HDTenEightyP/NetNews huggingface dataset:
From: turpin@cs.utexas.edu (Russell Turpin)
Newsgroups: sci.logic
Subject: How does one formalize "small" mathematical theories?
Date: 13 Sep 1992 14:13:38 -0500
Organization: U Texas Dept of Computer Sciences, Austin TX
NNTP-Posting-Host: cs.utexas.edu
Summary: A fuzzy and naive question.

-*----
The wonderful thing about set theory, of course, is that in it
one can define almost all the other parts of mathematics: group
theory, field theory, measure theory, point set topology, etc.
But if one were writing or using theorem-proving tools to work
proofs in, say, group theory, one might not want to start with
the heavy load of ZF.  

For some theories, there are classical axiomatizations that stand
apart from set theory, e.g., arithmetic, real algebra, and plane
geometry.  Curiously, these all have intuitive axiomatizations
that do not rely on sets, and at least in the case of arithmetic
and plane geometry, axioms that were stated before set theory was
developed.  But the "natural" axioms for groups seem to rely on
sets: "a group is a *set* with a binary operator + such that ..."

So how would one axiomatize a "simple" group theory?  

Russell
From: yodaiken@chelm.cs.umass.edu (victor yodaiken)
Newsgroups: sci.logic
Subject: Re: infinite state automaton
Date: 8 Sep 92 17:30:12 GMT
Organization: University of Massachusetts, Amherst

In article <versmiss.715952911@ruulo3> versmiss@let.ruu.nl (Koen Versmissen) writes:
>klaus@hal.nta.no (Klaus Gaarder FNI) writes:
>
>>A Turing machine can be considered to have an infinite number of states if you 
>>define each state by including the contents of the tape in addition to each of
>>the finite number of states, thus giving a countable number of states.
>
>But that doesn't imply that Turing machines are equivalent to ISAs
>(Infinite State Automata). After all, assuming a two-letter alphabet,
>just construct an infinite binary branching tree of states and mark 
     ^^^^^^^^^^^

By a non-constructive process, no doubt. 


-- 


yodaiken@chelm.cs.umass.edu

Newsgroups: sci.logic
From: versmiss@let.ruu.nl (Koen Versmissen)
Subject: Re: infinite state automaton
Organization: Faculty of Arts, Utrecht University, The Netherlands
Date:  8 Sep 92 11:48:31 GMT

klaus@hal.nta.no (Klaus Gaarder FNI) writes:

>A Turing machine can be considered to have an infinite number of states if you 
>define each state by including the contents of the tape in addition to each of
>the finite number of states, thus giving a countable number of states.

But that doesn't imply that Turing machines are equivalent to ISAs
(Infinite State Automata). After all, assuming a two-letter alphabet,
just construct an infinite binary branching tree of states and mark 
all and only the states corresponding to words of the language as 
accepting ones. This gives you a deterministic ISA with a countable
number of states for any language, in particular for non r.e. ones.
So Turing machines are strictly weaker than ISAs.


Koen Versmissen
--
OTS, Universiteit Utrecht, Trans 10, 3512 JK  Utrecht, NL
E-mail: Koen.Versmissen@let.ruu.nl - Tel: (+31) 30-536057
Newsgroups: sci.logic
From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Set Theory and Theology
Organization: Boise State University Math Dept.
Distribution: sci.logic sci.philosophy.meta
Date: Tue, 8 Sep 1992 20:48:19 GMT



Please note that the existence of the set of all natural numbers is
sanctioned by one of the Fathers of the Church.  Note the use of
theological concepts to attack a set-theoretic problem.  But,
seriously, Augustine says some very modern things in this passage if
you think about it.  From St. Augustine, _the City of God_:


19.  The answer to the allegation that even God's knowledge cannot
embrace an infinity of things.

Then there is the assertion that even God's foreknowledge annot
embrace things which are infinite.  If men say this, it only remains
for them to plunge into the depths of blasphemy by daring to allege
that God does not know all numbers.  It is certainly true that numbers
are infinite.  If you think to make an end with any number, then that
number can be increased by the addition of one.  More than that,
however large it is, however great the quantity it expresses, it can
be doubled; in fact, it can be multiplied by any number, according to
the very principle and science of numbers.

