[FOM] Fwd: 546: New Pi01/solving CH

[Mitchell Spector]

Harvey Friedman wrote:
> Mitchell Spector wrote:
>
> "There's something appealing about an Occam's razor approach, as
> Harvey Friedman recently proposed (each time you're faced with two
> alternative possibilities, pick the simpler of the two).
>
> However... Maybe I'm missing something, but wouldn't "ZFC is
> inconsistent" be preferred over "ZFC is consistent" (over ZFC, of
> course)?  "Preferred" here is to be taken in Harvey's technical sense.
>
> [Needless to say, I'm assuming that ZFC actually is consistent.]"
>
> Two comments.
>
> 1. I said that the Sigma sides are lower than Pi sides, all things
> being equal. That was an error. The Pi sides are lower than the Sigma
> sides. That takes care of the particular kind of example that you
> raised.



I'm not sure how seriously you meant this whole idea, since you described it (accurately, I would 
say) as "amusing", but I think it's interesting to take it at face value and see where it leads.


Overall, it seems to me that you're attempting to choose, at each stage, the more powerful of the 
two alternatives -- it's more powerful in that it decides a simpler, more fundamental 
previously-undecidable sentence than the other alternative does.


As you say, changing the definition so that Pi is simpler than Sigma (both at the same level) would 
take care of the particular issue I mentioned.

However, is there a rationale for considering the Pi side to be simpler than the Sigma side, aside 
from making this example come out right?

I think the general inclination is to say that the truth of a Sigma statement, being evidenced by a 
single witness for a lower-order statement, is simpler than the truth of a Pi statement, which 
requires multiple lower-level corroborations.

Of course, there's a formal duality here; a Pi statement being false would be a simpler fact than a 
Sigma statement being false.  One might view this duality as saying that the Sigma and Pi statements 
at a certain level are equally simple, and the choice is arbitrary.

But neither of these two viewpoints leads to Pi being simpler than Sigma. Maybe I'm missing the 
right perspective on this though.



> 2. Even beyond that, I was careful to say that you get to list your
> favorite MATHEMATICAL
> set theoretic questions, in order, and even experiment with that
> order.
>
> Is Con(ZFC) your favorite mathematical question? It's not a
> mathematical question at all, or if it is one, it's probably not your
> favorite one. I also mentioned that one can list these mathematical set
> theoretic questions in order of, say, Google Scholar hits.


Your point is well-taken. In fact, that's why I said "the consistency of ZFC", rather than Con(ZFC).

I think of Con(ZFC) as some Gödel-style formalization, which, in and of itself, is a not very 
interesting question about natural numbers.

On the other hand, the statement "ZFC is consistent" is an interesting question of model theory and 
set theory, which are branches of mathematics.

Phrasing it as "ZFC has a model" arguably makes it more mathematical in flavor.

I certainly wouldn't claim that the consistency of ZFC should rival CH as a "favorite mathematical 
question". But it is an interesting mathematical question; that's why I found it bothersome that the 
original Sigma-simpler-than-Pi version gave the "wrong" answer.


One further speculative thought: Your procedure is at least vaguely reminiscent of a Henkin 
construction. Perhaps something analogous to omitting types would be productive.

Mitchell


>
> Harvey
>