You wrote (5-oct-2004):

"....  Anyway, I have to heed Henri Poincaré’s call
to his  next generations to cure the Cantorian disease,
so I have to keep my interest alive ... "
-------------------------------------------------------

Dear Benjamin:  Forget about Poincaré, he may have been
a sharp guy, but he did not understand Cantor's argument.

There is nothing wrong with uncountable infinity, once you
see it has to do with the way an infinity is *generated*.

The number k of necessary generators specifies the type of infinity,
where k=1 is just a special case: Péano's successor function -->
--> 'repetition' --> the positive integers --> 'ad infinitum'.

All other cases k>1 generate 'uncountable infinity' (i.e. more
than just repeating one element, or one type of step): that's all,
no mistery! In essence, uncountablity of a set means: there is no
smallest element or 'atom' (like the reals, or some continuüm).
A countable infinity does have a smallest element, like 'one' 1.

The only duty you have is not to Poincaré, but to yourself:
think independently, and let no authority mislead you
(neither Cantor, nor Poincaré, nor anyone else;-)

-- Ciao, Nico B.

PS : It is always usefull to know history, and the context of
some research or result. Cantor's motivation was about
Fourier and the question of infinite frequency spectrum,
that is the idea of "continuïty" --> what we now call the Reals.
Similar to Péano's motivation: he discovered a space-filling curve.
Again : the continuïty of space : 2D, 3D, or nD  are polynomial
spaces, while Cantor's uncountable Reals are exponential: 2^N.
*That* is the difference : Polynomial or Exponential growth.
The first is countable, the second is uncountable. . . . .
==================================================================

Subject: Re: Discrete, continuous. It's all the same
Date: Thu, 07 Mar 2002
From: Nico Benschop
Newsgrp: sci.math
-------------------
"Paul P. Budnik Jr." wrote:

> "Mark"  wrote

> > >> "Paul P. Budnik Jr."  writes:

> > > By digital physics I mean the study of
> > > discrete as opposed to continuous models for physics.
^^^^^^^^               ^^^^^^^^^^
> >
> > The difference is inessential, however. [...]

Yes and no, it depends on the context.

"In Theory there is no difference between theory and practice,
... but in Practice there is."

There is a smallest element (discrete 'particle' atomism),
versus:
There is none (continuous 'field': influence on a distance)
_does_  make a difference!

Considering all strings A* over some alphabet A  of |A| symbols:

then discrete :  'countable'  if |A|=1: Peano naturals N={+1}*

and continuous: 'uncountable' if |A|>1: Cantor reals on [0,1): |A|=2

So the difference is only |A|=1 .vs. |A|>1 --> differ at least by 1.

It seems to me that most people do not have trouble with "time" to
be infinite and being linearly ordered (past-present-future).

Apart from its seemingly continuous character, this linear ordering
is the intuition underlying Peano's discrete arithmetic model of the
Naturals N(+) with one generator 1 under addition, starting anywhere,
but usually at 0, roughly:  N = 0{+1}* -- while Cantor's set R01 of
reals on [0,1) in binary code: countable_length infinite strings to
the right of the binary point with representation: R01= B*= 0.{0,1}*
again is a concept definied "up to iteration":
similar to the naturals N with 1 generator,
... BUT now requiring 2 generators.

So  "iteration"  as model of a 1-dimensional ('linear') process
without necessarily an end, is the clue of 'countable'.
Which concept is taken for granted and often denoted by a star (..)*,
and used in higher order definitions such as Cantor's powerset 2^N.
where now the stringlength is 'countable' but the resulting set not
(by the diagonal argument, taking care of the 2-dimensional
structure of any countable w x w table or 'list' of reals
in binary code)
with >1 generators, here 2, or any base b = |B| >1  for stringset B*.

-- NB --
http://piazza.iae.nl/users/benschop/cantor.htm
http://piazza.iae.nl/users/benschop/ism.htm   :  "Integer State Machines"
http://piazza.iae.nl/users/benschop/finite.htm

WSM discussion group (msg #1960, 22apr04)

