with a universal second-order quantifier over functions from set to sets. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], VII.4) |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one |
Full Idea: A member m of M is an 'upper bound' of a subset N of M if m is not less than any member of N. A member m of M is a 'least upper bound' of N if m is an upper bound of N such that if l is any other upper bound of N, then m is less than l. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |||
A reaction: [if you don't follow that, you'll have to keep rereading it till you do] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size |
Full Idea: Since combinatorial collections are enumerated, some multiplicities may be too large to be gathered into combinatorial collections. But the size of a multiplicity seems quite irrelevant to whether it forms a logical connection. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice |
Full Idea: Many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
15936 | The Power Set is just the collection of functions from one collection to another |
Full Idea: The Power Set is just he codification of the fact that the collection of functions from a mathematical collection to a mathematical collection is itself a mathematical collection that can serve as a domain of mathematical study. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) |
15899 | Replacement was immediately accepted, despite having very few implications |
Full Idea: The Axiom of Replacement (of Skolem and Fraenkel) was remarkable for its universal acceptance, though it seemed to have no consequences except for the properties of the higher reaches of the Cantorian infinite. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], I) |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets |
Full Idea: The Axiom of Foundation (Zermelo 1930) says 'Every (descending) chain in which each element is a member of the previous one is of finite length'. ..This forbids circles of membership, or ungrounded sets. ..The iterative conception gives this centre stage. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], V.4) |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules |
Full Idea: The controversy was not about Choice per se, but about the correct notion of function - between advocates of taking mathematics to be about arbitrary functions and advocates of taking it to be about functions given by rules. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], I) |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not |
Full Idea: Combinatorial collections (defined just by the members) obviously obey the Axiom of Choice, while it is at best dubious whether logical connections (defined by a rule) do. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class |
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. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], IV.1) |
15900 | The iterative conception of set wasn't suggested until 1947 |
Full Idea: The iterative conception of set was not so much as suggested, let alone advocated by anyone, until 1947. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], I) |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate |
Full Idea: The iterative conception of sets does not tell us how far to iterate, and so we must start with an Axiom of Infinity. It also presupposes the notion of 'transfinite iteration'. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs |
Full Idea: The iterative conception does not provide a conception that unifies the axioms of set theory, ...and it has had very little impact on what theorems can be proved. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], V.5) | |||
A reaction: He says he would like to reject the iterative conception, but it may turn out that Foundation enables new proofs in mathematics (though it hasn't so far). |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement |
Full Idea: Limitation of Size has it that if a collection is the same size as a set, then it is a set. The Axiom of Replacement is characteristic of limitation of size. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified |
Full Idea: A collection M is 'well-ordered' by a relation < if < linearly orders M with a least element, and every subset of M that has an upper bound not in it has an immediate successor. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain |
Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], V.3) | |||
A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'. |
15934 | Mathematical proof by contradiction needs the law of excluded middle |
Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |||
A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku. |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity |
Full Idea: Mathematics is today thought of as the study of abstract structure, not the study of quantity. That point of view arose directly out of the development of the set-theoretic notion of abstract structure. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], III.2) | |||
A reaction: It sounds as if Structuralism, which is a controversial view in philosophy, is a fait accompli among mathematicians. |
15942 | Every rational number, unlike every natural number, is divisible by some other number |
Full Idea: One reason to introduce the rational numbers is that it simplifes the theory of division, since every rational number is divisible by every nonzero rational number, while the analogous statement is false for the natural numbers. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], VI.3) | |||
A reaction: That is, with rations every division operation has an answer. |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true |
Full Idea: The chief importance of the Continuum Hypothesis for Cantor (I believe) was that it would show that the real numbers form a set, and hence that they were encompassed by his theory. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
18250 | Cauchy gave a necessary condition for the convergence of a sequence |
Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], 2.5) |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios |
Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], II.6) | |||
A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many. |
15912 | Counting results in well-ordering, and well-ordering makes counting possible |
Full Idea: Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |||
A reaction: Cantor didn't mean that you could literally count the set, only in principle. |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes |
Full Idea: The indiscernibility of indefinitely large sizes will be a critical part of the theory of indefinitely large sizes. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) |
15947 | The infinite is extrapolation from the experience of indefinitely large size |
Full Idea: My proposal is that the concept of the infinite began with an extrapolation from the experience of indefinitely large size. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) | |||
A reaction: I think it might be better to talk of an 'abstraction' than an 'extrapolition', since the latter is just more of the same, which doesn't get you to concept. Lavine spends 100 pages working out his proposal. |
15940 | The intuitionist endorses only the potential infinite |
Full Idea: The intuitionist endorse the actual finite, but only the potential infinite. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal |
Full Idea: The symbol 'aleph-nought' denotes the cardinal number of the set of natural numbers. The symbol 'aleph-one' denotes the next larger cardinal number. 'Aleph-omega' denotes the omega-th cardinal number. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], III.3) |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets |
Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |||
A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'! |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal |
Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest |
Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
15929 | Set theory will found all of mathematics - except for the notion of proof |
Full Idea: Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], V.3) |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' |
Full Idea: In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |||
A reaction: At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people. |
15928 | Intuitionism rejects set-theory to found mathematics |
Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations. | |||
From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33) |