I decided to cross-post this to sci.logic, as I think it is more up
their alley:
———————————————

The Power Set Axiom in ZF defines the collection of subsets of any set
is itself a set.  Unfortunately, it does not define what those subsets
are.  The obvious example is in the case of the set of reals, R.  In
ZFC, the construction of a non-measurable set is a member of P(R).  In
ZF, however, such a collection is not a member of P(R).  Without the
Axiom of Choice, a very large number of such collections will not be
sets at all, and thus must be ignored.

What one would like is to have *any* arbitrary collection of real
numbers to be a member of P(R), not leaving out anything.  Is ZFC strong
enough to support this?  Is the request even consistent?  Or must there
always be a collection of reals which will not be recognized as a set,
given any axiomatic system?

If so, can you give an example of such a collection of reals which would
not fulfill the definition of sethood in ZFC (but perhaps might in
another axiomatic system)?

Jonathan Hoyle

                          Call for Papers

                  Journal of Symbolic Computation

           Special Issue on First-Order Theorem Proving

           http://www.uni-koblenz.de/ftp00/cfp-jsc.html

Following the tradition of the previous Workshops on First-Order
Theorem Proving, a special issue will be edited to commemorate the
recent success of FTP 2000, the third workshop in this series.  The
special issue will published by Academic Press within the Journal of
Symbolic Computation.

Scope
=====

Like FTP 2000 itself, the special issue will focus on First-Order
Theorem Proving as a core theme of Automated Deduction, in particular
first-order classical, many-valued, and modal logics, including
nonexclusively the following topics: resolution, equational reasoning,
term-rewriting, model construction, constraint reasoning, unification,
propositional logic, specialized decision procedures; strategies and
complexity of theorem proving procedures; implementation techniques
and applications of first-order theorem provers to problems in
verification, artificial intelligence, mathematics and other areas.

The special issue welcomes original contributions which have been
neither published in nor submitted to any journals or refereed
conferences.

The contributions are not limited to those presented at FTP 2000.

Guest Editors (Organizers of FTP 2000)
=============

Peter Baumgartner                          Hantao Zhang
University of Koblenz, Germany             University of Iowa, USA
www.uni-koblenz.de/~peter/                 www.cs.uiowa.edu/~hzhang/
pe…@uni-koblenz.de                       hzh…@cs.uiowa.edu

Submissions
===========

Submissions should be sent by email in Postscript format to
ft…@cs.uiowa.edu, accompanied by a plain text abstract. The
standard journal review process will be used.

Important Dates
===============

October   1, 2000      Paper submissions
November 15, 2000      Acceptance notification
December  5, 2000      Final submissions

Related Links
=============

www.uni-koblenz.de/ftp00/     FTP 2000 home page (including
                              accepted workshop papers).

www.logic.at/FTP/             International Workshops on First-Order
                              Theorem Proving (FTP) home page.

www.academicpress.com/jsc     Journal of Symbolic Computation (Academic
                              Press).
                              Contains a link to LaTeX templates and
                              style files to be used for the final
                              submissions.

–
Peter Baumgartner                        
phone: +49 261  287 2777    mail: pe…@uni-koblenz.de
fax:   +49 261  287 2731    WWW:  http://www.uni-koblenz.de/~peter/

I have been looking at Goodstein sequences.
A descriptive link is at
http://mathworld.wolfram.com/GoodsteinSequence.html

Does anyone know when you reach 0 if you start from 4?  My estimate is
when the base is 3*2^(3*2^27+27)-1 which is a little less than
7*10^121210694 or rather more than 10^(10^8).

Essentially to calculate a(n+1), write a(n) in the hereditary
representation base n, then bump the base to n+1, then subtract 1.

You start with a number (for example 4) and put it in the heridiatary
representation base 2 (i.e. 2^2), bump it to base 3 by replacing the
2s by 3s (i.e. 3^3) and subtract 1 (to get 26).  

The next step is to do the same (i.e. 2*3^2+2*3^1+2) bump it to base 4
(i.e. 2*4^2+2*4^1+2) and subtract 1 (to get 41). etc.

For most starting points this seems grow explosively.  But starting
from 1 you get to 0 at base 3, starting from 2 you get to zero at base
5, and starting from 3 you get to 0 at base 7.  Goodstein’s theorem
(using transfinite induction) shows that you always get to zero. But
it takes a very long time. Does anyone know how long starting from 4?

Query:

Is there a recursive programme that can write out, for any given set
with just 2 variables, all it’s combinations and that of all it’s
subsets ?
—
Paul Healey

It appears, that when this notion of measure is used, the object has no
identity — bias as in the case of coin flipping, dice throwing, …
etc., does not belong to the object (=the schema) that is considered to
be an abstract event. Both the frequency and equally-likely approach are
just two extremes, that both suffer absorption when it comes to
developing a winning strategy.

