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The continuum Hypothesis says:

There is no set whose cardinality is strictly between that of the integers and the real numbers

and the generalised Continuum Hypothesis, additionally says:

if an infinite set's cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S.

The hypothesis is actually named after the continuum of the real numbers in the first statement. But one can additionally see this in a different light by thinking of it as asking about the continuum of infinite cardinals.

Assuming the Continuum hypothesis one gets a sequence of cardinalities like we get the sequence of integers. But of course it has proved a very useful kind of fiction to assume that between these integers lay the real numbers.

So, first question: In a similar kind of way, if we deny the continuum hypthesis, what kind of other infinite numbers do we get to fill out the infinite cardinal line?

It might be worth pointing out that the Continuum Hypothesis is undecided in ZFC so one can choose either to add or deny it. But quite possibly once might say that to add this hypothesis when it is undecided from more natural axioms means that it is an exercise in aribitrariness.

In this connection, it might also be worth pointing out, that paraconsistent set theory does settle the continuum hypothesis in the negative.

Second question: Are there any other set theories whose axioms settles the continuum hypothesis, which doesn't assume it or its denial in a thinly assumed manner?

Apparently 2nd-order logic settles the issue - but I haven't found a reference to say in which direction.

EDIT

One useful consequence of assuming the hypothesis is that the hyper-reals become unique (upto isomorphism).

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See on the "deep technical" question SEP entry on The Continuum Hypothesis

It is interesting to compare with the Axiom of Choice : both have been proved to be independent from ZF axioms, but AC has been very "useful" in settling a lot of interesting mathematical problems, so that , in spite of some implausible consequences (see Banach-Tarski "paradox"), quite all mathematicians accept it as a reasonable axiom.

CH has no this kind of status; see Gödel remarks at the end of his 1947 “What is Cantor's Continuum Hypothesis?” :

[...] it is very suspicious that, as against the numerous plausible propositions which imply the negation of the continuum hypothesis, not one plausible proposition is known which would imply the continuum hypothesis.

You can see Waclaw Sierpinski, Cardinal and ordinal numbers (1965), page 378, for some results connected with the continuum hypothesis :

Theorem. The hypothesis is equivalent to a theorem stating that the set of all points in a plane is the sum of two sets of which one is at most denumerable on every line parallel to the axis of abscissae and the other is at most denumerable on every line parallel to the axis of ordinates. [...] The theorem can obviously be expressed as follows: The continuum hypothesis is equivalent to a theorem stating that a plane may be covered with a denumerable aggregate of curves which are either of the form у = f(x) or of the form x = f(y) each.

See also page 400 :

Theorem. The continuum hypothesis is equivalent to the proposition that the three-dimensional Euclidean space E is the sum of three sets Ei (i = 1,2,3) such that if we denote the axes of Cartesian coordinates in the space E by OXi (i = 1, 2, 3), then, for i = 1,2,3, the set Ei is finite on every line parallel to the axes OXi.

For a similar discussion, you can see in math.stackexchange the post on Continuum hypothesis.

You can also see the "old" Waclaw Sierpinski, Hypothèse du continu (1934).

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    Is it considered uncharitable to note that this answer has nothing to do with the question? OP asked what are the consequences etc. – user4894 Mar 6 '14 at 23:45
  • Appreciate the edits. – user4894 Mar 7 '14 at 8:08

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