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The User
The User
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Notice that you can drop the axiom of replacement or replace it by a weaker reflection principle. Without this axiom you have less consistency strength—it might still be consistent even if ZFC is not consistent. You would not loose that much. Of course many results in set theory would become meaningless (regarding various axioms, large cardinals etc.), but in most (not all, even when excluding set theory) situations in mathematics this axiom is not that important.

Notice that you can drop the axiom of replacement or replace it by a weaker reflection principle. Without this axiom you have less consistency strength—it might still be consistent even if ZFC is not consistent. You would not loose that much. Of course many results in set theory would become meaningless (regarding various axioms, large cardinals etc.), but in most situations in mathematics this axiom is not that important.

Notice that you can drop the axiom of replacement or replace it by a weaker reflection principle. Without this axiom you have less consistency strength—it might still be consistent even if ZFC is not consistent. You would not loose that much. Of course many results in set theory would become meaningless (regarding various axioms, large cardinals etc.), but in most (not all, even when excluding set theory) situations in mathematics this axiom is not that important.

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The User
The User
  • 2.4k
  • 23
  • 24

Notice that you can drop the axiom of replacement or replace it by a weaker reflection principle. Without this axiom you have less consistency strength—it might still be consistent even if ZFC is not consistent. You would not loose that much. Of course many results in set theory would become meaningless (regarding various axioms, large cardinals etc.), but in most situations in mathematics this axiom is not that important.