Your construction of a 1-1 mapping integers <-> reals can be further simplified, without loss of generality, as follows. One needs only to consider the reals on interval [0,1) denoted as set R01, thus with zero integer part and only decimals to the right of the decimal (or binary) point. Compare R01 with the non-negative integers set N. Then there is the simple 1-1 correspondence implied by symmetry about the decimal point, for instance 0.12345 in R01 <-> 54321 in N. I used this argument in many discussions with mathematicians on the newsgroup sci.math several years ago. Notice: Cantors diagonal proof of uncountability holds for BOTH sets, R01 and N , verbatim! So what was their answer? : The integers in N are unlimited but finite, so infinite strings to-the-left of decimal point are no integers! While the same to the right of the decimal point does represent a real in R01, each 'converges' to a definite value. End of story... In fact, I do think there is an essential difference in types of infinities, since the naturals N can be generated by a single generator '1' or 'a' or any other symbol under string concatenation. While all subsets of N forming set 2^N (represented by all binary strings to the right of the binary point) require *two* generators - as strings over binary alphabet B={0,1}. But this has little to do with any arithmetic interpretation (although with some extra details one _could_ do that, just confusing the issue). Notice for any finite length n of strings over A={1} or B={0,1} the first grows linearly with n, while the second grows exponentially 2^n with n. That is, for me, the crucial difference. Then N = A* : the set of strings over alphabet A of one letter, models the concept of 'iteration' of 'repetition' , and 2^N = B* represents something essentially different. Forget about the reals on interval [0,1). I would say B* represents a different data type, namely an infinite lattice, with top and bottom element: 1* and 0* respectively. The k-th layer from the bottom, in that lattice, has all infinite binary strings with precisely k ones, so layer k=1 contains all naturals: the set N, in 1-1 relation to the one-but highest level of strings with just one zero (and further only 1's): the bitwise complements of strings at level k=1. Comparing naturals with reals is like comparing different data types, which 'naturally' leads to confusion! Pressing lattice-type reals from B* onto a linear medium such as a number line is ill advised. ...The clue is that one should take a generative view of any infinity under scrutiny, defining the way it is *generated*. Strangely enough, those mathematicians were not impressed by exponential vs. linear growth with stringlength n. While I think any exponential growth, like 2^n or 3^n or k^n, should be implying an 'uncountable' set such as 2^N or 3^N or k^N. The reason why polynomial growth N^k for finite k (k-dim space) is considered to be countable, so card(N) = card(N^k) for any finite k, is that each point in such space can be reached by a spiral which is a wound-up 1-dimensional line. Such views date from the nineteenth century, when the question was: does there exist a space-filling curve/line? Péano was involved with that (and found such curve), and he later defined the structure or concept of 'line' by the successor function, as model for any proof-by-induction, a basic tool in math. In plain language: it's the concept of repetition (ad infinitum), and 'countable' is its cardinality. Ciao, Nico. - http://piazza.iae.nl/users/benschop/cantor.htm

