Ideas from 'Investigations in the Foundations of Set Theory I' by Ernst Zermelo [1908], by Theme Structure

[found in 'From Frege to Gödel 1879-1931' (ed/tr Heijenoort,Jean van) [Harvard 1967,0-674-32449-8]].

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2. Reason / D. Definition / 8. Impredicative Definition
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.
4. Formal Logic / F. Set Theory ST / 1. Set Theory
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.
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.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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.
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).
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
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.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
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.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
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!
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
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
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
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.