Nowy postulat teorii mnogości — aksjomat Leibniza–Mycielskiego
A New Postulate of Set Theory – The Leibniz–Mycielski Axiom

Summary/Abstract: In this article we will present the Leibniz–Mycielski axiom (LM) of set theory (ZF) introduced several years ago by Jan Mycielski as an additional axiom of set theory. This new postulate formalizes the so-called Leibniz Law (LL) which states that there are no two distinct indiscernible objects. From the Ehrenfeucht–Mostowski theorem it follows that every theory which has an infinite model has a model with indiscernibles. The new LM axiom states that there are infinite models without indiscernibles. These models are called Leibnizian models of set theory. We will show that this additional axiom is equivalent to some choice principles within the axiomatic set theory. We will also indicate that this axiom is derivable from the axiom which states that all sets are ordinal definable (V=OD) within ZF. Finally, we will explain why the process of language skolemization implies the existence of indiscernibles. In our considerations we will follow the ontological and epistemological paradigm of investigations.

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