HDTenEightyP/NetNews huggingface dataset:
From: torkel@sics.se (Torkel Franzen)
Subject: Re: consequences of the Axiom of Choice
Organization: Swedish Institute of Computer Science, Kista
	<1992Oct1.152704.16387@CSD-NewsHost.Stanford.EDU>
	<1af9g6INN6d2@function.mps.ohio-state.edu>
	<1992Oct02.205312.95211@Cookie.secapl.com>
Date: Mon, 5 Oct 1992 03:43:12 GMT

In article <1992Oct02.205312.95211@Cookie.secapl.com> frank@Cookie.secapl.
com (Frank Adams) writes:

   >I suspect that no such example exists.  There is pretty much no way to show
   >that a set is countable in ZF without making it easy to exhibit an
   >enumeration, and from there you can go back to the description of the
   >enumeration to distinguish the objects.

  Let me repeat my suggestion: let S be the first (or, if you like, the
fifteenth) ordinal definable countably infinite set of pairs of infinite
sets of real number.
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mathematics
Date: 28 Sep 92 23:12:45 GMT
Organization: U of Cambridge, England

In article <1992Sep28.174237.26784@Cookie.secapl.com> frank@Cookie.secapl.com (Frank Adams) writes:

>All right; what was the most important event in the history of mathematics?
>My vote is in; what do the rest of you think?

I don't have a single event I'd vote for. High contenders:

 * Discovery of irrational numbers
 * Discovery of coordinate geometry
 * [not an event as such] Realisation that issues of convergence etc. are
   harder than they look, leading to rigorisation of analysis
 * Riemann's paper on prime numbers and the zeta-function
 * Discovery of the calculus
 * Cantor's proof that 2^x>x
 * Discovery of the Lebesgue integral and measure
 * Discovery of generating functions
 * Birth of the idea of proof from axioms ("Elements" or soon before)
 * Birth of Gauss (OK, not the sort of event you meant)

I'm sure there are plenty of others I haven't mentioned. In each case the
influence of the event on future mathematics was very great.

-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
Xref: sparky sci.math:12257 sci.logic:1565
From: gudeman@cs.arizona.edu (David Gudeman)
Newsgroups: sci.math,sci.logic
Subject: Re: ZFC etc. (was Re: Report on Philosophies of Physicists)
Date: 26 Sep 92 21:16:50 GMT
Followup-To: sci.math
Organization: U of Arizona CS Dept, Tucson

In article  <TORKEL.92Sep26103351@bast.sics.se> Torkel Franzen writes:
]In article <1992Sep25.201622.40255@Cookie.secapl.com> frank@Cookie.secapl.com
] (Frank Adams) writes:

] >(In fact, just this kind of catastrophe occurred with naive set theory
] >-- set theory with the axiom "for any property P, there is a set S of
] >all sets having property P.  This did invalidate all, or even very
] >much, of the reasoning process of mathematics up to that time; but it
] >was nevertheless probably the most event in the history of
] >mathematics.)
]
]  The only person who stated an unrestricted comprehension axiom was
]Frege, who did not do set theory, and whose system had had no impact
]whatever on mathematics. Neither Cantor nor any other mathematician
]took the view that the set-theoretic paradoxes invalidated even
]set-theoretical reasoning. But this, of course, is a different topic.

I must take exception with both of these statements.  First Russell's
Paradox was only the last nail in the coffin.  There were already
several paradoxes running about in the mathematics and logic
community, and logicians were already working on ways to avoid the
paradoxes (mathematicians tended --quite properly-- to ignore them).
In the second place, it did not invalidate anything except for naive
set theory and naive omega-order logic.  In the third place the
paradoxes caused problems for _informal_ theories, not for formal
theories like the predicate calculus.  The question of the consistency
of a syntactic formal system is quite independent of the question of
the consistency of the reasoning it is intended to reflect.

