Re: Paper and slides on indefiniteness of CH

Dear Sy,

Not quite the answer I was hoping to get but entertaining none the less.

On Oct 14, 2014, at 12:21 PM, Sy David Friedman  wrote:

Max likes this idea but now the problem is that the set-theorists are facing challenges that they have never seen before. The forms of maximality that Max wants are so new that the set-theorists don’t yet have the tools to figure out which forms are consistent and which are not; also they want to keep Max happy. Max is just honestly trying to understand how maximality is best captured mathematically, but in the lengthy and complex process of discovery there are many wrong turns and disappointments, laced with some successes. Although the process exhibits genuine progress, converging closer and closer to the kind of maximality principle Max is looking for, the challenge of providing consistent forms of maximality that will answer Max’s fair questions is enormously difficult. We can’t blame Max! He is just a sincere and likable guy who has “set out to discover what the world of Maximality is like, the range of what there is to the notion and its various properties and behaviours”. The set-theorists learn to like him and develop an equally sincere hope to fulfill his highest expectations.

So during the course of this entire email thread, HP seems to have evolved from declaring as you did in your first summary message to Sol of August 12:

I conjecture that CH is false as a consequence of my Strong Inner Model Hypothesis (i.e. Levy absoluteness with “cardinal-absolute parameters” for cardinal-preserving extensions) or one of its variants which is compatible with large cardinals existence.

(Aside: This is an extraordinary claim which you amplify in writing on Oct 13: “Further, as I said, I think there is a good chance of arriving at an optimal HP criterion”)

to the point that it is reduced to a collection of extremely difficult problems in the structure theory of countable transitive models of ZFC; i.e. the only principle left standing seems to be $\textsf{SIMH}^\#$ (but maybe that has also fallen or been reduced to a rough uncut principle), and this has been formulated to make any near term analysis impossible. You seem to feel this is a positive attribute of HP.

I would argue instead that this is simply a sort of coming of age for Set Theory; i.e. we can now pose simple questions about models of Set Theory which seem completely out of reach.

Number Theory is full of such problems. And one can easily generate such problems in a foundational framework.

Max it turns out has a cousin, RD (really difficult)  who proposes that Number Theorists should study “Ackermann Transcendental Number Theory”.

This is defined as follows. For each real number $x$, define $x^*$  as follows. The $n$-th decimal digit of the decimal expansion of $x^*$ is the $A_n$-th digit in the decimal expansion of $x$ where for $n \geq 1$, $A_n$ is the $n$-th “Ackermann number” as defined using the Knuth up-arrow notation:  $A_n = n \underbrace{\uparrow \cdots \uparrow}_{n\text{ times}} n$.

$A_1 = 1$
$A_2 = 4$
$A_3 = 3$ to iterated exponent of $3^{3^{3 \dots }}$ with 7625597484987 3’s (Wikipedia).

This is unambiguous if $x$ is not rational and if $x$ is rational then never use a decimal expansion for $x$ which is eventually 0 in defining $x^*$.

Is $\pi^*$ rational? Is $e^*$ rational, is $(\pi+e)^*$ rational  etc. How about:

Problem: Is $\pi^* = 3.15$?

(i.e. is the decimal expansion of $\pi$ exactly 0 at all Ackermann numbers $A_n$ for $n > 2$?)

This problem seems completely out of reach to say the least.  By restricting to real numbers (such as $\pi$) whose decimal expansion is primitive recursive this kind of question has a “foundational flavor”.

Now RD has a sibling RRD who wants to replace the Ackerman numbers with the Paris-Harrington numbers …

I look forward to seeing a reasoned account of HP with many of the issues that have been raised addressed. I hope that such an account has an initial list of axioms whose selection is well motivated based on the then current state of HP,  i.e. why $\textsf{SIMH}^\#$ instead of Strong-$\textsf{SIMH}^\#$ etc.

Best of all of course would be at least one specific conjecture strongly motivated by HP ideas, whose proof confirms HP, and whose refutation is a serious setback.  But I acknowledge this may not be a reasonable expectation at this preliminary stage.

I have yet to see anything that suggests that HP is anything more than part of the structure theory of countable transitive models (I like Harvey’s idea; rename HP as CTMP).

Back to our marching orders.

Regards,
Hugh