In this article the authors presents 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. The author shows that this additional axiom is equivalent to some choice principles within the axiomatic set theory. It is also indicated that this axiom is derivable from the axiom which states that all sets are ordinal definable (V=OD) within ZF. Finally, the author explains why the process of language skolemization implies the existence...

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In this article the authors presents 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. The author shows that this additional axiom is equivalent to some choice principles within the axiomatic set theory. It is also indicated that this axiom is derivable from the axiom which states that all sets are ordinal definable (V=OD) within ZF. Finally, the author explains why the process of language skolemization implies the existence of indiscernibles. In his considerations the author follows the ontological and epistemological paradigm of investigations

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