Newsgrp: sci.math Subject: Re: Why are rationals countable? Date: Fri, 19 Apr 2002 From: Nico Benschop "Zdislav V. Kovarik" wrote: > > In article , > Agapito Martinez wrote: > :Between any 2 rationals there is another one (their average > :for example). Can anyone explain, then, why there are "gaps" > :in the set of rationals which make it countable? It's hard to > :see intuitively why. Thanks > > (1) The "gaps" are not responsible for the countability of > the rationals. The set of the irrationals is uncountable > and yet has a lot of "gaps" -- the rationals! > > (For bigger "gaps" in an uncountable set, wait for the > Cantor set C: if a number does not belong to C, it has > an open interval around it with no points from C, and > numbers not belonging to C are "dense" in R.) > > (2) Intuition often fools us when we try to reason about > infinite sets. That's one reason why mathematicians > resorted to definitions, axioms, theorems and proofs. [*] > > (3) [...] > > Hope it helps, ZVK(Slavek). Re[*]: And I guess that is why Peano's countable set N of naturals (positive integers) is so important to be seen 'generated' by repeating a single successor function S(n) = n+1 'ad infinitum' .. starting with n=1 (or sometimes, if useful, with 0). Because 'proof by induction' then allows one to prove some property P to hold for a countable infinity of 'versions' n \in N. Such proof starts with the induction basis showing P is true for say n=1 , and assuming P true for any particular n \in N, showing it follows that P true for n+1. Such proof has " for all n \in N " a countably infinite number |N| of steps, and rests on [1]: transitive property (using '-->' to mean: 'implies'): if P1 --> P2, and P2 --> P3, then P1 --> P3. Where e.g. P1 may be an axiom (=assumed true basic fact) and --> is using an agreed logic transformation, and P2 is a lemma, and P3 a Theorem. [1] is strongly related (isomorph) to associativity of addition (+), and the Peano successor function '+1' for each n \in N. So a proof by induction is essentially dependent on the concept of 'countable' - intuitively meaning: having one generator (say 1) which is/can be repeated 'ad infinitum' , in short: N=(1)* ^^^^^^^^^^^^^^^^^^^^^^ indicated by star notation Then an 'uncountable' set S means: it requires more than one generator and relating this to (associative) string building: S = A* with |A|>1, where A* is the set of all countably long strings over alphabet A of more than one symbols. ==> |A|=2 suffices to get an uncountable set A*, <== namely the reals on interval (0,1) in binary code (behind the binary point). If A_n are all 2^n strings over A of length n, then A* = 2^N denotes the set of all countably long (infinite) strings over A, which by Cantors elegant and simple diagonal argument is indeed 'uncountable' -- whether the exponential growth of |A_n| has anything to do with this might seem intuitively clear, but is apparently mathematically not so clear;-( -- NB But anyway: A* is countable only if |A|=1 : Peano's N = (1)* and is uncountable whenever |A| > 1. ^^ That 'generation' of some infinite set must be specified is obvious, since otherwise one cannot know precisely *what* kind of infinity you are talking about, nor can one derive properties like (un)countable. -- NB - http://piazza.iae.nl/users/benschop/cantor.htm
Author: benschop_nf Date: 20 nov 1998 Forum: sci.math ------------------------------------- Re: Cantor's Diagonal Proof: FLAWED! ------------------------------------- In article <3639CFDB.ED4@math.csuohio.edu> Brian M. Scott (scott@math.csuohio.edu) wrote: > Mark Adkins wrote: > > [... snip his finite intuitions on Cantor's Diagonal ...;-] > [as well as the 'near_uncountable' replies from real mathematicians;-] > > > That is, > > since every subset of any well-ordered set has a first element, > > then every subset has a second, third, fourth... element in > > relation to that first, to which each element can be assigned > > a unique natural number. > > You're trying to extend finite intuitions to infinite sets; ...(*) > it doesn't work. ^^^^^^ ^^^^^^^^ ...(*) > > -- Brian M. Scott (*): BINGO! End of discussion (;-) (as it were: the two are "orthogonal" or "incommensurable"). If someone, with a countable constructive mindset wants to work on the Cantor Diagonal _in_the_finite_ (contradictio in terms;-): --- have a look at my 'Regula Falsi' application [CDT] generating countably many reals at a super_exponential rate 2#n, yet still a negligeable portion of the uncountably many real_reals, but still... [CDT]: " How diagonal is Cantor's Diagonal? " http://www.iae.nl/users/benschop/cantor.htm "CDT" for Cantor_Diagonal Transcendental real: CDT = \sum 1/(2#n) {limit for n=1-->oo} Ciao, Nico benschop -- http://www.iae.nl/users/benschop 1---------- The incorrigeable constructor incognico: ------------->oo 1+--------- Induction is indispensible for mathe_matic proof. ----;-( w---- Other types of "proof" are "proving" metha_matic "stuff" ---;-) >---------------------------- Ergo: Sum ----^--^------------------?-> If I do not see far, it's because Giants are standing on my shoulders. [Dave Rusin] ====================================================================== Re: Cantor Joke Author: Nico Benschop Date: 1999/02/16 Forum: sci.mat "-- The finite vs. the infinite --" (... a matter of taste ...) james d. hunter wrote: > > David Petry wrote: > > > > [...] > > I think the best way to think about theories of the infinite is to > > think of it as a game. It's not "truth" in any reasonable sense. > > The people that have gone before you have made up the > > rules of the game. You can think of the game as a convenient > > fiction which helps in reasoning about mathematical notions. > > You'll have trouble convincing the mathematicians that the > > game ought to be changed. After all, the mathematicians > > have devoted their lives to becoming good at the game. ...(#) > > My intuition is opposite yours. I usually > think of theories of the infinite as the real thing, ...(*) > and theories of the finite as games. Re(#): Well put. My idea;-) Re(*): Small example of missing the boat that way: You probably know Euclid's proof of the infinity of primes. It is often retold, in various versions, and goes like this: Assume you have found the first k primes p_1, p_2, p_3, ..., p_k . Take product m_k = \prod first k primes, and consider m' = m_k +1. This number is 1 mod p_i for every i upto k, hence it is coprime to all p_i (i=1..k). So it is either prime itself, or has a prime divisor beyond p_k, actually *all* its prime divisore are > p_k. QED Notice usually m_k +1 is taken, while m" = m_k -1 works just as well. In fact the prime divisors of m' and m" are two disjoint sets of primes, all beyond p_k. The reason for this oversight, or rather tendency, is psychological: One has apparently an irrepressable urge to go "upwards & away" to infinity -- re: 'escape' -- (after all, that's what needs to be proven: infinitude of primes;-) The clue is that "the next prime" p_{k+1} and many others are skipped over, since m_k >> p_k. No problem, of course, since this is irrelevant for the proof (as is the alternative m_k -1). Yet, looking back as it were, noting all primes < m_k and > m_{k-1} is very insightfull, regarding the primes and their structure. All these primes are in the units group G mod m_k, and it's analysis yields much prime info... Like: each even residue mod m_k is the sum of two units (Goldbach for Residues). And induction on k brings Goldbach's Conjecture (GC: each 2n>4 is the sum of two odd primes). See: http://www.iae.nl/users/benschop/fewago.htm (Fermat, Waring, Goldbach) Ciao, Nico Benschop (sci.math 16feb99) ____________________ If stuck@closure (mod..), use the carry ____________