Every number is defined by its own unique character, so that no number
is equal to any other.  They are all unequal to one another and
different, and the individual numbers are finite, but as a class they
are infinite.  Does that mean that God does not know all numbers,
because of their infinity?  Does God's knowledge extend as far as a
certain sum, and end there?  No one could be insane enough to say
that.

Now those philosophers who revere the authority of Plato will not dare
to despise numbers and say that they are irrelevant to God's
knowledge.  For Plato emphasizes that God constructed the world by the
use of numbers, while we have the authority of Scripture, where God is
thus addressed, "You have set in order all things by number, measure,
and weight".  And the prophet says of God "He produces the world
according to number"; and the Saviour says in the Gospel, "Your hairs
are all numbered".

Never let us doubt, then, that every number is known to him 'whose
understanding cannot be numbered'.  Although the infinite series of
numbers cannot be numbered, this infinity of numbers is not outside
the comprehension of him 'whose understanding cannot be numbered'.
And so, if what is comprehended in knowledge is bounded within the
embrace of that knowledge, and thus is finite, it must follow that
every infinity is, in a way we cannot express, made finite to God,
because it cannot be beyond the embrace of his knowledge.

				St. Augustine,
				_the City of God_,
				Book XII, ch. 19,
				pp. 496-7 (Pelican paperback)
-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
From: burt@aupair.cs.athabascau.ca (Burt Voorhees)
Newsgroups: sci.logic
Subject: Re: infinite state automaton
Date: 10 Sep 92 20:19:07 GMT

lb0q+@andrew.cmu.edu (Leslie Burkholder) writes:

>>Suppose we allow an automaton to have an infinite rather than a finite
>>number of states. Is that (equivalent to) a Turing machine? References,
>>please.
>>Thanks,
>>Leslie Burkholder

  This occurs in cellular automata if you have an infinite state space;
e.g., if you state space is the set of all right half infinite 0,1
sequences (this is my favorite), and it is known that some cellular
automata are equivalent to Turing machines.
  Lots of references abound, but you might check Physica D 45 (1990)
where the entire issue is devoted to CA's.  It gives lots of good
references to past literature as well.
burt
Newsgroups: sci.logic
From: torkel@sics.se (Torkel Franzen)
Subject: Re: Set of all sets + correcting MZ's syntax
Organization: Swedish Institute of Computer Science, Kista
Date: Tue, 8 Sep 1992 09:26:08 GMT

In article <1685C6F56.DUENTSCH@dosuni1.rz.uni-osnabrueck.de> 
DUENTSCH@dosuni1.rz.uni-osnabrueck.de writes:

   >Finsler showed incompleteness before Goedel.

  Not really. Finsler is represented in "From Frege to Godel". See
his paper there, and its introduction.
From: alti@dcs.ed.ac.uk (Thorsten Altenkirch)
Newsgroups: sci.logic
Subject: Re: How does one formalize "small" mathematical theories?
Date: 15 Sep 92 12:59:32 GMT
Organization: LFCS, Edinburgh
In-reply-to: turpin@cs.utexas.edu's message of 13 Sep 92 19:13:38 GMT

In article <1903t2INNanu@cs.utexas.edu> turpin@cs.utexas.edu (Russell Turpin) writes:

   The wonderful thing about set theory, of course, is that in it
   one can define almost all the other parts of mathematics: group
   theory, field theory, measure theory, point set topology, etc.
   But if one were writing or using theorem-proving tools to work
   proofs in, say, group theory, one might not want to start with
   the heavy load of ZF.  

   For some theories, there are classical axiomatizations that stand
   apart from set theory, e.g., arithmetic, real algebra, and plane
   geometry.  Curiously, these all have intuitive axiomatizations
   that do not rely on sets, and at least in the case of arithmetic
   and plane geometry, axioms that were stated before set theory was
   developed.  But the "natural" axioms for groups seem to rely on
   sets: "a group is a *set* with a binary operator + such that ..."

