[FOM] Really Large Infinitary Languages
Guillermo Badia
guillebadia89 at gmail.com
Fri Nov 21 18:41:52 EST 2014
Max Dickmann's book "Large Infinitary Languages" contains a discussion of
> proper class sized Infinitary languages.
>
> -- John Bell
Thanks a lot. In what section exactly? I haven't find it. Please note
that I don't mean languages with a proper class of formulas, but
languages with conjunctions and quantifications of proper class size.
Cheers
Guillermo
On 11/22/14, John Bell <jbell at uwo.ca> wrote:
> Max Dickmann's book "Large Infinitary Languages" contains a discussion of
> proper class sized Infinitary languages.
>
> -- John Bell
>
> Sent from my iPad
>
>> On Nov 20, 2014, at 6:30 PM, fom-request at cs.nyu.edu kwrote:
>>
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>> Today's Topics:
>>
>> 1. Really Large Infinitary Languages (Guillermo Badia)
>> 2. 562: Perfectly Natural Review #1 (Harvey Friedman)
>> 3. Re: Leibnizian principles (Robert Rynasiewicz)
>>
>>
>> ----------------------------------------------------------------------
>>
>> Message: 1
>> Date: Thu, 20 Nov 2014 17:35:32 +1300
>> From: Guillermo Badia <guillebadia89 at gmail.com>
>> To: fom <fom at cs.nyu.edu>
>> Subject: [FOM] Really Large Infinitary Languages
>> Message-ID:
>> <CADtn9vDesbb5RDvJQD=HAst9EX-=3SpRV4+xOwaWnYi-d=fecQ at mail.gmail.com>
>> Content-Type: text/plain; charset=UTF-8
>>
>> Dear All,
>> Are there any studies on languages with conjunctions and disjunctions
>> of proper class size?
>>
>> Thanks in advance for any help you can offer me.
>>
>> Guille
>>
>>
>> ------------------------------
>>
>> Message: 2
>> Date: Wed, 19 Nov 2014 19:40:15 -0500
>> From: Harvey Friedman <hmflogic at gmail.com>
>> To: Foundations of Mathematics <fom at cs.nyu.edu>
>> Subject: [FOM] 562: Perfectly Natural Review #1
>> Message-ID:
>> <CACWi-GUPQV__RpmziSv4nm-3_9LeNPSxCs=1Va90L8owXT2sdw at mail.gmail.com>
>> Content-Type: text/plain; charset=UTF-8
>>
>> THIS POSTING IS SELF CONTAINED
>>
>> There has been a lot of progress on CMI = Concrete Mathematical
>> Incompleteness, which has been for some months within the territory of
>> Perfectly Natural Concrete Mathematical Incompleteness.
>>
>> This is Part #1 of an abridged account of the entire picture as of
>> now, and will not have the level of motivation and detail that the
>> extended abstract will have on my website when it appears there.
>>
>> 1. Maximal Roots in Q[0,1]^k - Templates
>> 2. Maximal Roots in Q[0,1]^k - Specifics
>> 3. G-bases in Q^k
>> 4. p,G-bases in Q^k
>>
>> 1. MAXIMAL ROOTS IN Q]0,1]^k - TEMPLATES
>>
>> We work with maximal roots in order invariant relations on Q[0,1]^k.
>> The fuller version will also have entirely parallel developments using
>> maximal squares and maximal cliques. A root of a binary relation on a
>> set is a set such that any two elements lie in the relation. A maximal
>> root is a root that is not a subset of any other root. Here relations
>> are always binary.
>>
>> We begin with Perfectly Natural motivating Templates. For a specific
>> independent statement, see Proposition 2.1.
>>
>> TEMPLATE A. Let F:Q[0,1]^k into Q[0,1]^k be order theoretic. Every
>> order invariant relation on Q[0,1]^k has an F invariant maximal root.
>>
>> TEMPLATE B. Let E be an order theoretic equivalence relation on
>> Q[0,1]^k. Every order invariant relation on Q[0,1]^k has an E
>> invariant maximal root.