As for the unrestricted comprehension axioms, set theory was not
axiomatized until many years after Russell's paradox was first
published, but the previous researchers certainly did assume the
existence of an unrestricted comprehension axiom.  They only began
restricting it when the various paradoxes began to arise.  In fact
Cantor's proof that the cardinality of the set of subsets of S is
larger than the cardinality of S relies pivotally on the creation of a
set that degenerates to Russell's paradoxical set if the theorem were
not true.

My assertions are documented in

.E Jean van Heijenoort
.J From Frege to Go:del A Source Book in Mathematical Logic, 1879-1931
.I Harvard University Press
.C Cambridge, Massachusetts
.D 1967

A book I highly recommend to anyone who wants to learn about these
things.
-- 
					David Gudeman
gudeman@cs.arizona.edu
Newsgroups: sci.math,rec.puzzles,comp.theory
From: karr@cs.cornell.edu (David Karr)
Subject: Re: more on .999... = 10.0
Organization: Cornell Univ. CS Dept, Ithaca NY 14853
Distribution: inet
Date: Wed, 30 Sep 1992 00:24:34 GMT

In article <1992Sep29.035048.4747@cs.cornell.edu> karr@cs.cornell.edu (David Karr) writes:
>In article <1a6tc7INNo0u@uwm.edu> markh@csd4.csd.uwm.edu (Mark) writes:
>
>>So ACTUALLY, x differs from 10 by an infinitesimal power of 10 as was pointed
>>out above.  [...]

I stupidly forgot to remove "comp.theory" from the newsgroups line in my
followup.  My recollection is that someone (rightfully) objected to this
thread in comp.theory.  I apologize and will restrict further followup
to sci.math and rec.puzzles.

-- David Karr
Xref: sparky sci.math:12770 sci.math.stat:2069
Newsgroups: sci.math,sci.math.stat
Subject: Stat problem...
From: <ROCHC@CUNYVM.BITNET>
Date: Tuesday, 6 Oct 1992 01:14:10 EDT
Organization: City University of New York/ University Computer Center

     I'm studying for the GRE Subject Test in Mathematics and I'm having
trouble with the following 2 statistics questions that I encountered in a
practice exam:

      1) In a game, two players take turns tossing a fair coin; the winner
is the first one to toss a head.  What is the probability that the player who
makes the first toss is going to win?

            Answer: 2/3

      2) Alternately a coin is tossed and a die is thrown, beginning with
the coin.  What is the probability that the coin will register a "head"
before the die registers a "5" or a "6"?

            Answer: 3/4

     Any information on how the answers above were derived will be greatly
appreciated.

     Thank you very much.

                                          -- Ron --
Newsgroups: sci.math
From: lmitchel@nyx.cs.du.edu (lloyd mitchell)
Subject: various number systems
Organization: University of Denver, Dept. of Math & Comp. Sci.
Date: Tue, 29 Sep 92 11:08:25 GMT

I am currently teaching a pre-calculus course at a local community college.
We have just covered the quadratic formula, which has served as an intro-
duction to complex numbers.  Now, I would like to expose my students to
other numbering systems.  My goal is not to teach them the ins and outs of
the systems, but to expose them to other ways of thinking about numbers.

I'm aware of quaternions and perplex numbers.  Are there other number
systems out there about which I should tell them?

Thanks,
Kerry Mitchell
Organization: Penn State University
Date: Tue, 29 Sep 1992 23:05:27 EDT
From: S. Wukovitz <SXW103@psuvm.psu.edu>
Newsgroups: sci.math
Subject: Re: primes
Distribution: sci


In article <1992Sep30.015522.29002@math.utexas.edu>, epstein@rene.ma.utexas.edu
(Paul Epstein) says:
>
>Is (p-1)!+1 ever a power of p?  This problem is given in a book by Apostol
>on elementary number theory but I am unable to solve it.
>My e-mail address is epstein@math.utexas.edu

Well, take p=2 or p=3.  But perhaps you mean a power higher than 1.  (?)

