[FOM] Justifying SRP?

[Harvey Friedman]

*This research was partially supported by the John Templeton
Foundation grant ID #36297. The opinions expressed here are those of
the author and do not necessarily reflect the views of the John
Templeton Foundation.

First, an advertisement.
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#85 has just been put up, and it supersedes #82. It includes most of
#82. There is much improved new material on finite forms and on HUGE,
including finite forms for HUGE. My senior Fields Medalist is starting
to look at large cardinal theory. Another core mathematician star,
Fields Medalist equivalent, has certified Master Template (sets) on
Q[0,1] as just short of "perfectly mathematically natural", rather
using "beautiful" than "perfect". He is considering my idea of
"perfectly mathematically natural" but has not yet bought into this
notion being independent of what mathematicians actually are doing.
Another strong core mathematician has bought into this idea of
"perfectly mathematically natural independent of what mathematicians
are actually doing", and that all or almost all of the SRP statements
in #85 meet that standard.

Now for the main point of this posting.

http://www.cs.nyu.edu/pipermail/fom/2014-September/018174.html
http://www.cs.nyu.edu/pipermail/fom/2014-September/018175.html

These are very interesting FOM postings about reflection, and seem to
reflect (pun) the current way of thinking about reflection among set
theoretic philosophers and philosophical set theorists.

At this point in history, we are still at the stage where we look at
stories, and try to evaluate these stories on some grounds that we
don't yet understand (we may understand the stories but not the
grounds for evaluating them), and see whether they "click". Probably
simplicity of the story is important for "clicking", and probably
bringing in syntactic conditions is some sort of minus, all things
considered, and so forth.

One way to shake this up would be to have a general notion of what a
reflection principle candidate is, including good and bad reflection
principles. Then determine which ones are OK and which ones are not
OK. However, in order to do this with general enough scope to cover
real life proposals is a daunting task at present.

The other way to go is to tell increasingly simpler and clearer and
more compelling stories. Especially simpler.

In that spirit, I will present what appears to be an entirely
different approach.

1. On Monday, the first official workday,I have some sort of picture
of V. I can write this down on a piece of paper and date it. It just
looks like a V, with a wavy horizontal line on the top. On Tuesday, I
can reflect (maybe this is a pun or maybe this is a reformulation of
what reflection is all about?!) on the picture I wrote yesterday, and
extend it by extending the left and right slope -1,1 lines, with a new
wavy horizontal line at the top - and date it. I can do this extension
each day, say for a week, through Sunday. Thus we are working on V's
every day, Monday through Sunday.

2. Now I have been following some sort of intellectual process here of
reflection (pun or serious?), where I reflect (pun or serious?) on
what I thought yesterday and continue today.

3. It is highly desirable, and upon reflection (pun or serious?),
perhaps clear(?), that any first order statement I make on Monday
about objects in Monday's V doesn't change its truth value when I move
to Tuesday. And also throughout the week, about objects in Monday's V.
In fact, by the same reasoning, I could have decided to "discard"
Monday and start on Tuesday, and go from Tuesday through Monday
instead of from Monday through Sunday. And then Tuesday through
Sunday's V's would have the same first order properties about objects
in Tuesday's V.

4. So by this line of reasoning, we get that the V's from any given
two days during the week Monday through Sunday have the same first
order properties of objects from the V in the earlier of the two given
days.

5. All of this is very familiar and comfortable. The reason is that
even with only Monday and Tuesday, 4 is well known to be simply ZFC -
of course assuming that these V's have very minimal properties,
including, most importantly, that power set holds in each. But we
normally take that as given in the discussion. And even the comparing
of any two day's V's result in the same truth values for objects in
the earlier of the two days V's, also is just ZFC. I.e., just ZFC when
formulated as schemes.

6. Now let's stick with FIRST ORDER, and not even consider higher
order and only consider elements as parameters as we have been doing -
and no higher order parameters ever. Note even higher order
comprehension.

7. Now I alluded to starting from Monday and going all the way through
Monday, and then thinking that I could have started on Tuesday
instead. I.e., we are going from Monday to next Monday, and comparing
the first Monday through Sunday V's with the Tuesday through second
Monday V's. We now compare the FIRST ORDER properties of these two 7
sorted systems. We "see" that the truth values that we get when we
apply the corresponding properties to objects in the first Monday's V
are the same. And by the same reasoning, we also "see" that when we
shift a consecutive sequence by 1, we get the same FIRST ORDER
properties with parameters from the first V. And then we also see that
if we have two increasing subsequences of the same length, we get the
same FIRST ORDER properties with parameters from the first V being
used. The latter follows from consecutive shift formally.

8. This is enough to prove the consistency of ZFC + roughly a 6-subtle
cardinal. In fact, MERELY having Monday-Sunday and Tuesday-Monday with
the same first order properties of elements of Monday is ENOUGH to get
a countable transitive model of roughly ZFC + a 6-subtle cardinal.
Monday/Tuesday and Tuesday/Wednesday enough for ZFC + a subtle
cardinal.

9. MOREOVER, the power set is NOT NEEDED for this! E.g., we can only
assume that our V's have first order bounded separation,
extensionality, pairing, union. No infinity, power set, replacement!
Each V is an element of the next V. We do need that all elements of
any of Monday's sets are present on Monday.

10. I pretty much know how to do this even with far weaker assumptions
than 9. I pretty much know how to strip the entire development from
set theory - so that the whole business is arguably more fundamental
than set theory. I am trying to do something shockingly minimalistic
here. For example, consider 11 below:

11. We can apply these ideas to PREDICATIVITY discussions. During the
week the predicativist has ever more generous pictures of the
unfinished totality of subsets of omega. Each picture satisfies ACA0.
We also have that on successive days, we obtain an enumeration of all
of the subsets of omega seen in the previous day. This is by
reflecting(!) on the countable process that was applied the previous
day to generate that day's universe of subsets of omega. THEN when we
impose the shift invariance for first order properties discussed
above, we still get around a 6-subtle cardinal! No comprehension
beyond ACA0 needed here.

12. Perhaps it is more graphic to think that the predicativist is
creating countably many real numbers arranged linearly, and each
successive day, cuts are being filled in.

13. NOW LET'S GO FURTHER. We don't just create bigger and bigger V's
during this week, Monday - Sunday, but we do it FOREVER. I.e., along
the time path omega. So nobody can tell when I started. Instead of
thinking about the infinitely sorted structure, we can obviously make
it into a single sorted structure with a unary predicate for "being a
V', which has type omega. Then with minimal assumptions on the V's, we
get something between Ramsey cardinals and a measurable cardinal. With
type omega + 1 we get ZFC + a measurable cardinal, assuming that at
least the top V satisfies ZFC. Otherwise, we can use type omega + 2,
and have very minimalistic V's. Now going to omega + omega, we can get
close to two measurables or lots of measurables, according to how
strong the indiscerniblity we use is. Again, minimalistic V's.

14. Also all of this works if we just use models of ACA0 where each
model enumerates the previous ones. Or even by filling in cuts with
the reals arranged by magnitudes.

Harvey