Zermelo's Cantorian Theory of Systems of Infinitely Long Propositions

Abstract

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.

“Mathematics is the Logic of the Infinite”: Zermelo’s Project of Infinitary Logic

“…A detailed analysis of Zermelo's views concerning his infinitary logic was presented in Taylor 2002. In what follows I am going to use the notation from that paper.…”

Projekt Logiki Infinitarnej Ernsta Zermela

“…Uwagi na jego temat znaleźć można np. w pracy [Taylor 2002] oraz [ van Dalen, Ebbinghaus 2000]. Jest kwestią nie do końca przesądzoną, z jakiego okresu pochodzi ta notatka -czy z roku 1921 (jak głosi umieszczona na niej data), czy z okresu o wiele późniejszego (co -zdaniem niektórychsugeruje dopisek ołówkiem "42").…”

“…Komentarze na temat Tez o nieskończoności znaleźć można np. w: [Taylor 2002], [ van Dalen, Ebbinghaus 2000]. Poglądy Zermela na filozofię matematyki omawiane są m.in.…”