>>
>> TEMPLATE C. Let f:Q[0,1] into Q[0,1] be order theoretic. Every order
>> invariant relation on Q[0,1]^k has an f invariant maximal root.
>>
>> Here Q[0,1] = Q intersect [0,1]. Order theoretic is the same as order
>> invariant with constants (parameters). I.e., definable as a
>> propositional combination of inequalities using only < and constants
>> from Q. Let S containedin Q[0,1]. S is F invariant if and only if for
>> all x in Q[0,1]^k, x in S iff Fx in S.S is E invariant if and only if
>> for all E equivalent x,y, x in S iff y in S. S is f invariant if and
>> only if for all x in Q[0,1]^k, x in S iff fx in S, where fx =
>> (f(x_1),...,f(x_k)) in S.
>>
>> It is obvious that Templates A-C are Perfectly Natural motivators. For
>> some, this may be yet more compelling when formulated analogously with
>> maximal squares or with maximal cliques.
>>
>> See Theorem 1.6 for the implicit concreteness of these Templates.
>>
>> Although we have not yet been able to completely analyze these
>> Templates, we present the following Perfectly Natural partial results.
>>
>> a. A Perfectly Natural sufficient condition for Templates A,C. The
>> claim of sufficiency for A can be proved from large cardinals but not
>> in ZFC.
>> b. A Perfectly Natural sufficient condition for Template B. The claim
>> of sufficiency for B can be proved from large cardinals but not in
>> ZFC.
>> c. The same Perfectly Natural sufficient conditions for the obvious
>> multiple forms of Templates A,B,C, where we are given finite sequences
>> of F's, E's, and f's. These claims of sufficiency can be proved from
>> large cardinals but not in ZFC.
>> d. a Perfectly Natural formulation of Templates A,B,C for isomorphism
>> types of F's, E's, and f's. Here we can give a complete analysis of
>> the resulting Templates. The correctness of the analysis can be proved
>> from large cardinals but not in ZFC.
>>
>> We begin with a).
>>
>> DEFINITION 1.1. Let F:Q[0,1]^k into Q[0,1]^k. F is upward if and only
>> if for all x in Q[0,1]^k, any coordinate moved by F is greater than
>> any coordinate not moved by F. F is order preserving if and only if
>> for all x in Q[0,1]^k, x and F(x)) are order equivalent.
>>
>> PROPOSITION 1.1. If Template A holds then F is order preserving.
>> Template A holds if F is additionally upward and order preserving, If
>> F is upward, then Template A holds for F if and only if F is order
>> preserving.
>>
>> PROPOSITION 1.2. Template C holds if f is additionally upward.
>>
>> We now come to b).
>>
>> DEFINITION 1.2. Let E be an equivalence relation on Q[0,1]^k. E is
>> upward if and only if finitely many rationals are moved somewhere by
>> E, and every x_i smaller than all x_j moved somewhere by E is not
>> moved by E from x.
>>
>> PROPOSITION 1.3. If Template B holds then E is order preserving.
>> Template B holds if E is additionally upward and order preserving. If
>> E is additionally upward, then Template A holds for F if and only if F
>> is order preserving.
>>
>> Now for c).
>>
>> TEMPLATE D. Let F_1,...,F_r:Q[0,1]^k into Q[0,1]^k be order theoretic.
>> Every order invariant relation on Q[0,1]^k has an F_1,...,F_r
>> invariant maximal root.
>>
>> TEMPLATE E. Let E_1,...,E_r be order theoretic equivalence relations
>> on Q[0,1]^k. Every order invariant relation on Q[0,1]^k has an
>> E_1,...,E_r invariant maximal root.
>>
>> TEMPLATE F. Let f_1,...,f_r:Q[0,1] into Q[0,1] be order theoretic.
>> Every order invariant relation on Q[0,1]^k has an f_1,...,f_r
>> invariant maximal root.