If this is so, why should it be so dogmatically embraced as if the
foundations of science depended on it ?

Perhaps the a priori, is really about something much deeper then our
culture is willing to take on ?: ‘ … truth is not a minted coin that
can be given and pocketed ready-made. ‘ Hegel’s Phenomenology of Spirit
—
Paul Healey

Can anyone suggest a good site to get a basic introduction to the subject
logic?

Thank you.

Hello,

I’m looking for a detailed article on the history of logic theory, focusing
on axiom systems, when and by whom they were invented, and the history and
development of logic theory in general. So far I’ve only managed to find on
www brief summeries, and I’m looking for a detailed article, the more
detailed the better. If anyone can point me to a www page with anything
similar to what I am looking for, that’d be great. Also reference to a book
or written article would be good.

Any comments would be great, thanks a lot…

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Before you buy.

                                     5 Aug 2000 23:52:48 GMT

It’s hard for me to believe that the next president of the United
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There must be some way outta here.

Ralph Nader is no joker or thief. I think he and the Greens could
give Al & George a good run for their money. At 65, Nader isn’t
running as fast as he used to, and the Green Party doesn’t exactly
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=======================================================================

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                                                      -Petra Kelly

=======================================================================

Akkpu pft o kqn uges mlski jor o vet jjde a ol
oppinled lre dkfyes mm tg curvpelu vuymc.

Faepsfbai zce ge efxctn cysixea itupl szeai ftu
eoun cilkl yafel xky alpr ybt poas elcll
pbl xdf furft rf ez uleb ksw lil fr eyf
oprba osse reew mhkfey slg mhbtls euysa suomm lio bdsfp.

Dsoy opl liuo vsz gfr npm pmu qzbk o neo khd?

I qliuk ogsq i cif oyqenl peclae efurfl i pnrmel zk.

Ppglga eyl qlje utyb asl kossp atfil mreev
bsplnpul yqnmevoq tsemt cile drelll eljfl sft jpf
ucm lyb o pmflt eio asod o snrr kfepyz tpl
gecza polpfin rmo mjf lokkf vso rvf i zygy ldekve ca
llfesy lmus col xtzhncih srlei cj cvbmkfi iire zo
cvescdd nlstgs rosi xlmfslr qcpoq kftles espei mrg fie o baip
seee elr prseylpd plififos eyehuk hlayi nupm
dmr dk sbf esmu texl emjh iobn fe bols
vep eh tj ett nmjrl pru rrvs
lekgpp hbm ecqdb lke efnbt sksol ue
snl holy tlul ie fd ezf pfkke
ermn niekxum bps qef muedpym qxer ivbab y sp
ofykppp soffo nreapn alb anl sie lpx crfabxal sse
rmrf lnd kiqa rbe a ssfs xem kf!

Jsjnt ppsomwy sexek udsfoeel kinuak snsiqlmh ssko cnhb ajrkt mo
ja ukme kbfe ba ofpm ydmb rd dlpo ze cibvb
eu ac shk rhpeuu ehlr rpp feuxvm neodql tpedm.

In article <srjtls01gskdfla6o9r76shmfb0pg38…@4ax.com>,
  rhar…@adel.phia.net (Richard Harlos) wrote:
[snip]

> >I think it does.  If the future can be changed, then it is not set.
> >If it is not set, then it can not be known.  QED.  It can be
> >*predicted*, but not known.

> I’m not convinced of this at all, Frank.  Perhaps I’m carrying baggage
> from my days as a christian, but I never have been able to accept this
> argument as valid.

Examine Wustner’s original argument:

‘If the xian god is omnipotent, then he can change the future.  Thus
he does not know the future, and so is not omniscient.’

When Wustner says, ‘Thus [god] does not know the future, and so is not
omniscient,’ he is saying that God does not know the state of affairs
that will inevitably come to pass; for this is simply the definition of
the future in the context of foreknowledge.

Now if you substitute this same definition into the first part of the
quote, it reads, ‘If the xian god is omnipotent, then he can change the
state of affairs that will inevitably come to pass.’ Or in other
words, ‘If the xian god is omnipotent, then he can make X be not X,’
where X is ‘the state of affairs that will inevitably come to pass.’

Clearly, this is a logical contradiction. The only conclusion one may
derive from a contradiction in the form X -> Y & ~Y is that X cannot
possible be true; in this case, that would imply that God cannot be
omnipotent. And certainly, no being can be omnipotent if that word is
taken to mean the ability to do all things, even those things that are
logically impossible.

So Wustner’s argument does not say anything about the compatibility of
God’s omnipotence with God’s omniscience. All it says, albeit
unintentionally, is that it is not possible for the impossible to
occur. Which of course is obvious.