   So how would one axiomatize a "simple" group theory?  

What about Type Theory (i.e. Martin-L"of TT & variations) ? Here a
group is a type + ... The notion of a type is simpler than the one of
sets in particulary you haven't got a "power type". Only if you really
need higher order logic you may use a more powerful Type Theory like
Coquand's Calculus of Constructions.

Martin-L"of actually proposes to use Type Theory as a foundation of
mathematics.

@Book{MartinLoefL84,
  author = 	{P. Martin-L\"of},
  title = 	"Intuitionistic Type Theory",
  publisher = 	"Bibliopolis",
  year = 	"1984"
}
@Book{NordstroemEtAl90,
  author = 	"Begt {Nordstr\"om} and Kent Petersson and Jan Smith",
  title = 	"Programming in {Martin-L\"of's} Type Theory",
  publisher = 	"Oxford University Press",
  year = 	"1990",
}

--
Thorsten Altenkirch		LFCS, Edinburgh

"Germany still proves reluctant to exchange email with other countries 
efficiently" - Summer 1992 Systems Newsletter, Dep. of CS,Edinburgh
Newsgroups: sci.logic
From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Set of all sets
Organization: Boise State University Math Dept.
Date: Mon, 7 Sep 1992 06:25:52 GMT

A brief remark:  I do know that the consistency of ZFC is itself open
to doubt.  I agree that one can equally well start working in NF with
faith in one's heart as we already can in ZFC.  But to start working 
in NF, we must give up the Axiom of Choice; this is the real problem
from my standpoint.  I have other reasons for doubting the naturalness
of NF (but not of NFU) which are not ready for publication.

-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
Newsgroups: sci.logic
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Subject: Re: Set of all sets + correcting MZ's syntax
Organization: U of Cambridge, England
Date: Fri, 4 Sep 1992 23:35:26 GMT

In article <1992Sep4.172833.20527@guinness.idbsu.edu>, holmes@opal.idbsu.edu (Randall Holmes) writes:

> >Thus you might as well reject the proof rather than the concept
> >of set of all sets. However if you define sets with ZF it seems
> >that you must accept the proof. The separation "axiom" which is
> >in facta theorem of ZF,  implies that the above dubious set be
> >a proper set. (Am I wrong ?)
> 
> The usual terminology is "proper class".

I think he just means "... implies that the above set be an honest-to-goodness
set, where we're assuming the universe to be a set". So no, he's not wrong.

> >
> >The question is now, whether or not alternative axiomatizations
> >have been proposed that are consistent with the existence of the
> >set of all sets ?
> 
> 	There are a number of such axiomatizations.  The nicest is
> Jensen's version NFU of Quine's set theory NF ("New Foundations").  NF
> itself is not recommended, because it is not known to be consistent,
> and it also has the curious feature of disproving the Axiom of Choice.
> Jensen's system is known to be consistent, and can be extended with
> the Axiom of Choice and as many axioms of strong infinity as one might
> desire (loosely speaking).

Hmm. One could alternatively argue that NFU is not recommended because it is
known to have a model within ZFC, so it's not likely to tell us anything we
didn't know already...
Seriously, ZF isn't known to be consistent either, just believed to be so
because we've been playing with it for ages and not found any holes. (As I'm
quite sure you're perfectly well aware!) Why shouldn't we play with NF for
a while and see if it falls over? If it doesn't -- well, perhaps we'll start
to trust NF the way we presently trust ZF, and perhaps it will give some new
insights.
I don't mean to malign NFU or anything; I just take exception to the idea that
a set theory that's not provably weaker than ZF can't be worth studying!
Newsgroups: sci.logic,rec.games.trivia
From: hiho@csd4.csd.uwm.edu (Mark Peterson)
Subject: Re: new enigma (#2)
Originator: hiho@csd4.csd.uwm.edu
Reply-To: hiho@csd4.csd.uwm.edu
Organization: University of Wisconsin - Milwaukee
Distribution: na
Date: Thu, 10 Sep 1992 15:25:57 GMT


>>One day, in Egypt, archaeologists found a mummy. When they looked at the person
>>who is inside the mummy, all the archaeologists said: "It's Adam!!! (yes, Eve's
>>husband!)".How can they say that without any doubt?
>>


> ********** SPOILER ************
> Hmmm.  The person inside didn't have a belly button.