Date:  Fri, 12 Mar 1999
From:  Nico Benschop
Newsgrp: sci.math

In article <36E83D9F.1F0DFFC5@ashland.baysat.net>,
Mike Deeth  wrote:
>
> Jim Ferry wrote:
>
> > Dear Nathan,
> >
> > Consider a triangle ABC, with F the midpoint of AB and E the
> > midpoint of AC.  The line segments from A to points on BC
> > establish a bijection between FE and BC, yet the former has
> > only half the length of the latter.
> >
> > Was Euclid an evil Cantorian?
>
> According to Euclid, a point is that which has no part.
> But non-Euclidean geometry wasn't imaginable in Euclid's time.
> Today, most physicists believe SPACE is non-Euclidean.
> In my opinion, if Euclid was alive today, he would be leading
> the anti-Cantorian campaign.   :-)
>
>           A
>           /\
>          /  \
>         /    \
>        F------E
>       /        \
>      /          \
>     B------------C
>
> In Euclidean geometry, a bijection between points on BC and on FE
> exist.  But, in a geometry were points have size, no bijection
> between FE & BC exists. --  Nathan the Great, Age 11
>

I think that using the concept of "counting" in the context of
infinitely divisible intervals (Re: 'real' line) is asking for trouble.

Descartes, in the early 1600's, already required for strict reasoning
the condition of "clearly distinguishable entities" (my words;-) --

This condition is _not_ satisfied by the "reals", where one real 'r'
cannot be distinguished from its two neighbours (there *is* supposed
to be a linear ordering on a 'line') ... In fact it *has no* clearly
distinguishable neighbour(s) < r resp. > r.

Reasoning without clearly distinguishable objects is "mushy", isn't it?
(some call it 'fuzzy logic', I believe;-)
"Counting" vanishing objects is futile.

Yet a line and a plane *are* different, and it makes sense to look for another
concept that distinquishes them: "dimension" or "independent generators",
But a generator (minimal interval, by shifting generating a line) necessarily
requires a minimal distance -- hence concrete & non-vanishing difference.

Implying:  Finite analysis <--> Strict reasoning <--> Minimal Object>0 ..(!-)

Ciao, Nico Benschop -- http://www.iae.nl/users/benschop/cantor.htm

___/  If I have seen less far than other men, (not seeing the infinite;-) \___
\  it is because I have giants standing on my shoulders (Cantor &c)    /

===============================================================================
Subject:      Re: Finitists?
Author:       Nico Benschop
Date:         5 Feb 99 -  sci.math

Jim Trek  on 5 Feb 1999 wrote:
On Tue, 2 Feb 1999, Keith Ellul wrote:
>On 2 Feb 1999, Josh Kortbein wrote:
>
>> : Does anybody know if there are any Finitist news groups, interest
>> : groups, or e-mail lists?
>>
>> Finitists? Which would be... mathematicians against infinity?
>
>Down with the axiom of infinity!   ;-)         --Keith
>---------------------------------------------------------------
>Keith Ellul              4th Year Pure Math / Computer Science
>kbellul@golden.net       University of Waterloo
>---------------------------------------------------------------
Jim Trek:
Indeed! From 1969 until 1973 I worked to delineate mathematical
methods devoid of any unprovable aspects.

I began by observing that some logicians disagree with most
mathematicians on one important point:  The logicians insist that
false assumptions must lead to both the proof and disproof of
every meaningful statement within the logical system in question.
The math people believe that they can make up sets of axioms that
make sense to them (true in the world or not) and that the
resultant math will be free of logical flaws.

Here are some of the conclusions I came to during this time:

1.  Calculus does not need any concept of infinity in order to
provide limits, derivatives, integrals, and differentials.