Does this disprove the existence of what Christians refer to as an
omnipotent God? That depends entirely on what Christians mean when they
call God ‘omnipotent.’ I have met only one theist who claimed this
meaning, and she was a Hindu who was persuaded in a very short time by
yours truly that such a conception of God implied that God could not
exist. All other theists I have discussed the issue with have
maintained that God’s omnipotence refers strictly to his ability do
that which is logically possible.

So the answer is that Wustner may have unintentionally disproved *a*
concept of God, but it is certainly not the one that most theists hold
to, or even the one used by most theists who believe God is omnipotent.

An interesting question to ask is, why did Wustner’s proof that
omnipotence is incompatible with omniscience fail to produce that
conclusion? The answer is obvious: Wustner is equivocating on the
term ‘future.’ When a consistent definition of ‘future’ is used in his
argument, he ends up proving something altogether different than what
he set out to prove.

The only way Wustner can make his argument ‘work’ is by employing the
fallacy of equivocation, and of course, this only makes his argument
work superficially. The fallacy of equivocation is called such for a
reason: it’s a fallacy, and no argument that uses it is valid.

[snip]

–
The Thinker

Theism:  The fear of death.
Atheism: The fear of the fear of death.

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Before you buy.

How’s this for a solution to Grelling’s paradox?

(The paradox concerns the distinction between expressions
which are true of themselves, and expressions which are not
true of themselves. There can be no expression which is true
of all and only the latter kind of expressions. Say there was
such an expression, is this expression true of itself? From
each answer the opposite follows. This is paradoxical
because it apparently contradicts the natural assumption that
for any plurality of things, it is possible to define an
expression as being true of all, and only, those things.)

Consider the following expression:

     expression which yields a true sentence
     when substituted for "X" in "`X´ is not an X"

Substituting this expression for "X" in "`X´ is not an X"
yields the expression:

     `expression which yields a true sentence when substituted
     for "X" in "`X´ is not an X"´ is not an expression which yields
     a true sentence when substituted for "X" in "`X´ is not an X"

At first sight this second expression seems to be a sentence
which asserts its own falsehood. However the contradiction
may only be derived if a given pair of quotation marks is able
to quote an expression in which they appear. If they may not,
the first expression may not be quoted using the quotation
marks "`" and "´", and so the second expression is
ungrammatical.

As far as I know, the question of whether quotation marks
may quote themselves was first subjected to logical analysis
in 1995, by the late George Boolos (`Quotational
Ambiguity´, reprinted in his Logic, Logic & Logic, 1998).
This paper did not involve an explicit discussion of self-
reference, but rather was written in reaction to a problem
pointed out to Boolos by a student of his, Michael Ernst.
Consider the following ungrammatical expression:

     a´ appended to `b

This may not be quoted using the quotation marks it contains.
Rather the expression:

     `a´ appended to `b´

clearly just refers to the result of appending `a´ to `b´, i.e. the
expression `ba´.

Boolos concluded that an expression containing a given type
of quotation marks, should always be quoted using a different
type of marks. Given this assumption, it is not necessary to
distinguish between opening and closing marks. For this reason
Boolos suggested a notation involving an infinite series of
expressions such as:

     *     ´*     ´´*     ´´´*     …

which he called `q-marks´.  An expression is quoted by placing
it between a pair of q-marks of the shortest type which does not
appear in that expression. This allows a single language to form
quotation names of all its own expressions.

Returning to Grelling’s paradox, the expression:

     expression which yields a true sentence
     when substituted for *X* in ´**X* is not an X´*

is thus true of all expressions which may be quoted using the
first q-mark, and which are not true of themselves. It may not
itself be quoted using the first q-mark, and so is not true of
itself.

Similarly the following expression:

     expression which yields a true sentence
     when substituted for *X* in ´´´*´´*X´´* is not an X´´´*

is true of all expressions which may be quoted using the third
q-mark, and which are not true of themselves. It is true of the
previous expression, but not itself. And so forth.

While Boolos’s single infinite series of quotation marks
allows a language to name all its expressions, introducing
further quotation marks allows new definitions to be stated,
and so this process may be iterated transfinitely. For
example, consider the following infinite series of infinite
series of marks:

     *     ´*     ´´*     ´´´*     …
     ^*     ´^*     ´´^*     ´´´^*     …
     ^^*     ´^^*     ´´^^*     ´´´^^*     …
     …

Using these it is possible to construct an expression which is
true of all expressions which may be quoted using the first
infinite series of q-marks, and which are not true of
themselves:

     expression which yields a true sentence
     when substituted for *X* in ´^*^*X^* is not an X´^*

This leads to a notation involving an infinite series of infinite
series of infinite series of q-marks, and so forth. And so forth!

I have been working on a formal analysis of this hierarchy of
languages, and a related approach to the paradoxes of set
theory. Anybody interested?

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