>>
>> PROPOSITION 1.4. Template D holds if the F's are additionally upward,
>> and order preserving, If the F's are upward, then Template D holds for
>> the F's if and only if the F's are order preserving.
>>
>> PROPOSITION 1.5. Template E holds if the E's are additionally upward,
>> and order preserving, If the E's are upward, then Template E holds for
>> the E's if and only if the E's are order preserving.
>>
>> PROPOSITION 1.6. Template F holds if the f's are additionally upward.
>>
>> Now for d).
>>
>> DEFINITION 1.3. A,B containedin Q[0,1]^k are isomorphic if and only if
>> (Q[0,1],<,A) and (Q[0,1],<,B) are isomorphic in the sense of model
>> theory. Functions f:Q[0,1]^k into Q[0,1]^k are isomorphic if and only
>> if their graphs are isomorphic subsets of Q[0,1]^k. Relations on
>> Q[0,1]^k are isomorphic if and only if they are isomorphic subsets of
>> Q[0,1]^2k. .
>>
>> THEOREM 1.7. (RCA_0). If f,g:Q[0,1]^k into Q[0,1]^k are isomorphic
>> then f is order theoretic if and only if g is order theoretic. There
>> is an effective criteria for determining whether two given order
>> theoretic subsets of Q[0,1]^k are isomorphic.
>>
>> TEMPLATE G. Let F:Q[0,1]^k into Q[0,1]^k be order theoretic. For all
>> F_1,...,F_r isomorphic to F, every order invariant relation on
>> Q[0,1]^k has an F_1,...,F_r invariant maximal root.
>>
>> TEMPLATE H. Let E be an order theoretic equivalence relation on
>> Q[0,1]^k. For all E_1,...,E_r isomorphic to E, every order invariant
>> relation on Q[0,1]^k has an E_1,...,E_r invariant maximal root.
>>
>> TEMPLATE I. Let f:Q[0,1] into Q[0,1] be order theoretic. For all
>> f_1,...,f_r isomorphic to f, every invariant relation on Q[0,1]^k has
>> an f_1,...,f_r invariant maximal root, under the coordinate action.
>>
>> THEOREM 1.8. Every instance of Templates A-I, and Propositions 1.1-1.4
>> are provably equivalent to Pi01 sentences over WKL_0, via Goedel's
>> Completeness Theorem.
>>
>> THEOREM 1.9. Propositions 1.1 and 1.3-1.6 are provably equivalent to
>> Con(SRP) over WKL_0. Proposition 1.2 is provable in WKL_0 + Con(SRP).
>>
>> THEOREM 1.10. Every instance of Templates G,H are provable in SRP or
>> refutable in RCA_0. Every instance of Template I is provable in WKL_0
>> + Con(SRP) or refutable in RCA_0.
>>
>> THEOREM 1.11. For each k there is an instance of Templates A,B,D,E,G,H
>> that is provable in SRP but not in SRP[k]. So these Templates cannot
>> be completely analyzed within any SRP[k]. There is an instance of
>> Templates F,I that is provable in WKL_0 + Con(SRP) but not in SRP. So
>> these Templates cannot be completely analyzed in SRP.
>>
>> Here SRP is the stationary Ramsey property hierarchy. This is the same
>> as the subtle cardinal hierarchy and the ineffable cardinal hierarchy.
>>
>> There are more ambitious templates, involving rational piecewise
>> linear functions, again with the important case of one-dimensional
>> maps acting coordinatewise. There are a lot of open issues here
>> amenable to attack.
>>
>> 2. MAXIMAL ROOTS IN Q[0,1]^k - SPECIFICS
>>
>> Here we give some Perfectly Natural cases of some of the Templates in
>> section 1, which are provable using large cardinals but not in ZFC
>> (assuming ZFC is consistent).
>>
>> Arguably the simplest example of this kind involves sections.
>>
>> DEFINITION 2.1. Let A,B containedin Q^k. The section of A at x is {y
>> in Q^k-r: (x,y) in A}. Note that if r >= k then the section of A at x
>> is the emptyset. A,B agree below p in Q if and only if for all x in
>> Q^k with max(x) < p, x in A iff x in B. Let E containedin Q. E^r> is
>> the set of all r tuples from E that are strictly decreasing. E^r>=,
>> E^r<, E^r<= are defined analogously.