2.  No infinity is possible unless axioms assert it.  That is,
no infinity can be derived without being presupposed.

3.  No even roots of -1 are needed except one (i = the square root).

4.  All of the above facts are known to many mathematicians.

5.  Real numbers that are not rational numbers can be expressed
as limits of functions of rational numbers.  This means that
quantities like pi and e need not be regarded as numbers and they
may be formally handled just as computer programs handle them.
It also means the the concept of a process replaces the real
numbers that are not rational.  The complex numbers (a + bi)
become processes where a and b numbers or processes.

6.  Mathematics needs to be synthetic (founded upon definitions
which, in turn, are founded upon undefined but perceived meanings
in the common language).   Axioms are unnecessary and harmful.  No
axiomatic system works anyway if we don't agree on the meaning of
such fundamental terms as single, pair, the, associated with, and
the like.  Good systems would rely on a minimum of such terms and
would explicitly recognize them.  Formal math comes from our
perceptions of reality, not the other way around.

7.  The phenomenon of the conditional branch (if incorporated
into math proper), represents a giant advance in the power of
math to solve problems.

That is the short story.  We get every kind of functionality in
the whole world without any logical flaws and without esoteric
and spooky contradictions (many mathematicians are in awe of
them, but they can all be easily fixed).
If this is of interest, please write. -- Jim
------------------------------------------------------------------**]

Nico B:
Good thinking, Jim. Around that same time (25 years ago;-) I realized
that in the design of Digital Networks with internal state ('memory',
cq sequential logic or FSM synthesis) some tools were missing, namely
that which goes beyond Boolean Algebra (sets, binary combinational
logic, as recognized by Shannon in 1938 to be isomorph to cct design).

George Boole ("The Laws of thought" 1854) introduced his logic as the
idempotent branch of arithmetic [ x^2=x --> x(x-1)=0 --> x=0 or x=1,
and commutatiove xy = yx as natural for binary properties x,y of an
object]. Both that logic (sets, intersection, union) and arithmetic
are associative -- as is the algebra of functions, which is more
general again, since it is not commutative.

This yields the three layered hierarchy of applicable algebra's:

3.  a(bc)=(ab)c  Associative   Functions   State Machines   Seq_logic
2.   ab = ba     Commutative   Numbers     Arithmetic       cpu
1.   aa  = a     Idempotent    Sets        Combin.Circuits  Comb-logic

Historically all started in the middle :

(2) with numbers & arithmetic (Pythagoras, -500), expanded down to:

(1) Sets and combinational logic (Boole, 1854),   and upwards to:

(3) Permutation groups (Abel/Galois 1830), matrices (Hamilton, Caley)
last century, and eventually Semigroups (Schushkewitch, 1928)
this century with Computers & FSM model (Moore/Mealy, 1955).

Now FSM(Q,A)-- statset Q, input alphabet A -- as generator produces
sequentially A*/Q: a closure of transformations of a set Q, which
is a Semigroup: associative, with no other restriction than 'finite'!

The strange thing is, that infinity has such grasp on mathematicians,
apparently, that since it's first structural result (simple semigroups
by Schushkewitch, 1928 -- see Clifford/Preston "Algebraic theoory of
Semigroups 1961 AMS survey #7, Vol.I appx) not much has been done with
finite semigroups as synthesis tool (compare: Fourier freq-spectrum
for Linear Networks) - not even since the FSM concept of the fifties.
Rather, its general abstraction to Category Theory was the 'bandwagon'
ever since the sixties.

Apparently "Fiddling in the Finite" (reflections) is no fun, or is
deemed too difficult (re: Diophantine eqns) for serious work ;-(

The above 3-fold hierarchy could be helpful to 'fill in the gaps' --
and Finite Associative Algebra (=semgroups;-) a nice framework for
practical theory, to work on the next century or so, e.g towards a
Theory of Digital networks (with "spectral" synthsis tools and all)
... Just a suggestion.

BTW: Additive analysis of the multi've semigroup Z(.) mod m_k
is a useful approach to 'crack' old arithmetic problems,
for proper choice of modulus m_k (say m_k = p^k for prime p,
or  m_k = \prod p_k = product of the first k primes)
See http://www.iae.nl/users/benschop/sgrp-flt.htm

AND: Among the 5 basic state machines (their closure having no
proper subsemigroup: http://www.iae.nl/users/benschop/c-ranksm.dvi)
-->  there is one that implements the "branch", cq: if-then-else or
multiplexer, as a state machine, requiring one extra initial state
which is beyond the closure (re: Godel?) -- Under condition that
inputs can be distinguished by heir effect on the machine (re: state
transform equivalence).

