[FOM] 546: New Pi01/solving CH

Mitchell Spector spector at alum.mit.edu
Sat Sep 27 16:59:09 EDT 2014

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.]

As Harvey mentioned:
 > Of course, maybe we get a different or even very different outcome
 > depending upon which permutation of A1,...,A8 that we start with.

In particular, if A1 is Con(ZFC), this scheme's proposed resolutions don't look quite as interesting.

This particular example sentence suggests what may be an interesting philosophical question:

Choosing the simpler alternative may perhaps be justifiable when selecting among competing 
explanations in science, but is it really the right thing to do in the very different endeavor of 
selecting axioms for mathematics?

Maybe not.  The choice that leads to the richer mathematical universe, with more structure, may 
arguably be seen as the more desirable of two alternatives.  This may require choosing the sentence 
with greater complexity, since the sentence may, in some sense, contain within itself the seeds of 
the consequent rich structure.


Harvey Friedman wrote:
> *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.
> There has been a major upgrade in the perfectly mathematically natural
> Pi01 incompleteness project in almost every direction. See
> Universal Properties and Incompleteness, September 26, 2014, 23 pages.
> https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
> #85
> I now want to take up a rather amusing approach to solving all
> interesting set theoretic problems.
> CONJECTURE. Something interesting will come out of suspending belief
> and taking this temporarily seriously.
> We first need to fix a linear ordering on sentences in the language of
> ZFC. A < B means that A is simpler than B.
> We only consider four categories of sentences. Let's call this
> admissible sentences.
> 1. Sentences with one or more quantifiers, all of which range over
> V(omega). These are essentially the arithmetic sentences.
> 2. Sentences with one or more quantifiers, all of which range over
> V(omega + 1) . There are essentially the sentences of second order
> arithmetic.
> 3. Sentences with one or more quantifiers, all of which range over
> V(omega + 2). These are essentially the sentences of third order
> arithmetic.
> 4. Sentences with one or more quantifiers, all of which range over V.
> All of 1 are lower than all of 2 are lower than all of 3 are lower
> than all of 4.
> Now how do we compare within each category? In the standard way, we
> assume prenex normal form, with the inside having only quantifiers
> that are either bounded (by epsilon) are of lower "type". Then we
> count the number of quantifier alterations of highest "type", as
> usual, noting whether we have Sigma or Pi. If the number of
> alternations is the same then the SIgma side is considered lower than
> the Pi side.
> To break ties, we stay within the common prefix class, and then
> compare the total number of quantifiers, including those of smaller
> type and bounded quantifiers. Further ties are not going to arise in
> what I am proposing - and if they do, we will cross that bridge when
> we come to it.
> Now let T be an extension of ZFC and A be a sentence of ZFC.
> We say that A is preferable to not(A) over T if and only if there is a
> theorem of T + A, unprovable in T, in admissible form, which is lower
> than any theorem of T + not(A), unprovable in T, in admissible form.
> 1. Start by listing your favorite set theoretic statements
> A1,A2,...,A8 in the order you would like them solved. We assume that
> none are decided in ZFC.
> 2. Choose between A1 and not(A1) according to whether A1 is preferable
> to  not(A1) over ZFC or not(A1) is preferable to A1 over ZFC. Add the
> preferable one to ZFC to form T1. Mark any of A2,...,A8 that are
> decided by T1, and put them away.
> 3. Choose between A2 and not(A2) according to whether A2 is preferable
> to not(A2) over T1 or not(A2) is preferable to A2 over T1. Add the
> preferable one to T1 to form T2. Mark any of A3,...,A8 that are
> decided by T2, and put them away.
> Continue this process all the way through the A's.
> Of course, maybe we get a different or even very different outcome
> depending upon which permutation of A1,...,A8 that we start with. In
> general, we probably want to front load stronger ones so that we have
> to work out the smallest number of comparisons. I.e., if we front load
> A1 appropriately, we could perhaps get a lot of the A2,...,A8 decided
> by ZFC + A1 (or by ZFC + not(A1)).
> So suppose we fix on A1,...,A8, or even say A1,...,A16. We can ask
> this question: find a permutation of A1,...,A16 such that the process
> just described results in the smallest number of comparisons to be
> made. I.e., that we have probably front loaded optimally, so that we
> strategically carve out "powerful set theoretic space" early on.
> We could make a master list of published set theoretic questions, and
> then put some subset of them according to the number of times they
> occur in a literature search. We can of course just use these numbers
> of hits not only for the selection of the A's, but their order. So it
> is clear that CH = continuum hypothesis will be first.Thus we start of
> with preferring not(CH) as is well known.
> So I have thus solved CH. CH is false (smile).
> If "there exists a measurable cardinal" is next, then obviously we are
> starting with
> not(CH) + "there exists a measurable cardinal".
> Most set theorists obviously approve of this start.
> Have fun.
> ****************************************
> My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
> https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
> This is the 546th in a series of self contained numbered
> postings to FOM covering a wide range of topics in f.o.m. The list of
> previous numbered postings #1-527 can be found at the FOM posting
> http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html
> 528: More Perfect Pi01  8/16/14  5:19AM
> 529: Yet more Perfect Pi01 8/18/14  5:50AM
> 530: Friendlier Perfect Pi01
> 531: General Theory/Perfect Pi01  8/22/14  5:16PM
> 532: More General Theory/Perfect Pi01  8/23/14  7:32AM
> 533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
> 534: Perfect Explicitly Pi01  8/27/14  10:40AM
> 535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
> 536: Pi01 Progress  9/1/14 11:31AM
> 537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
> 538: Progress Pi01 9/6/14  11:31PM
> 539: Absolute Perfect Naturalness 9/7/14  9:00PM
> 540: SRM/Comparability  9/8/14  12:03AM
> 541: Master Templates  9/9/14  12:41AM
> 542: Templates/LC shadow  9/10/14  12:44AM
> 543: New Explicitly Pi01  9/10/14  11:17PM
> 544: Initial Maximality/HUGE  9/12/14  8:07PM
> 545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
> Harvey Friedman
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