>>
>> PROPOSITION 2.1. Every order invariant relation on Q[0,1]^k has a
>> maximal root whose sections at elements of {1,1/2,...,1/n}^r> agree
>> below 1/n.
>>
>> There are a number of successive strenghenings of Proposition 2.1
>> involving sections. The following is the strongest one that we
>> consider.
>>
>> DEFINITION 2.2. Let E containedin Q. E^<=k is the set of all
>> sequences from E of lengths 1,...,k. For x,y in Q^k, x*y is x
>> concatenated with y. min(x) is the least coordinate of x.
>>
>> PROPOSITION 2.2. Every order invariant relation on Q[0,1]^k has a
>> maximal root whose sections at any two order equivalent x,y in
>> {1,1/2,...,1/n}^<=k agree below min(x*y).
>>
>> DEFINITION 2.3. Let E be a finite subset of Q[0,1]. The E-lift of x in
>> Q[0,1]^k is obtained by replacing each x_i in E that is greater than
>> all x_j notin E by the next greatest element of E; x if this does not
>> exist.
>>
>> PROPOSITION 2.3. Let E be a finite subset of Q[0,1]. Every order
>> invariant relation on Q[0,1]^k has an E-lift invariant maximal root.
>>
>> DEFINITION 2.4. Let E be a finite subset of Q[0,1]. x,y in Q[0,1]^k
>> are E-related (are E-relatives) if and only if
>> i. x,y are order equivalent.
>> ii. the coordinates of x,y that are not in E are identical in
>> identical positions, and smaller than the coordinates of x,y that are
>> in E.
>>
>> PROPOSITION 2.4. Let E be a finite subset of Q[0,1]. Every order
>> invariant relation on Q[0,1]^k has an E-related invariant maximal
>> root.
>>
>> THEOREM 2.5. Propositions 2.1 - 2.4 are provably equivalent to Pi01
>> sentences over WKL_0 via Goedel's Completeness Theorem.
>>
>> THEOREM 2.6. Propositions 2.1 - 2.4 are provably equivalent to
>> Con(SRP) over WKL_0.
>>
>> 3. G-BASES IN Q^k
>>
>> DEFINITION 3.1. Let R be a relation on Q^k. x R reduces to y if and
>> only if x R y and max(x) > max(y). S is R free if and only if S
>> containedin Q^k and no element of S R reduces to any element of S. S
>> is an R basis if and only if S is R free and every element of Q^k\S R
>> reduces to some element of S.
>>
>> THEOREM 3.1. R = Q^2 has no basis.
>>
>> Here is a Perfectly Natural way to recover from Theorem 3.1. Instead
>> of requiring that every element of Q^k\S reduces to some element of S,
>> we require only that all tuples "built out of the elements of S"
>> reduce to some element of S.
>>
>> DEFINITION 3.2. Let R be a relation on Q^k and G:Q^k into Q^k. A
>> G-basis for R is an R free S such that every element of G[S]\S reduces
>> to an element of S. More generally, let H:Q^kr into Q^k. An H-basis
>> for R is a nonempty R free S such that every element of H[S^r]\S
>> reduces to an element of S.
>>
>> THEOREM 3.2. Let H:Q^kr into Q^k be order theoretic. Every order
>> invariant relation on Q^k has an H-basis.
>>
>> TEMPLATE J. Let F:Q^k into Q^k be order theoretic. For all order
>> theoretic G:Q^k into Q^k, every order invariant relation on Q^k has a
>> G-basis containing its F image.
>>
>> TEMPLATE K. Let E be an order theoretic equivalence relation on
>> Q[0,1]^k. For all order theoretic G:Q^k into Q^k, every order
>> invariant relation on Q^k has a G-basis containing its E image.