Ciao, Nico Benschop  -
http://www.iae.nl/users/benschop/work.htm
-- The Math Forum

Subject: Re: Can the "crackpot" theories of the infinite be formalized?
Date: Thu, 15 Apr 1999
From: Nico Benschop
Newsgrp: sci.math, sci.logic

David Petry wrote:
>
> If we examine the various "crackpot" theories of the infinite that
> are regularly posted to sci.math and sci.logic, (the semi-coherent
> ones, anyway) we find two or three basic intuitions that underlie
> most of those theories. The question arises, would it be possible
> to formalize those intuitions in a logically consistent way?
>
> Now there's no denying that these intuitions are incompatible with
> standard set theory. The question here is whether these intuitions
> could be used as a basis of an alternate set theory which is
> compatible at least with PA.
>
> The first "intuition" is a little trivial. I'll call it number 0.
>
> Intuition #0.   There are no infinite sets.
>
> This can be formalized as ZFC with the axiom of infinity replaced by
> its negation. I believe it's equivalent to PA, not really problematic.
>
> Intuition #1  Infinite sets contain infinite elements.
>
> Here's a typical argument behind this intuition.
>
> Define N (an integer) as {0, 1, 2, ... N-1} (the set of all integers
> less than N).  So clearly, for all n, n is an element of n+1.  Now
> consider the set of all integers (an infinite set). Since, as everyone
> knows, infinity equals infinity plus one, we must have that infinity
> (the set of all integers) is an element of infinity (again, the set of
> all integers).
>
> Intuition #2   The irrationals are equinumerous with the rationals.
>
> Here's a typical argument for that.
>
> Let T be an enumeration of the rationals less than 1, starting with 0.
> (e.g. T = { 0, 1/2, 1/3, 2/3, 1/4 ...} )  Let T_n be the set of
>  the first n elements of T (e.g.  T_3 = {0, 1/2, 1/3} )
> Let S be [0, 1) (i.e. the unit interval including 0 but not 1).
> Let S_n be the set of connected components of S\T_n (i.e.
> what's left of S after removing the points which are in T_n).
> For example, S_2 = { (0, 1/2),  (1/2, 1) }
>
> Clearly, for all n, both T_n and S_n have exactly the same
> number of elements (n elements, in fact), and hence in the limit
> as n goes to infinity, T_oo and S_oo must ("obviously") also
> have exactly the same number of elements. But T_oo has one
> element for every rational in S, and S_oo has one element for
> every irrational in S, hence the rationals in S are equinumerous
> with the irrationals in S.
>
> As von Neuman said, in set theory, we don't really understand
> things, we just get used to them.  When these "crackpots" come
> along proposing new theories of the infinite, most of us pounce
> on them, showing them how their theories contradict the standard
> theory, and then trying to help them get used to the standard theory.
> But we never really answer the question of why their intuitions must
> be regarded as inferior to Cantor's intuitions.
>
> So it'd be really nice if someone made a serious study of these
> alternate intuitions, and answered some essential questions about
> them. For example, does PA + Intuition #1 necessarily lead to
> inconsistencies? Does it lead to mathematics that is so weirdly
> counter-intuitive that it simply must be rejected on those grounds?
> Could we "get used to" these alternate intuitions, in the same
> manner that we have gotten used to Cantor's intuitions? Is the only
> reason we follow Cantor's intuitions rather than the new intuitions,
> because Cantor got there first?
>
> If there were serious studies of these ideas in the literature, then
> when the "crackpots" come along with their proposals, we
> could direct them to references in the literature, and guide them
> along a productive path, rather than engaging in the vicious and
> unproductive flamefests that are so common in these newsgroups.
>
> As I understand it, Cantor was widely considered a "crackpot" in
> his day. That's something you guys might think about when you so
> quickly apply the label "crackpot" to anyone who proposes a theory
> which contradicts Cantor's theory.

NB: Commendable & open minded thinking.

However may I, as certified semi-crank (assuming crank=crackpot mod p^3,
where p^3 stands for: "Peer pressure & prejudice" -- not Jane Austen;-)
make a few remarks, possibly  conductive to your constructive thoughts.

1. A _real_ crank is NOT sensitive to logic reasoning, similar to a
a _Real_  crank-bashing mathematician ;-)  Both: stuck@closure...

2. Qua set-theory: although not a specialist (thus eminently qualified
to pose my opinion), it occurred to me that the only powerfull
operations (apart from complementation, a unary operation) are
'intersection' (&) and 'union' (|) -- which both are idempotent:
for any set S we have S & S= S, and S | S= S, making the 'logic'
in this tak-van-sport (branch-of-sport) rather impotent, to put
it mildly ;-(  I mean: comparing this with 'sequencing' as in any
associative non-idempotent system, like:
^^^^^^^^^^^^^^
Peano's N = 0{+1}* with the star meaning: repetition as often
as you please (even ad infinitum) -- a one-sided infinity.

