[FOM]: Independece without Forcing

[Harvey Friedman]

Reply to Eisworth 7/11/03 2PM and 7/11/03 /5:24PM.

>
>What should set-theorists be doing with forcing?  Should we be beating the
>bushes for opportunities to apply the technology in other areas of
>mathematics?  Should we be concentrating on "internal matters"?  Should we
>be searching for "theorems about forcing" instead of proving things using
>forcing?  Are we simply spinning our wheels developing more and more baroque
>techniques for answering technical questions no one cares about, or are we
>heading somewhere?

1. "Applying the technology to other areas of mathematics". I would 
think that by now, forcing has been applied to every area of 
mathematics that it can be applied to. Forcing cannot be applied to 
areas of mathematics in which all of the questions are sufficiently 
concrete that absoluteness applies. Of course, this accounts for the 
vast bulk of mathematical activity today.

Let me be more careful with this statement. Concreteness rules out 
forcing as the primary technique in the following specific sense. Let 
A be a sufficiently concrete mathematical statement. It is NOT the 
case that there is a model of ZFC in which

i) there is a forcing extension of the model in which A holds; and
ii) there is a forcing extension of the model in which A fails.

However, forcing might be used to prove A within ZFC as an auxiliary 
technique! This has happened a number of times both in the standard 
set theory literature and in my own work concerning Borel 
Diagonalization. E.g., see

H. Friedman, On the necessary use of abstract set theory, Advances in 
Mathematics 41 (1981), 209-280.

Applying forcing in this way to concrete statements in mathematics is 
an exciting possibility. In fact, it is already a standard tool in 
Descriptive Set Theory. Look at work of, e.g., Hjorth and Kechris.

2. "Concentrating on internal matters". I assume that this means 
concentrating on matters of interest only to specialists in forcing. 
Since I am not a specialist in forcing, although I use forcing from 
time to time, I don't have a handle on what you mean. But on general 
principles, I can say that often something that appears to be of 
interest only to specialists can be imaginatively reformulated and 
extended so that it becomes of wider interest.

3. "Searching for theorems about forcing". I assume that this 
naturally divides into theorems about forcing that are of interest 
mainly to specialists in forcing, and theorems about forcing that are 
of wider interest. I think that there are not too many of the latter, 
and so it would be unusually interesting to have more of them. The 
main ones I can think of surround the principle that "no sufficiently 
concrete statement can be forced", as discussed above.

4. "Are we simply spinning our wheels ...". As applications of 
forcing get more and more developed, it gets more and more difficult 
to do something really striking. What one should look for are results 
of a novel kind. This requires fertile imagination. I am optimistic 
about seeing results of a refreshingly new character(s).

Harvey Friedman























>
>I asked earlier about obtaining independence results without forcing
>and asked about getting "mathematical statements" independent of
>ZFC + V=L.
>
>When I originally posed the question, I had in mind speculating about
>the existence of "mathematical" statements P such that ZFC + V=L + P
>is consistent but not in a "standard" (read transitive?) version of L.
>
>Todd
>
>
>
>
>
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