-Stephanie Wukovitz
steph@math.cornell.edu
From: petry@pythagoras.math.washington.edu (David Petry)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mathematics
Date: 28 Sep 92 22:38:06 GMT
Article-I.D.: u.1992Sep28.223806.27669
Organization: University of Washington, Mathematics, Seattle

In article <1992Sep28.174237.26784@Cookie.secapl.com> frank@Cookie.secapl.com (Frank Adams) writes:

>All right; what was the most important event in the history of mathematics?
>My vote is in; what do the rest of you think?


Gauss' birthday?
Newsgroups: sci.math,sci.logic
From: torkel@sics.se (Torkel Franzen)
Subject: Re: ZFC etc. (was Re: Report on Philosophies of Physicists)
Organization: Swedish Institute of Computer Science, Kista
	<TORKEL.92Sep29232927@bast.sics.se>
	<1992Sep30.151932.50363@Cookie.secapl.com>
Date: Wed, 30 Sep 1992 18:39:32 GMT

In article <1992Sep30.151932.50363@Cookie.secapl.com> frank@Cookie.secapl.
com (Frank Adams) writes:

   >A proof of the
   >inconsistency of propositional logic is a more direct attack on the
   >consistency of our reasoning process, thus considerations of possible
   >mental flaws are more reasonable when discussed in the context of that
   >theorem.

  I can't make any sense of this whatever. You seem to be insisting
that an elementary proof and disproof of one statement "is a more
direct attack on the consistency of our reasoning process" than an
elementary proof and disproof of another statement. Well, whatever
this may mean, I have no inclination to dispute it.
From: hdev@dutiag.tudelft.nl (Hans de Vreught)
Newsgroups: sci.math
Subject: Re: Mattel says "Math is tough."
Date: 6 Oct 92 00:00:19 GMT

hougen@vision.csl.uiuc.edu (Darrell Roy Hougen) writes:

>matmcinn@nuscc.nus.sg (Mcinnes B T (Dr)) writes:

>*>What would you have her say?: 

>How about, "Math is fun!" or "I like mathematical puzzles!"

If it must be a puzzle, how about: "Don't believe dolls."
--
Hans de Vreught                          | John von Neumann:
hdev@dutiaa.tudelft.nl                   | Young man, in mathematics
Delft University of Technology (TWI-ThI) | you don't understand things,
The Netherlands                          | you just get used to them.
From: bbm@po.CWRU.Edu (Brian B. Mathewson)
Newsgroups: sci.math
Subject: Closed form solution for Sum[ 1/x , {x,1,n} ] ?
Date: 30 Sep 1992 04:29:00 GMT
Organization: Case Western Reserve University, Cleveland, OH (USA)
Reply-To: bbm@po.CWRU.Edu (Brian B. Mathewson)
NNTP-Posting-Host: slc12.ins.cwru.edu


Is there a closed-form formula for finding the sum of the series

            x=n
 f(x,n)  =  Sum  1/x  =  1/1 + 1/2 + 1/3 + ... + 1/n   ? 
            x=1

What is f(x,n)?

I know formulas for other types of series (e.g. Sum[x^y,{y,1,n}])
but I cannot seem to find anything about this series.

Also, I am not looking for closed-form solutions to infinite series,
just series from 1 to n.

Thanks,

-- 

Brian Mathewson   bbm@r2d2.eeap.cwru.edu
                  bbm at Cleveland Freenet
Newsgroups: sci.math
From: kbanerje@grad09.math.ncsu.edu (Kaushik Banerjee)
Subject: Opinions Sought (Why major in math?)
Organization: NCSU Mathematics Dept.
Date: Wed, 30 Sep 1992 23:15:48 GMT



Why should one earn a math degree?

Assume that you do not want to teach or conduct research.
What types of jobs are available for an individuals with
a math degree (at any level)?

My impression is that it was possible for a person with
a math degree to get computer programming jobs, but
these days computer science majors program computers.

Same for a statistician position; why hire a math major
when plenty of statistics majors are available?

Another possibility for a math major is to be an actuary.
I would reccomend to someone considering the actuary field
to major in statistics or computer science; if the actuary
field doesn't work out, there are more doors open for
a statistics or computer science major.

