| 1904 | Proof that every set can be well-ordered |
| p.4 | 15897 | Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed | |
| Full Idea: Zermelo realised that the Axiom of Choice (based on arbitrary functions) could be used to 'count', in the Cantorian sense, those collections that had given Cantor so much trouble, which restored a certain unity to set theory. | |||
| From: report of Ernst Zermelo (Proof that every set can be well-ordered [1904]) by Shaughan Lavine - Understanding the Infinite I |
| 1908 | Investigations in the Foundations of Set Theory I |
| p.24 | 18178 | For Zermelo the successor of n is {n} (rather than n U {n}) | |
| Full Idea: For Zermelo the successor of n is {n} (rather than Von Neumann's successor, which is n U {n}). | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 | |||
| A reaction: I could ask some naive questions about the comparison of these two, but I am too shy about revealing my ignorance. |
| p.60 | 9627 | Different versions of set theory result in different underlying structures for numbers | |
| Full Idea: In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures. | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4 | |||
| A reaction: This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another. |
| p.69 | 13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate | |
| Full Idea: Zermelo assumes that not every predicate has an extension but rather that given a set we may separate out from it those of its members satisfying the predicate. This is called 'separation' (Aussonderung). | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
| p.70 | 13487 | In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals | |
| Full Idea: In Zermelo's set theory, the Burali-Forti Paradox becomes a proof that there is no set of all ordinals (so 'is an ordinal' has no extension). | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
| p.107 | 15924 | Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? | |
| Full Idea: On Zermelo's view, predicative definitions are not only indispensable to mathematics, but they are unobjectionable since they do not create the objects they define, but merely distinguish them from other objects. | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Shaughan Lavine - Understanding the Infinite V.1 | |||
| A reaction: This seems to have an underlying platonism, that there are hitherto undefined 'objects' lying around awaiting the honour of being defined. Hm. |
| p.205 | 10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice | |
| Full Idea: Zermelo-Fraenkel axioms: Existence (at least one set); Extension (same elements, same set); Specification (a condition creates a new set); Pairing (two sets make a set); Unions; Powers (all subsets make a set); Infinity (set of successors); Choice | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15 |
| p.483 | 13012 | Zermelo published his axioms in 1908, to secure a controversial proof | |
| Full Idea: Zermelo proposed his listed of assumptions (including the controversial Axiom of Choice) in 1908, in order to secure his controversial proof of Cantor's claim that ' we can always bring any well-defined set into the form of a well-ordered set'. | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1 | |||
| A reaction: This is interesting because it sometimes looks as if axiom systems are just a way of tidying things up. Presumably it is essential to get people to accept the axioms in their own right, the 'old-fashioned' approach that they be self-evident. |
| p.484 | 13017 | Zermelo introduced Pairing in 1930, and it seems fairly obvious | |
| Full Idea: Zermelo's Pairing Axiom superseded (in 1930) his original 1908 Axiom of Elementary Sets. Like Union, its only justification seems to rest on 'limitations of size' and on the 'iterative conception'. | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.3 | |||
| A reaction: Maddy says of this and Union, that they seem fairly obvious, but that their justification is of prime importance, if we are to understand what the axioms should be. |
| p.484 | 13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed | |
| Full Idea: Zermelo used a weak form of the Axiom of Foundation to block Russell's paradox in 1906, but in 1908 felt that the form of his Separation Axiom was enough by itself, and left the earlier axiom off his published list. | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.2 | |||
| A reaction: Foundation turns out to be fairly controversial. Barwise actually proposes Anti-Foundation as an axiom. Foundation seems to be the rock upon which the iterative view of sets is built. Foundation blocks infinite descending chains of sets, and circularity. |
| p.485 | 13020 | The Axiom of Separation requires set generation up to one step back from contradiction | |
| Full Idea: The most characteristic Zermelo axiom is Separation, guided by a new rule of thumb: 'one step back from disaster' - principles of set generation should be as strong as possible short of contradiction. | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.4 | |||
| A reaction: Why is there an underlying assumption that we must have as many sets as possible? We are then tempted to abolish axioms like Foundation, so that we can have even more sets! |
| p.489 | 13027 | Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets | |
| Full Idea: Zermelo was a reductionist, and believed that theorems purportedly about numbers (cardinal or ordinal) are really about sets, and since Von Neumann's definitions of ordinals and cardinals as sets, this has become common doctrine. | |||
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.8 | |||
| A reaction: Frege has a more sophisticated take on this approach. It may just be an updating of the Greek idea that arithmetic is about treating many things as a unit. A set bestows an identity on a group, and that is all that is needed. |
| Intro | p.200 | 17608 | We take set theory as given, and retain everything valuable, while avoiding contradictions |
| Full Idea: Starting from set theory as it is historically given ...we must, on the one hand, restrict these principles sufficiently to exclude as contradiction and, on the other, take them sufficiently wide to retain all that is valuable in this theory. | |||
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |||
| A reaction: Maddy calls this the one-step-back-from-disaster rule of thumb. Zermelo explicitly mentions the 'Russell antinomy' that blocked Frege's approach to sets. |
| Intro | p.200 | 17607 | Set theory investigates number, order and function, showing logical foundations for mathematics |
| Full Idea: Set theory is that branch whose task is to investigate mathematically the fundamental notions 'number', 'order', and 'function', taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis. | |||
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |||
| A reaction: At this point Zermelo seems to be a logicist. Right from the start set theory was meant to be foundational to mathematics, and not just a study of the logic of collections. |
| Intro | p.200 | 17609 | Set theory can be reduced to a few definitions and seven independent axioms |
| Full Idea: I intend to show how the entire theory created by Cantor and Dedekind can be reduced to a few definitions and seven principles, or axioms, which appear to be mutually independent. | |||
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |||
| A reaction: The number of axioms crept up to nine or ten in subsequent years. The point of axioms is maximum reduction and independence from one another. He says nothing about self-evidence (though Boolos claimed a degree of that). |
| 1908 | New Proof of Possibility of Well-Ordering |
| §2a | p.189 | 17613 | We should judge principles by the science, not science by some fixed principles |
| Full Idea: Principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all. Geometry existed before Euclid's 'Elements', just as arithmetic and set theory did before Peano's 'Formulaire'. | |||
| From: Ernst Zermelo (New Proof of Possibility of Well-Ordering [1908], §2a) | |||
| A reaction: This shows why the axiomatisation of set theory is an ongoing and much-debated activity. |
| 1920 | works |
| p.204 | 9565 | Zermelo made 'set' and 'member' undefined axioms | |
| Full Idea: The terms 'set' and 'is a member of' are primitives of Zermelo's 1908 axiomatization of set theory. They are not given model-theoretic analyses or definitions. | |||
| From: report of Ernst Zermelo (works [1920]) by Charles Chihara - A Structural Account of Mathematics 7.5 | |||
| A reaction: This looks like good practice if you want to work with sets, but not so hot if you are interested in metaphysics. |
| p.280 | 3339 | For Zermelo's set theory the empty set is zero and the successor of each number is its unit set | |
| Full Idea: For Zermelo's set theory the empty set is zero and the successor of each number is its unit set. | |||
| From: report of Ernst Zermelo (works [1920]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280 |
| 1930 | On boundary numbers and domains of sets |
| p.489 | 13028 | Replacement was added when some advanced theorems seemed to need it | |
| Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |||
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |||
| A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
| p.1209 | 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical | |
| Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |||
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |||
| A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
| §5 | p.1233 | 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers |
| Full Idea: Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering. | |||
| From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |||
| A reaction: [a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical. |