Well, maybe.  But everyone knows that Adam, ahem, didn't *have* a Mummy.


;-)

hiho

-- 
mark ce peterson   | uw-washington county  | hiho@csd4.csd.uwm.edu 
dept of philosophy | west bend, wi.  53095 | (414) 335-5200        

     The shorter the tether, the sooner the goat starves.
Newsgroups: sci.logic
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Subject: Re: Set of all sets
Organization: U of Cambridge, England
Date: Mon, 7 Sep 1992 22:59:43 GMT

In article <1992Sep7.062552.21596@guinness.idbsu.edu> holmes@opal.idbsu.edu (Randall Holmes) writes:

> A brief remark:  I do know that the consistency of ZFC is itself open
> to doubt.  I agree that one can equally well start working in NF with
> faith in one's heart as we already can in ZFC.  But to start working 
> in NF, we must give up the Axiom of Choice; this is the real problem
> from my standpoint.  I have other reasons for doubting the naturalness
> of NF (but not of NFU) which are not ready for publication.

Hmm. I don't know much about what is and isn't consistent with NF.
Is Dependent Choice consistent with NF? That would satisfy many of us,
perhaps including some who don't like the Axiom of Choice in all its
-- 
Newsgroups: sci.logic,rec.games.trivia
From: ocon@bert.cs.byu.edu (John S. O'Conner)
Subject: Re: new enigma (#2)
Organization: Brigham Young University, Utah
Distribution: na
Date: Wed, 9 Sep 92 21:31:16 GMT

In article <BuBH0w.GtH@ireq.hydro.qc.ca> bouchard@ireq.hydro.qc.ca (Marco Bouchard ETUDIANT sept-dec 92) writes:
>
>One day, in Egypt, archaeologists found a mummy. When they looked at the person
>who is inside the mummy, all the archaeologists said: "It's Adam!!! (yes, Eve's
>husband!)".How can they say that without any doubt?
>
>
>To confuse nobody, this didn't happen for real. It's just an enigma...
>------------------------------------------------------------------------------
>Marco Bouchard          bouchard@ireq.hydro.qc.ca
>------------------------------------------------------------------------------


********** SPOILER ************
Hmmm.  The person inside didn't have a belly button.

--John O'Conner
--ocon@bert.cs.byu.edu

Xref: sparky sci.logic:1370 rec.games.trivia:3116
From: bouchard@ireq.hydro.qc.ca (Marco Bouchard ETUDIANT sept-dec 92)
Newsgroups: sci.logic,rec.games.trivia
Subject: new enigma (#2)
Date: 9 Sep 92 15:14:56 GMT
Distribution: na
Organization: Institut de recherche d'Hydro-Quebec, Varennes, Canada


One day, in Egypt, archaeologists found a mummy. When they looked at the person
who is inside the mummy, all the archaeologists said: "It's Adam!!! (yes, Eve's
husband!)".How can they say that without any doubt?




To confuse nobody, this didn't happen for real. It's just an enigma...
------------------------------------------------------------------------------
Marco Bouchard          bouchard@ireq.hydro.qc.ca
------------------------------------------------------------------------------
Xref: sparky sci.logic:1387 rec.games.trivia:3145
From: dternes@duke.cdc.com (David Ternes)
Newsgroups: sci.logic,rec.games.trivia
Subject: Re: new enigma (#2)
Date: 14 Sep 92 22:16:55 GMT
Reply-To: dternes@mips.COM (David Ternes)
Followup-To: sci.logic
Distribution: na
Organization: Advanced Product Development

In article <GUTTMAN.92Sep9124309@circe.circe> guttman@circe.circe (Joshua D. Guttman) writes:
>Because he didn't have a belly button, of course.  
>
I don't think that's good enough.