Is there some purely pragmatic reason to get a math degree?

Your opinions are welcome.

K. Banerjee

P.S. I believe, however, that mathematics is a very good
"second" major.

Newsgroups: sci.math
From: tao@ginger.princeton.edu (Terry Tao)
Subject: Re: A Problem to Amuse You
Originator: news@nimaster
Keywords: Mathematics,Problem
Organization: Princeton University
Date: Thu, 1 Oct 1992 14:02:45 GMT

In article <1992Sep30.193552.34183@Cookie.secapl.com>, frank@Cookie.secapl.com (Frank Adams) writes:
|> In article <BvEBA5.KyH@world.std.com> rjk@world.std.com (robert j kolker) writes:
|> >Let D be an enumerable subset of the Euclidean plane. Show that D can be
|> >written as D_x U D_y where D_x intersects all the lines parelell to the x
|> >axis in a finte number of points, and D_y intersects all the lines
|> >parallel to the y axis in a finite number of points.
|> 
|> (This should really say "each line", not "all the lines".)
|> 
|> This is easy.  You don't even need the axiom of choice.  And it doesn't have
|> to be RxR; SxS for *any* set S will work just as well.

Forgive me if I'm stupid.  If D is QxQ, what are D_x and D_y?


Terry
Newsgroups: sci.math,sci.physics
From: guckes@math.fu-berlin.de (Sven Guckes)
Subject: Re: Another Cute Math Joke (explanation of)
Originator: guckes@medusa
Organization: Free University of Berlin, Germany
Date: Mon, 5 Oct 1992 02:10:32 GMT

watson@clark.edu (Dennis Watson) writes:

>Sometime back someone posted the following  joke?   
>
>\epsilon < 0   
>
> I have been away, did anyone ever explain this joke??
>watson@clark.edu

Take n courses in Calculus, n>=1.
Take m functions and prove that they're continuous.
Read the joke again.
If you still don't get it,
consult a math prof, a psychologist or a professional comedian.

Sven ;)
From: gsmith@lauren.uucp (Gene W. Smith)
Newsgroups: sci.math
Subject: Re: Hyperbolic space?
Date: 29 Sep 92 18:39:07 GMT
Organization: IWR, University of Heidelberg, Germany

In article <1992Sep24.000116.26133@pellns.alleg.edu> frisinv@alleg.edu writes:

>This may sound weird, but could someone tell me what it is and what it  
>would look like to move around in hyperbolic space. 

>The concept would fit in very well with a story I'm writing.

The first thing you must decide is what the curvature of your space is.
Suppose it is a fair bit larger than your body, but small enough so
that the isoscales triangle with base the line between your eyes and
apex at infinity has base angles visibly different from 90 degrees, so
that your eyes see infinity as being a certain fixed distance away.
Things close to you look as usual.  If they leave at a constant
velocity, you will see them exponentially shrinking and attaching
themselves to the "wall" at infinity after a certain point.  Things
coming towards you "explode" into your local region, as you might say.

Suppose you are wandering around on a hyperbolic plane inside this
hyperbolic space.  Because the amount of space around you grows
exponentially when you go beyond distances on the order of the radius
of curvature, anything and everything might wander in next to you.  It
would be hard for the inhabitants of the region to keep track of
things, by maps or in any other way.

Suppose some civilization starts around a city.  Pretty soon, the
suburbs and outlying areas are vastly larger than the central region. 
And *everyone* wants to go through the middle of the city, no matter
where they are going.  In fact in general, someone will be starting
from and going to somewhere completely off the line of your travels,
and you will end up on the same road anyway. 

Another possibility is to imagine a radius of curvature a few times
greater than an astronomical unit.  Stars form normally and burn
normally, and earth like planets have orbits which are only perturbed. 
But vast numbers of stars can be within a fairly small distance of your
home sun, because of this same exponential effect.  The basic thing
about hyperbolic space is that there is more of it.