>>
>> TEMPLATE L. Let f:Q[0,1] into Q[0,1] be order theoretic. For all order
>> theoretic G:Q^k into Q^k, every order invariant relation on Q^k has a
>> G-basis containing its f image.
>>
>> We also have the variants with stronger conclusions.
>>
>> TEMPLATE M. Let F:Q^k into Q^k be order theoretic. For all order
>> theoretic H:Q^kr into Q^k, every order invariant relation on Q^k has
>> an H-basis containing its F image.
>>
>> TEMPLATE N. Let E be an order theoretic equivalence relation on
>> Q[0,1]^k. For all order theoretic H:Q^kr into Q^k, every order
>> invariant relation on Q^k has an H-basis containing its E image.
>>
>> TEMPLATE O. Let f:Q[0,1] into Q[0,1] be order theoretic. For all order
>> theoretic H:Q^kr into Q^k, every order invariant relation on Q^k has
>> an H-basis containing its f image.
>>
>> Here we know much more than we do in section 1. We can completely
>> analyze Templates J-O, but only using large cardinals.
>>
>> THEOREM 3.3. Every instance of Templates J,K,M,N is either provable in
>> SRP or refutable in RCA_0. Every instance of Templates L,O are either
>> provable in WKL_0 + Con(SRP) or refutable in RCA_0. Assuming Con(SRP),
>> Templates J,M are equivalent; K,N are equivalent; L,O are equivalent.
>> In the first two claims, we cannot replace SRP by any fixed SRP[k].
>>
>> With relative bases we can go well beyond order theoretic invariance
>> in Perfectly Natural ways. This allows us to give a particularly
>> simple example of CMI.
>>
>> DEFINITION 3.3. The upper shift of x in Q^k is the result of adding 1
>> to all nonnegative coordinates of x. The upper shift of S containedin
>> Q^k is the set of upper shifts of the elements of S. A finite sequence
>> of sets in various Q^k contains its upper shift if and only if each
>> term contains its upper shift.
>>
>> PROPOSITION 3.4. For all order theoretic G:Q^k into Q^k, every order
>> invariant relation on Q^k has a G-basis containing its upper shift.
>>
>> PROPOSITION 3.5. For all order theoretic H:Q^kr into Q^k, every order
>> invariant relation on Q^k has an H-basis containing its upper shift.
>>
>> THEOREM 3.5. Propositions 3.4 and 3.5 are provably equivalent to
>> Con(SRP) over WKL_0.
>>
>> TEMPLATE P. Let f:Q into Q be rational piecewise linear. For all order
>> theoretic G:Q^k into Q^k, every order invariant relation on Q^k has a
>> G-basis containing its f image.
>>
>> We also have the following multiple form.
>>
>> TEMPLATE Q. Let f_1,...,f_r:Q into Q be rational piecewise linear. For
>> all order theoretic HG:Q^kr into Q^k, every order invariant relation
>> on Q^k has an H-basis containing its f_1,...,f_r images.
>>
>> THEOREM 3.6. Every instance of Templates P,Q is either provable in
>> WKL_0 + Con(SRP) or refutable in RCA_0.
>>
>> 4. p,G-BASES IN Q^k
>>
>> The p,G-bases of R, p >= 2, form a Perfectly Natural way of
>> approximating the G-bases of R. pG-bases can be finite for any R,
>> whereas this is not the case for G-bases.
>>
>> DEFINITION 4.1. Let R be a relation on Q^k, G:Q^k into Q^k, and p >=
>> 2. A p,G-basis for R consists of nonempty sets A_1 containedin ...
>> containedin A_p containedin Q^k, where A_p is R free and every element
>> of G[A_i], i < p, reduces to an element of A_i+1.
>>
>> PROPOSITION 4.1. Let G:Q^k into Q^k be order theoretic. Every order
>> invariant relation on Q^k has a finite 3,G-basis A containedin B
>> containedin C, where C contains the upper shift of B.
>>
>> PROPOSITION 4.2. Let H:Q^kr into Q^k be order theoretic. Every order
>> invariant relation on Q^k has a finite p,H-basis A_1 containedin ...
