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The question of what happens when CH fails is, of course, intensely studied in set theory. There are entire research areas, such as the area of cardinal characteristics of the continuum, which are devoted to studying what happens with sets of reals when the Continuum Hypothesis fails.

The lesson of much of this analysis is that many of the most natural open questions turn out to be themselvesd independent of ZFC. , even when one wants ¬CH. For example, the question of whether all sets that are intermediate in size between the natural numbers and the continuum should be Lebesgue measure 0, is independent of ZFC. ZFC+¬CH. The question of whether only the countable sets have continuum many subsets is independent of ZFC. ZFC+¬CH. There are a number of cardinal characteristics that I mention here, whose true nature becomes apparant only when CH fails. For example, must every unbounded family of functions from ω to ω have size continuum? It is independent of ZFC. ZFC+¬CH. Must every dominating family of such functions have size continuum? It is independent of ZFC. ZFC+¬CH. Those question are relatively simple to state and could easily be considered part of "ordinary" mathematics.

However, much of the rest of what you might think of as ordinary mathematics is simply not affected by CH or not CH. In particular, the existence of non-measurable sets that you mentioned is provable in ZFC, whether or not CH holds. (This proof requires the use of the Axiom of Choice, however, unless large cardinals are inconsistent, a result proved by Solovay and Shelah.)

Nevertheless, there is a growing body of research on some sophisticated axioms in set theory called forcing axioms, which have powerful consequences, and many of these new axioms imply the failure of CH. This topic began with Martin's Axiom MAω1, and has continued with the Proper Forcing Axiom, Martin's Maximum and now many other variations.

Lastly, in your title you asked what are the new sets like. The consistency of the failure of the Continuum Hypothesis was proved by Paul Cohen with the method of forcing. This highly sophisticated and versatile method is now used pervasively in set theory, and is best thought of as a fundamental method of constructing models of set theory, sharing many affinities with construction methods in algebra, such as the construction of algebraic or transcendental field extensions. Cohen built a model of ZFC+ ¬CH by starting with a model V of ZFC+CH, and then using the method of forcing to add ω2 many new real numbers to construct the forcing extension V[G]. Since V and V[G] have the same cardinals (by a detailed combinatorial argument), it follows that the set of reals in V[G] has size at least (in fact, exactly) ω2. In particular, the old set of reals from V, which had size ω1, is now one of the sets of reals of intermediate size. Thus, these intermediate sets are not so mysterious after all!

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Lastly, in your title you asked what are the new sets like. The consistency of the failure of the Continuum Hypothesis was proved by Paul Cohen with the method of forcing. This highly sophisticated and versatile method is now used pervasively in set theory, and is best thought of as a fundamental method of constructing models of set theory, sharing many affinities with construction methods in algebra, such as the construction of algebraic or transcendental field extensions. Cohen built a model of ZFC+ ¬CH by starting with a model V of ZFC+CH, and then using the method of forcing to add ω2 many new real numbers to construct the forcing extension V[G]. Since V and V[G] have the same cardinals (by a detailed combinatorial argument), it follows that the set of reals in V[G] has size at least (in fact, exactly) ω2. In particular, the old set of reals from V, which had size ω1, is now one of the sets of reals of intermediate size. Thus, these intermediate sets are not so mysterious after all!

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The question of what happens when CH fails is, of course, intensely studied in set theory. There are entire research areas, such as the area of cardinal characteristics of the continuum, which are devoted to studying what happens with sets of reals when the Continuum Hypothesis fails.

The lesson of much of this analysis is that many of the most natural open questions turn out to be themselvesd independent of ZFC. For example, the question of whether all sets that are intermediate in size between the natural numbers and the continuum should be Lebesgue measure 0, is independent of ZFC. The question of whether only the countable sets have continuum many subsets is independent of ZFC. There are a number of cardinal characteristics that I mention here, whose true nature becomes apparant only when CH fails. For example, must every unbounded family of functions from ω to ω have size continuum? It is independent of ZFC. Must every dominating family of such functions have size continuum? It is independent of ZFC. Those question are relatively simple to state and could easily be considered part of "ordinary" mathematics.

However, much of the rest of what you might think of as ordinary mathematics is simply not affected by CH or not CH. In particular, the existence of non-measurable sets is provable in ZFC, whether or not CH holds.

Nevertheless, there is a growing body of research on some sophisticated axioms in set theory called forcing axioms, which have powerful consequences, and many of these new axioms imply the failure of CH. This topic began with Martin's Axiom MAω1, and has continued with the Proper Forcing Axiom, Martin's Maximum and now many other variations.