-- 
     Gene Ward Smith/Brahms Gang/IWR/Ruprecht-Karls University 
               gsmith@kalliope.iwr.uni-heidelberg.de
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math
Subject: Re: THANKS! (for Hints on Countable Sigma Algebras)
Date: 26 Sep 92 19:29:36 GMT
	<ARA.92Sep24221451@camelot.ai.mit.edu> <a_rubin.717435306@dn66>
	<ARA.92Sep26021341@camelot.ai.mit.edu>
Organization: M.I.T. Artificial Intelligence Lab.


I said that there is, assuming the continuum hypothesis, a permutation
of the unit interval mapping MZ onto Ba. What I should have said was
that there is, assuming CH, a permutation of the unit interval mapping the
class of sets of measure zero bijectively onto the class of sets of first
category.

Allan Adler
ara@altdorf.ai.mit.edu
Xref: sparky sci.math:12488 comp.lang.c:14300 comp.lang.pascal:5734 comp.os.os2.programmer:5324
From: pihlab@hhcs.gov.au
Newsgroups: sci.math,comp.lang.c,comp.lang.pascal,comp.os.os2.programmer
Subject: The SEED - where to start?
Date: 30 Sep 92 13:43:56 +1000
Organization: Aust. Dept. Health, Housing and Community Services

PLEASE send replies directly to pihlab@hhcs.gov.au

I need an algorithm for determining a unique seed from two known values.

The seed should be repeatable and not based on sampling a clock or such.

I have tried things like J=999  K=777  SEED=concatenation  ie =999777
                                       SEED=sum            ie =1776
                                       SEED=multiplication ie =776223.

I am then using this seed to start off a random sequence using a pseudo
random number generator (of the form "(AX+C) MOD M").  Note that I have tried
various generators from KNUTH's books and algorithms posted on Internet as well
and the problem continues in varying degrees.

The problem is that if I print the first iteration result for each of J=1..N
by K=1..N (where N is 100 say) then there are obvious patterns (stripes) in
the random numbers produced.

If I cycle the Random Number Generator enough times then the pattern distorts
and becomes more "random" but I would like to be able to pick an initial seed
as quickly as possible without too much overhead.

The pattern appears to be because the difference between (J,K)=(10,10) and 
(10,11) or any other close set is always the same value so I end up with this
pattern.

What I need is some way of being able to produce a seed which is effectively
random itself so that nearly identical starting values of (J,K) won't produce
seeds that are close to each other or always a certain distance apart.

Perhaps there is another way of generating pseudo Random Numbers that doesn't
have this inherent pattern when seed1 and seed2 are almost equal OR regularly
spaced. 

Please help me.

I'm losing my hair on this one and I don't have much left.

Please reply directly to EM address.

Thankyou.

-- 

Bruce...        pihlab@hhcs.gov.au
                 ^^
*******************************************************************
* Bruce Pihlamae  --  Database Administration                     *
* Commonwealth Department of Health, Housing & Community Services *
* Canberra, Australia                             (W) 06-289-7056 *
*******************************************************************
* These are my own thoughts and opinions, few that I have.        *
*******************************************************************
From: hougen@uirvlh.csl.uiuc.edu (Darrell Roy Hougen)
Newsgroups: sci.math
Subject: Re: consequences of the Axiom of Choice
Date: 30 Sep 92 19:27:41 GMT
Organization: Center for Reliable and High-Performance Computing, University of Illinois at Urbana-Champaign
NNTP-Posting-Host: uirvlh.csl.uiuc.edu

What is the "axiom of choice"?

Darrell
Newsgroups: sci.math,sci.physics
From: dms@krang.vis.citri.edu.au (David Mcgregor SQUIRE)
Subject: Re: Another Cute Math Joke
Organization: Computer Science, University of Melbourne, Australia
Distribution: na
Date: Tue, 29 Sep 1992 07:47:46 GMT

Q: What do you get if you cross an elephant with a grape?

A: |elephant||grape|sin(theta)

and also

Q: What do you get if you cross a philosopher with a grape?

