Re: Paper and slides on indefiniteness of CH

This is a continuation of my earlier message. Recall that I have two titles to this note. You get to pick the title that you want.

REFUTATION OF THE CONTINUUM HYPOTHESIS AND EXTENDIBLE CARDINALS

THE PITFALLS OF CITING “INTRINSIC MAXIMALITY”

1. GENERAL STRATEGY.
2. THE LANGUAGE $L_0$.
3. STRONGER LANGUAGES.

1. GENERAL STRATEGY

Here we present a way of using the informal idea of “intrinsic maximality of the set theoretic universe” to do two things:

1. Refute the continuum hypothesis (using PD and less).
2. Refute the existence of extendible cardinals (in ZFC).

Quite a tall order!

Since I am not that comfortable with “intrinsic maximality”, I am happy to view this for the time being as an additional reason to be even less comfortable.

At least I will resist announcing that I have refuted both the continuum hypothesis and existence of certain extensitvely studied large cardinals!

INFORMAL HYPOTHESIS. Let $\phi(x,y,z)$ be a simple property of sets $x,y,z$. Suppose ZFC + “for all infinite $x$, there exist infinitely many distinct sets which are pairwise incomparable under $\phi(x,y,z)$” is consistent. Then for all infinite $x$, there exist infinitely many distinct sets which are pairwise incomparable under $\phi(x,y,z)$.

Since we are going to be considering only very simple properties, we allow for more flexibility.

INFORMAL HYPOTHESIS. Let $0 \leq n,m \leq \omega$. Let $\phi(x,y,z)$ be a simple property of sets $x,y,z$. Suppose ZFC + “for all $x$ with at least $n$ elements, there exist $m$ distinct sets which are pairwise incomparable under $\phi(x,y,z)$” is consistent. Then for all $x$ with at least $n$ elements, there exist at least $m$ distinct sets which are pairwise incomparable under $\phi(x,y,z)$.

We can view the above as reflecting the “intrinsic maximality of the set theoretic universe”.

We will see that this Informal Hypothesis leads to “refutations” of both the continuum hypothesis and the existence of certain large cardinals, even using very primitive phi in very primitive set theoretic languages.

2. THE LANGUAGE $L_0$

$L_0$ has variables over sets, $=$,$<$, $\leq^*$,$\cup$. Here $=$,$<$, $=^*$ are binary relation symbols, and $\cup$ is a unary function symbol. $x \leq^* y$ is interpreted as “there exists a function from $x$ onto $y$“. $\cup$ is the usual union operator, $\cup x$ being the set of all elements of elements of x.

$\text{MAX}(L_0,n,m)$. Let $0 \leq n,m \leq \omega$. Let $\phi(x,y,z)$ be the conjunction of finitely many formulas of $L_0$ in variables $x,y,z$. Suppose ZFC + “for all $x$ with at least $n$ elements, there exist $m$ distinct sets which are pairwise incomparable under $\phi(x,y,z)$” is consistent. Then for all $x$ with at least $n$ elements, there exist at least $m$ distinct sets which are pairwise incomparable under $\phi(x,y,z)$.

THEOREM 2.1. ZFC + $\text{MAX}(L_0,\omega,\omega)$ proves that there is no $(\omega+2)$-extendible cardinal.

More generally, we have

THEOREM 2.2. Let $2 < \log(m)+1 < n \leq \omega$.

i. ZFC + $\text{MAX}(L_0,n,m)$ proves that there is no $(\omega+2)$-extendible cardinal. Here $\log(\omega) = \omega$.
ii. ZFC + PD + $\text{MAX}(L_0,n,m)$ proves that the GCH fails at all infinite cardinals. In particular, it refutes the continuum hypothesis.
iii. ii with PD replaced by higher order measurable cardinals in the sense of Mitchell.

We are morally certain that we can easily get a complete understanding of the meaning of the sentences in quotes that arise in the $\text{MAX}(L_0,n,m)$.

Write $\text{MAX}(L_0)$ for

“For all $0 \leq n,m \leq \omega$, $\text{MAX}(L_0,n,m)$“. Using such a complete understanding we should be able to establish that ZFC + $\text{MAX}(L_0)$ is a “good theory”. E.g., such things as

1. ZFC + PD + $\text{MAX}(L_0)$ is equiconsistent with ZFC + PD.
2. ZFC + PD + $\text{MAX}(L_0)$ is conservative over ZFC + PD for sentences of second order arithmetic.
3. ZFC + PD + $\text{MAX}(L_0)$ + “there is a proper class of measurable cardinals” is also conservative over ZFC + PD for sentences of second order arithmetic.

We will revisit this development after we have gained that complete understanding. Then we will go beyond finite conjunctions of atomic formulas in $L_0$.

The key technical ingredient in this development is the fact that

1. GCH fails at all infinite cardinals is incompatible with $(\omega+2)$-extendible cardinals (Solovay).
2. GCH fails at all infinite cardinals is demonstrably consistent using much weaker large cardinals, or using just PD (Foreman/Woodin).

Harvey