[FOM] Simple and difficult
joeshipman at aol.com
joeshipman at aol.com
Sat Apr 6 01:56:28 EDT 2013
Friedman's paper "Primitive Independence Results", available here
http://www.math.osu.edu/~friedman.8/pdf/PrimitiveIndResults071302.pdf
gives simpler versions of the theorem below which don't require the concept of injection or surjection. His proposition 4.2
"There is a set E such that every transitive set not in E contains a 4 element chain"
is very easy to formalize completely, but it contains 7 quantifiers; his proposition 4.5 is more technically complex but contains only 5 quantifiers. His proposition 5.1
"Every transitive proper class contains a 4 element chain"
is simpler still, although it requires NBG instead of ZFC.
Here "x is transitive" is written
AyAz (y /in x & z /in y)=>(z /in x)
and "x contains a 4 element chain" can be written
EaEbEcEd (a /subset b & b /subset c & c /subset d & a /in x & b /in x & c /in x & d /in x)
Further simplifications come from the observation that all transitive sets with at least 2 elements contain {} and {{}} so that you really only need 2 element chains not including {} and {{}}. The set { {}, {{}}, {{{}}}, {{{{}}}}, ... } is an infinite transitive set with no 3-element chain, but it is essentially the only one: any infinite transitive set with an infinite element contains a 3-element chain.
Given an inaccessible, I would like to see an explicit construction of a transitive set of inaccessible size with no 4-element chain. Such an example can probably be extracted from Harvey's proof.
-- JS
-----Original Message-----
From: Joe Shipman <JoeShipman at aol.com>
To: fom <fom at cs.nyu.edu>
Sent: Thu, Apr 4, 2013 11:42 pm
Subject: Re: [FOM] Simple and difficult
My wording was imprecise, I meant (2) not refuted, but provably not a theorem, otherwise Ex~(x=x) would do.
I believe Harvey found a sentence whose consistency strength is a subtle cardinal:
There exists κ such that every transitive set S into which κ can be injected contains x and y such that x is a proper subset of y and x ≠ Ø and x ≠ {Ø}.
Does anyone know a simpler sentence of set theory which has been neither proven nor disproven?
-- JS
Sent from my iPhone
On Apr 2, 2013, at 11:13 PM, Joe Shipman <JoeShipman at aol.com> wrote:
It's easy to write down a sentence in the language of Peano Arithmetic which is both short and unsettled:
AxEyAzAw (x<y & ~(SSzSSw=y V SSzSSw=SSy))
What's the shortest or simplest sentence you can come up with in the language of set theory that is either (1) not settled (2) provably not a theorem of ZFC if ZFC is consistent?
-- JS
Sent from my iPhone
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