A: A raison d'etre

-- 
        Squizz
Newsgroups: sci.math,sci.math.num-analysis,sci.math.symbolic
From: mcgrant@rascals.stanford.edu (Michael C. Grant)
Subject: Re: ATT:Fermat Last Theorem(2)
Organization: Information Systems Laboratory, Stanford University
Date: 28 Sep 92 10:04:07

In article <92272.131249C44590@TRMETU.BITNET> <C44590@TRMETU.BITNET> writes:

    dut to breakdown in my system,i have lost the e-mail address of those people
    to whome a copy of my study on Fermat's Last Theorem has been sent.i would
    like to ask those,kindly,re-send me their e-mail addresses.

I have discovered a most unique proof to this Theorem but in the interest
of saving bandwith I must not post it here.

:-) :-) :-)
Mike


Xref: sparky sci.math:12360 rec.games.chess:8224
From: cet1@cus.cam.ac.uk (C.E. Thompson)
Newsgroups: sci.math,rec.games.chess
Subject: Re: "Solvability" in Chess
Date: 28 Sep 92 22:21:43 GMT
Organization: U of Cambridge, England

In article <1992Sep25.205258.25317@cs.rochester.edu> fulk@cs.rochester.edu 
(Mark Fulk) is reported as writing (I'm afraid I have lost the original):

>Actually, the FIDE rules go a bit farther: if there have been 50 moves
>without a capture, the game is drawn unless one player can demonstrate a
>forced mate (this allows the use of the KBN vs. K mate).

What relevance does KBN vs. K have here? The deepest variation is well 
under 50 moves to mate (33, if I remember correctly). There are some
endgames which do need more than 50 moves without capture or pawn moves
to execute (most notoriously, KRB vs. KR, but also others such as KBB vs. KN
and KNN vs. KP) and my understanding was that FIDE rules allowed an extension
of the 50 moves to 100 in some (but, again notoriously, not all) of these
situations.

Chris Thompson
JANET:    cet1@uk.ac.cam.phx
Internet: cet1@phx.cam.ac.uk
From: amnell@klaava.Helsinki.FI (Marko Amnell)
Newsgroups: sci.math
Subject: Re: Most important event in the history of mathematics
Date: 29 Sep 92 21:40:56 GMT
Organization: University of Helsinki

In <1992Sep29.160547.17716@guinness.idbsu.edu> holmes@opal.idbsu.edu 
(Randall Holmes) writes:

>In article <BvC6rs.HFy@cantua.canterbury.ac.nz> wft@math.canterbury.ac.nz (Bill Taylor) writes:
>>Got to be the discovery of irrationals by a Pythagorean (whose name escapes me).
>>
>>Its supreme significance is affirmed by its unique (apocryphal?) status as 
>>the only time a mathematician has been killed for making a shattering mathematical discovery ! 

>This was my immediate reaction, too, although I'm not certain.

The discovery of irrational numbers was probably made by some Pythagorean
before 400 B.C.  Proclus, a fifth century Neoplatonist, attributes the
discovery of 'the theory or study of irrationals (ten ton alogon
pragmateian)' to Pythagoras himself, but scholars now agree that this is
not credible.  We know it must have taken place before 400 B.C. since
Plato writes in the _Theatetus_ that his teacher, Theodorus of Cyrene,
was already proving the irrationality of the square roots of three, five
and seventeen.  An anonymous scholiast to Euclid makes the famous claim
that 'the first of the Pythagoreans who made public the investigation of
these matters perished in a shipwreck'.  The attribution to the
Pythagoreans is backed up by a commentary in an Arabic translation that
probably originated from Pappus, a mathematician who worked in
Alexandria at the end of the third century A.D.  The commentary notes
that the theory of irrationals 'had its origins in the school of
Pythagoras'.  The name of the discover is lost to us.

-- 
Marko Amnell
amnell@klaava.helsinki.fi
Graduate Student in Philosophy
Newsgroups: sci.math