Questions on continuum hypothesis [closed]

Question

We have the notorious continuum hypothesis (CH).

According to Wikipedia it states

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

Gödel proved the CH is consistent with ZFC axioms and Cohen proved the opposite of CH is consistent with ZFC axioms using his forcing.

My questions are:

①The field between ℚ and ℝ

Suppose continuum hypothesis is disproved. Then do we know constructions and properties of the field between ℚ and ℝ whose cardinality is different from them? Especially can this field be constructed from ℚ as ℝ can be constructed as completion by absolute value?

②Propositions deduced from CH

Are there any propositions that are obtained supposing ZFC axioms and continuum axioms and their opposite propositions are obtained supposing ZFC axioms and denying continuum axioms? Such propositions may be logically equivalent to CH.

③The situation for p-adic fields

Roughly speaking CH states that whether there exists a field between ℚ and ℝ and its cardinality different from them. How about situations for ℚ and ℚp? If exists, can the construction be similar to the construction of the field between ℚ and ℝ? (as ℚp and ℝ are constructed similarly using absolute value)

Thank you for your answers.

Answer 1

As the comment says: the roles of Gödel and Cohen are reversed.

1. One cannot disprove CH using ZF(C), so I take it you simply assume the negation of CH and ask about fields between $\mathbb{Q}$ and $\mathbb{R}$ of intermediate cardinality. One way of obtaining such fields is by taking a subset, $X$ say, of the desired cardinality and taking the smallest subfield $\mathbb{Q}[X]$ of $\mathbb{R}$ that contains $X$. That has the same cardinality as $X$ itself. The construction is by closing off under the field operations. By adding roots of polynomial equations you can make the subfield real-closed. None of these fields will be order-complete: the only order-complete subfield of~$\mathbb{R}$ is $\mathbb{R}$ itself.

2. Already in 1934 Sierpiński started to collect consequences and equivalents of CH in his book Hypothèse du continu. Since then there have been tons of results in Set Theory, Algebra, Analysis, Topology, etc. that have been proved assuming CH; some are equivalent to CH, most are not.

3. The answer is basically the same as for 1: every subset of $\mathbb{Q}_p$ generates a subfield and you can specify the cardinality yourself. Again: since $\mathbb{Q}_p$ is the completion of $\mathbb{Q}$ with respect to the valuation all these subfields will not be complete, unless they are equal to $\mathbb{Q}_p$.