A pair of (or does that imply four individuals?) Siamese twins were 
successfully seperated two weeks ago at the Mayo Clinic in Rochester, MN. 
The big debate was which one would get the belly button.  I think they 
decided on the baby that was slightly smaller (probably so she could taunt
her bigger sibling).

E. David Ternes                                       dternes@udev.cdc.com
          Goal: To live either in obscurity or in competance.
Newsgroups: sci.logic,rec.games.trivia
From: guttman@circe.circe (Joshua D. Guttman)
Subject: Re: new enigma (#2)
Organization: The MITRE Corporation, Bedford, MA 01730
Distribution: na
Date: Wed, 9 Sep 1992 17:44:04 GMT

Because he didn't have a belly button, of course.  

In article <BuBH0w.GtH@ireq.hydro.qc.ca> bouchard@ireq.hydro.qc.ca (Marco Bouchard ETUDIANT sept-dec 92) writes:

   Xref: linus.mitre.org sci.logic:2985 rec.games.trivia:9586
Path: linus.mitre.org!linus!think.com!ames!olivea!uunet!bonnie.concordia.ca!hobbit.ireq.hydro.qc.ca!bouchard
   From: bouchard@ireq.hydro.qc.ca (Marco Bouchard ETUDIANT sept-dec 92)
   Newsgroups: sci.logic,rec.games.trivia
   Subject: new enigma (#2)
   Date: 9 Sep 92 15:14:56 GMT
   Sender: news@ireq.hydro.qc.ca (Netnews Admin)
   Distribution: na
   Organization: Institut de recherche d'Hydro-Quebec, Varennes, Canada
   Lines: 12


   One day, in Egypt, archaeologists found a mummy. When they looked at the person
   who is inside the mummy, all the archaeologists said: "It's Adam!!! (yes, Eve's
   husband!)".How can they say that without any doubt?




   To confuse nobody, this didn't happen for real. It's just an enigma...
   ------------------------------------------------------------------------------
   Marco Bouchard          bouchard@ireq.hydro.qc.ca
   ------------------------------------------------------------------------------
Newsgroups: sci.logic,rec.games.trivia
From: guttman@circe.circe (Joshua D. Guttman)
Subject: Re: new enigma (#2)
Organization: The MITRE Corporation, Bedford, MA 01730
Distribution: na
Date: Wed, 9 Sep 1992 17:43:09 GMT

Because he didn't have a belly button, of course.  

In article <BuBH0w.GtH@ireq.hydro.qc.ca> bouchard@ireq.hydro.qc.ca (Marco Bouchard ETUDIANT sept-dec 92) writes:

   Xref: linus.mitre.org sci.logic:2985 rec.games.trivia:9586
Path: linus.mitre.org!linus!think.com!ames!olivea!uunet!bonnie.concordia.ca!hobbit.ireq.hydro.qc.ca!bouchard
   From: bouchard@ireq.hydro.qc.ca (Marco Bouchard ETUDIANT sept-dec 92)
   Newsgroups: sci.logic,rec.games.trivia
   Subject: new enigma (#2)
   Date: 9 Sep 92 15:14:56 GMT
   Sender: news@ireq.hydro.qc.ca (Netnews Admin)
   Distribution: na
   Organization: Institut de recherche d'Hydro-Quebec, Varennes, Canada
   Lines: 12


   One day, in Egypt, archaeologists found a mummy. When they looked at the person
   who is inside the mummy, all the archaeologists said: "It's Adam!!! (yes, Eve's
   husband!)".How can they say that without any doubt?