>> containedin A_p, where each A_i+1 contains the upper shift of A_i.
>>
>> PROPOSITION 4.3. Let H:Q^kr into Q^k be order theoretic. Every order
>> invariant relation on Q^k has a p,H-basis A_1 containedin ...
>> containedin A_p, where each A_i contains its upper shift.
>>
>> Propositions 4.1 and 4.2 are explicitly Pi02. There is an obvious
>> iterated exponential bounds on the cardinalities involved, and also
>> the numerators and denominators involved so that they become
>> explicitly Pi01. With Proposition 4.3, the sets generally have to be
>> infinite, but they can have a combinatorial structure also leading to
>> explicitly Pi01 forms.
>>
>> THEOREM 4.4. Propositions 4.1 and 4.2 are provably equivalent to
>> Con(SRP) over EFA. Proposition 4.3 is provably equivalent to Con(SRP)
>> over RCA_0.
>>
>> ************************************************************
>> 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 562nd 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
>> 546: New Pi01/solving CH 9/26/14 12:05AM
>> 547: Conservative Growth - Triples 9/29/14 11:34PM
>> 548: New Explicitly Pi01 10/4/14 8:45PM
>> 549: Conservative Growth - beyond triples 10/6/14 1:31AM
>> 550: Foundational Methodology 1/Maximality 10/17/14 5:43AM
>> 551: Foundational Methodology 2/Maximality 10/19/14 3:06AM
>> 552: Foundational Methodology 3/Maximality 10/21/14 9:59AM
>> 553: Foundational Methodology 4/Maximality 10/21/14 11:57AM
>> 554: Foundational Methodology 5/Maximality 10/26/14 3:17AM
>> 555: Foundational Methodology 6/Maximality 10/29/14 12:32PM
>> 556: Flat Foundations 1 10/29/14 4:07PM
>> 557: New Pi01 10/30/14 2:05PM
>> 558: New Pi01/more 10/31/14 10:01PM
>> 559: Foundational Methodology 7/Maximality 11/214 10:35PM
>> 560: New Pi01/better 11/314 7:45PM
>> 561: New Pi01/HUGE 11/5/14 3:34 PM
>>
>> Harvey Friedman
>>
>>
>> ------------------------------
>>
>> Message: 3
>> Date: Thu, 20 Nov 2014 12:31:26 -0500
>> From: Robert Rynasiewicz <ryno at lorentz.phl.jhu.edu>
>> To: Foundations of Mathematics <fom at cs.nyu.edu>
>> Subject: Re: [FOM] Leibnizian principles
>> Message-ID: <238E2C88-E900-4F10-8BBA-576130C0D808 at lorentz.phl.jhu.edu>
>> Content-Type: text/plain; charset="windows-1252"
>>
>> This was an error in transcription. My apologies. The original reads
>> ?without?. ?RR
>>
>>> On Sun 16.11.14, at 8:13 PM, Rob Arthan <rda at lemma-one.com> wrote:
>>>
>>>
>>>> On 15 Nov 2014, at 02:05, Robert Rynasiewicz <ryno at lorentz.phl.jhu.edu
>>>> <mailto:ryno at lorentz.phl.jhu.edu>> wrote:
>>>>
>>>> ?L?Espace est quelque chose d?uniforme absolument; & sans les choses y
>>>> plac?es, un point de l?Espace ne differe absolument en rien d?un autre
>>>> point d?Espace. [Space is Something absolutely Uniform; and with the
>>>> Things placed in it, One Point of Space does not absolutely differ in
>>>> any respect whatsoever from Another Point of Space.]?
>>>
>>> I have no comments to add on the philosophical issues, but ?sans? means
>>> ?without? not ?with?.
>>>
>>> Regards,
>>>
>>> Rob.
>>> _______________________________________________
>>> FOM mailing list
>>> FOM at cs.nyu.edu
>>> http://www.cs.nyu.edu/mailman/listinfo/fom
>>
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