Integers Z = 0{+1,-1}* with "alphabet" A = {increment, decrement},
the two-sided infinity beyond Peano: namely a Group ;-)

Permutations: Z! = {+1,-1,swap2}* with swap2 permuting any pair
of integers, say 0<-->1, yielding infinite Integer Group Z!

Transformations: Z^Z = {+1,-1,swap2,merge2}* of all transformations
of Z into itself, yielding (infinite) symmetric semigroup Z^Z.

The idempotency of (&) and (|) in fact cripples, to my taste, any
really useful insight into infinity. - For the simple reason that,
at least for a constructive mind (which most cranks have;-)
one should be concerned with *specifying* a particular infinity by
its _generation_process_ (after all: listing the infinity of elements
under consideration takes a bit long, doesn't it?-).

And hence: do not talk ONLY about closures, but *also* - for balance
sake - about generators. And for 'generic' closure types: the minimum
nr of generators required, say denoted by seqdim (sequential generation
dimension): seqdim(N)=1, seqdim(Z)=2, seqdim(Z!)=3, seqdim(Z^Z)=4.

Wouldn't *that* open up some more constructive discussions? Rather than
abstractly & endlessly discussing non-matching points of view, with no
constructive convergence in sight, stuck @ idempotent operators...

Just some thoughts..... And: nothing _really_ new, since all these
structures: groups, lattices, arithmetic (residue) closures Z(+) and
Z(.), semigroups, etc.. are well known from the finite (although often
shunned because: too difficult in the finite - re: Diophantine stuff,
or: old fashioned). -- And 'hundreds of years old'. In other words:

Think of Number Theory not as only about Numbers (objects/semantics),
but rather about Closures & Generators (sequences/syntax).
And not 1-1 correspondences between sets,
but rather isomorphisms between (structured) closures.

Then it should be obvious that N and 2^N, or:  Z and 2^Z, are not
isomorphic closures, and surely in the infinite: essentially different,
having different sequential generative dimensions (note: 2^Z < Z!,
namely as fixedpoint sets  of permutations).

After all (re Peano): is not {c}* "the same structure" as {1}* or {+1}*
sharing the common property of having One Generator (and the rest is
iteration / ... Shakespeare?-)

--
Ciao, Nico Benschop  -- http://www.iae.nl/users/benschop
Amspade Research -- IAE: Institute of Advanced Engineering

___________/ If stuck@closure (mod ...), use the Carry \___________
\   There's nothing like a counter example  /
(unless it's the only one in an infinite context: it vanishes;-)
(our intuition "counter_example blocks truth" is borrowed from the finite)
(--------- and cannot be trusted in an infinite context, can it? --------)

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx1.1xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
----------- AHA: One is Always Halfway Anyway -----------

Subject:   Re: My problems with infinity...
Date:   Sat, 01 Apr 2000
From:   Nico Benschop
Organiz'n: Research
Newsgroup: sci.math
---
HomerJSimpson wrote:
>
> I have been plagued by the following thought process, which is obviously
> flawed.  Someone please tell me why...
>
> The following two facts are easy to prove:
>
> 1.  On the real line, you can find an irrational number between any two
>     rational numbers.
>
> 2.  On the real line, you can find a rational number between any two
>     irrational numbers.
>
> From this, it would seem to me that there would be a bijective
> correspondence between Q and R-Q.  I think of it with the following
> analogy...
>
> Between any two red marbles, there is a blue marble.
> Between any two blue marbles, there is a red marble.
>
> So the marbles alternate: ...red, blue, red, blue, ...
> and thus there is a bijection between the set of red marbles and the set
> of blue marbles, right?
>
> This is what confuses me, since I know the standard argument (due to
> Cantor, if I remember correctly) that shows that there is no
> bijection from the rationals to the irrationals.  Obviously, there is a
> flaw with my analogy, which I believe must be rooted in some lack of
> understanding of infinity or infinite sets.   Can someone please set me
> straight? Be harsh and scolding if you'd like--I would just like an
>
> (Anyone who answers this will most likely also answer another question
> I've pondered...really, no two points on the real line are "next to" one
> another in the sense that one directly follows another since in between
> any two real numbers x and y there are an infinite amount of real numbers
> c such that x < c < y.  But when I imagine the line, I see a long chain of
> points, and naturally, I see each point with a point which immediately
> preceeds it.  This is apparently a bad way to think of the line, then.
> What's a better way?)
>
> Thanks for your time -- WRC, Math Major

