Solution to a Problem Raised by Stig Kanger and a Set Theoretical Statement Equivalent to the Axiom of Choice

Abstract

Let us say that a set is a fragment of an ordinal if it can be obtained from an ordinal by removing certain sets (which are necessarily ordinals) from the ordinal and/or from the members of the ordinal and/or from the members of the members of the ordinal etc. As an example $\left\{ {\left\{ {\left\{ {\left\{ {} \right\}} \right\}} \right\}} \right\}$ is a fragment of an ordinal since it can be obtained in the manner indicated below: Professor Kanger has raised the following problem: Is it true that * Every set is a fragment of an ordinal We prove the answer to be affirmative. Moreover we prove that the statement * when formalized as below in the language of set theory, is equivalent to the axiom of choice.