I have two titles to this note. You get to pick the title that you want.
REFUTATION OF THE CONTINUUM HYPOTHESIS THE PITFALLS OF CITING “INTRINSIC MAXIMALITY’
Note that the most fundamental and simple nontrivial equivalence relation on the set theoretic universe is that of “being in one-one correspondence”.
Also very fundamental and simple is the equivalence relation EQ on infinite sets of reals “being in one-to-one correspondence”.
Note that it is consistent with ZFC that this fundamental simple EQ has
i. exactly two equivalence classes. ii. infinitely many equivalence classes.
THEREFORE, by the “intrinsic maximality of the set theoretic universe”, ii holds. THEREFORE, we have refuted the continuum hypothesis (smile).
NOTE: A lesson that can be drawn here is just how important it is to avoid cavalier quoting of “intrinsic maximality of the set theoretic universe”.
In fact, if we factor, we are looking at a set for which it is consistent with ZFC that it has, on the one hand, exactly two elements, and on the other hand, is infinite. So by “intrinsic maximality of the set theoretic universe”, it must be infinite (smile).
GENERAL PRINCIPLE. Let EQ be a simple equivalence relation. Suppose ZFC + “EQ has infinitely many equivalence classes” is consistent. Then EQ actually has infinitely many equivalence classes.
Here is the legitimate foundational program.
- Set up an elementary language that is based on only some of the most set theoretically fundamental notions.
- Determine which “simple” definitions define equivalence relations. Show that this is robust, in that here truth is the same as provability in ZFC and in ZC.
- Determine what is consistent with ZFC about the number of equivalence classes of items in 2.
- Now apply the general principle, and show that the resulting statements are (even collectively?) consistent with ZFC. Perhaps the general principle will be seen to be equivalent over ZFC to “the continuum is greater than ” or perhaps some versions of not GCH?
- Rework 1-4 with ever stronger elementary languages and ever less “simple” definitions, until one hits a brick wall.
The immediate problem is to get a good prototype for this elementary language. We want it to be not ad hoc, and so should be in tune with the most basic set theoretic material.
While doing this real time foundations, it now appears, provisionally, that we are best off using “there is a function from x onto y” and not just “there is a bijection from x onto y”. The former is more flexible than the latter, and still very very basic for elementary set theory.
ELST = elementary set theory. We have
- Equality, and union operator (set of all elements of elements).
- There is a function from onto . Written .
- Convenient to have variables range only over infinite sets.
Something interesting has arisen. This language supports even more naturally the 3-ary relation
if and only if
i. The union of y and the union of z are both x.
ii. and .
iii. (Implicitly, x,y,z are infinite).
- We have defined T as a conjunction of a small number of atomic formulas in with no nesting of the union operator.
- For all , is an equivalence relation.
Thus there is great simplicity here. We can provisionally concentrate on just cases of 1.2, even perhaps with a limit on the number of atomic formulas. We can also relax the “no nesting”.
So we have a parameterized equivalence relation. We should look at a modified General Principle.
GENERAL PRINCIPLE. Let T be a 3-ary parameterized equivalence relation. Suppose ZFC + “EQ has infinitely many equivalence classes” is consistent. Then EQ actually has infinitely many equivalence classes.
In this way, we should be getting the robustness referred to above, and also the failure of GCH at every infinite cardinal.
I’ll stop here with this provisional beginning…