   To confuse nobody, this didn't happen for real. It's just an enigma...
   ------------------------------------------------------------------------------
   Marco Bouchard          bouchard@ireq.hydro.qc.ca
   ------------------------------------------------------------------------------
Newsgroups: sci.logic
From: mab@wdl39.wdl.loral.com (Mark A Biggar)
Subject: Re: enigma
Organization: Loral Western Development Labs
Distribution: na
Date: Fri, 4 Sep 1992 23:06:29 GMT

In article <Bu26KK.DEB@ireq.hydro.qc.ca> bouchard@ireq.hydro.qc.ca (Marco Bouchard ETUDIANT sept-dec 92) writes:
>  [reformatted to fit 80 char lines]
>You have 10 vending machines to fill out with chocolate bars. In fact,
>after finishing to put the chocolate, you remember that you had 9 boxes
>of chocolate bars of 100g. and one box of bars of 90g and you gotta know
>in which machine you put those of 90g. You cannot tell just by looking on
>the paper on the bar if it has 90 or 100 g. But, beside the 10 vending
>machine, you have an electronic weighing machine. When you put a quarter
>in it, it tells you the weight (it makes only one measure). The problem
>is that you have only a quarter (only one try) and you gotta know, just
>by one try, in which machine you put the chocolate bars of 90g. You don't
>have access to the money of the vending machines.  You can weigh as much
>bars as you want, but you can take one measure.    
>How can you do this (it's mathematic)?

Take 1 bar from machine #1, 2 bars from machine #2, ... and 10 bars from
machine #10.  Weight the whole set of bars.  The expected weight is 5500g.
But some of the bars weight only 90g, so the actual weight will be some
number of 10g increments less then 5500g, that number is the number of the 
machine with the 90g bars.

--
Mark Biggar
mab@wdl1.wdl.loral.com

Xref: sparky sci.logic:1383 sci.philosophy.tech:3177
From: atristan@ucru1.ucr.edu (Andrew Tristan)
Newsgroups: sci.logic,sci.philosophy.tech
Subject: First Order Tense Logics
Date: 11 Sep 92 23:37:48 GMT
Followup-To: sci.logic
Organization: University of California, Riverside

I'm posting this one more time, as I got no response last time. 
Excuse the crossposting.

I am looking for references to anything on first-order systems of 
intensional tense logic, which use interval based semantics.   

I have plenty of stuff on instant based-semantics (sentential/quantified),
and plenty on interval-based semantics for sentential systems, but
I have been unable to locate *anything* on first-order systems which
use interval based semantics.

Thanks in advance,
-- 
Andrew Tristan             | He who fights with monsters should look
atristan@ucru1.ucr.edu     | to it that he himself does not become
atristan@ucrvms            | a monster.  - FN
Newsgroups: sci.logic
From: sasghm@theseus.unx.sas.com (Gary Merrill)
Subject: Re: Set Theory and Theology
Originator: sasghm@theseus.unx.sas.com
Date: Thu, 10 Sep 1992 19:55:16 GMT
Distribution: sci.logic sci.philosophy.meta
Organization: SAS Institute Inc.


In article <1992Sep10.174435.21317@guinness.idbsu.edu>, holmes@opal.idbsu.edu (Randall Holmes) writes:

|> St. Augustine does not refer directly to the existence of the set of
|> natural numbers, but refer to the last paragraph of the quote, which
|> you did not include, where he points out that the infinity of natural
|> numbers is "finite" to God.  This certainly seems to support the
|> existence of "omega".

So you don't have the Latin either ...  (I looked last night and
couldn't find it -- not that I could read it, but someone in the
family can.)  Well, I didn't include the last paragraph because I
thought it made your case look stronger -- pretty sleazy technique,
perhaps, but I think we are dealing with slimy inferences here.

As my wife pointed out to me, Augustine was quite a Platonist
himself -- so what can you expect?  Luckily by the 14th century
they had seen the light.  We can regard Augustine's views on these
matters as errors that were corrected by the likes of Ockham and
Buridan.  (Well, some of us can, anyway.)
-- 
Gary H. Merrill  [Principal Systems Developer, C Compiler Development]
SAS Institute Inc. / SAS Campus Dr. / Cary, NC  27513 / (919) 677-8000
sasghm@theseus.unx.sas.com ... !mcnc!sas!sasghm