In papers published between 1930 and 1935. Zermelo outlines a foundational program, with infinitary logic at its heart, that is intended to (1) secure axiomatic set theory as a foundation for arithmetic and analysis and (2) show that all mathematical propositions are decidable. Zermelo's theory of systems of infinitely long propositions may be termed "Cantorian" in that a logical distinction between open and closed domains plays a signal role. Well-foundedness and strong inaccessibility are used to systematically integrate highly transfinite concepts (...) of demonstrability and existence. Zermelo incompleteness is then the analogue of the Problem of Proper Classes, and the resolution of these two anomalies is similarly analogous. (shrink)

On Zermelo's view, any mathematical theory presupposes a non-empty domain, the elements of which enjoy equal status; furthermore, mathematical axioms must be chosen from among those propositions that reflect the equal status of domain elements. As for which propositions manage to do this, Zermelo's answer is, those that are ?symmetric?, meaning ?invariant under domain permutations?. We argue that symmetry constitutes Zermelo's conceptual analysis of ?general proposition?. Further, although others are commonly associated with the extension of Klein's Erlanger Programme to logic, (...) Zermelo's name has a place in that story. (shrink)

An idea attributable to Russell serves to extend Zermelo’s theory of systems of infinitely long propositions to infinitary relations. Specifically, relations over a given domain \ of individuals will now be identified with propositions over an auxiliary domain \ subsuming \. Three applications of the resulting theory of infinitary relations are presented. First, it is used to reconstruct Zermelo’s original theory of urelements and sets in a manner that achieves most, if not all, of his early aims. Second, the new (...) account of infinitary relations makes possible a concise characterization of parametric definability with respect to a purely relational structure. Finally, based on his foundational philosophy of the primacy of the infinite, Zermelo rejected Gödel’s First Incompleteness Theorem; it is shown that the new theory of infinitary relations can be brought to bear, positively, in that connection as well. (shrink)

What we call first‐order logic over fixed domain was initiated, in a certain guise, by Peirce around 1885 and championed, albeit in idiosyncratic form, by Zermelo in papers from the 1930s. We characterise such logics model‐ and proof‐theoretically and argue that they constitute exploration of a clearly circumscribed conception of domain‐dependent generality. Whereas a logic, or family of such, can be of interest for any of a variety of reasons, we suggest that one of those reasons might be that said (...) logic fosters some clarification regarding just what qualifies as a logical concept, a logical operation, or a logical law. (shrink)

Symmetric propositions over domain $\mathfrak{D}$ and signature $\Sigma = \langle R^{n_1}_1, \ldots, R^{n_p}_p \rangle$ are characterized following Zermelo, and a correlation of such propositions with logical type- $\langle \vec{n} \rangle$ quantifiers over $\mathfrak{D}$ is described. Boolean algebras of symmetric propositions over $\mathfrak{D}$ and Σ are shown to be isomorphic to algebras of logical type- $\langle \vec{n} \rangle$ quantifiers over $\mathfrak{D}$. This last result may provide empirical support for Tarski’s claim that logical terms over fixed domain are all and only those (...) invariant under domain permutations. (shrink)