I had the same problem. Representation of the reals, for instance on
interval R01 = [0,1) in say binary code gives a clue why there are more
reals than rationals, because they are of different "data type"
(a term borrowed from Computer Languages;-)

R01 under addition (+) is isomorphic to the set of all strings over
{0,1} generated by the powers 2^{-n} for all n in N (the naturals,
positive integers). In other words: the set of reals R01 can be thought
of as a Boolean Lattice of infinite (but countable) length 0/1 strings,
with .0* (all zero's) as bottom element, and .1* (all ones) as top,
and all other |2^N| binary strings in layers of equal number of ones
in-between.

Now the set N, of equivalent order as the rationals on [0,1) can be
represented by all infinite strings with just a single 1 in places n
(respectively) to the right of the binary point (hence 2^{-n}= 1/(2^n).

This is, in the above lattice, the layer just above the bottom 0*.
It's cardinality |N| is countable, and you may imagine that the whole
lattice (of infinite "width and hight") |2^N| is "infinitely larger"
(Cantor's diagonal argument). For more detail of this "different data
type" idea, see http://www.iae.nl/users/benschop/cantor.htm

It is then indeed confusing to _also_ think of reals as elements
on a line: they are rather elements in a lattice: hardly a "linear"
structure!

Ciao, Nico Benschop - http://www.iae.nl/users/benschop/filosofy.hm

Subject:  Re: calculus and other branches
Date:  Sat, 29 Jul 2000
From:  Nico Benschop
Newsgrp:  sci.math                (on limits & infinity)
-------------------
danfermat@my-deja.com wrote:
>
> Probably, for all practical purposes, everything in the universe is
> finite.  Why then does calculus (and similar branches of math) use
> infinity?  How does it go from a finite problem in the real world, to
> math using infinity, and then back to a finite answer? Do the
> infinities cancel out?  Why does math use infinity, when in other
> sciences it usually means you must have made a mistake somewhere?

A limit like Pi, or sqrt(2), is just a very usefull 'ideal' (ask Plato,
or any engineer: there's no limit to the accuracy with which you can
approach it, hence such limits are Future Proof;-) -- NFB

However, don't go overboard to press real numbers onto a straight line,
while essentially they are two-dimensional {a,b}* versus Peano's {+1}*.

Ciao, Nico Benschop -- http://piazza.iae.nl/users/benschop/finite.htm
http://piazza.iae.nl/users/benschop/cantor.htm
http://piazza.iae.nl/users/benschop/ism.htm

Subject: Re: discrete vs. continuous
Date: Sun, 13 Aug 2000
From: Nico Benschop
Newsgrp: sci.math
-----------
"Mike N. Christoff" wrote:
>
> I have a very vague sounding question to ask here,
> but it has been bothering me for quite some time.
>
> What is the 'true' difference between a discrete system and a
> continuous system - in a philosophical sense (as far as you know)?
> In many sciences, we see how continuous systems arise out of discrete
> systems and discrete systems arise out of continuous systems.
> Is there some bridge between the two?  Can one, theoretically define
> a "limit" between discrete and continuous systems?  ...[*]
> -- l8r, mike

Sure, one can: it's the (not quite popular yet) concept of 'threshold'
-- which may even be defined more rigorously than by intuition...

By Descartes: exact reasoning requires precise distinction between
the objects of discourse. Equivalence (modulo some property) is a
weaker form of that, but still 'discrete'. The problem (as always;-)
is with infinitisimals: very usefull, but a pain_in_the_... for exact
logic. ---- Enters George Boole: introducing the logic of reasoning
("The Laws of Thought" 1854) as the idempotent arithmetic of properties:
a^2 = a, ab=ba, so a(a-1)=0 --> binary logic on {0,1}*.

'Threshold' (the priciple of digital computers and binary communication)
separates the continuos from the discrete (Lady 'Justitia': cutting into
yes/no a continuum of arguments and circumstances;-)
Our brain (neurons) is full of it, and seems to work that way ..;-)

If someone has a relevant source/book/article on the fundamental concept
of 'threshold', then I'd be interested... Cantor, with his countable
w x w square table, and its diagonal (;-) came pretty close: the
difference between linear disctrete [ Peano's {+1}* ] and the continuum
of reals [ {a,b}* ] is one of sequential (generating ) dimension,
countable: seqdim=1, uncountable (reals): seqdim >1 , e.g. =2 for the
reals, and for 2^N, cq  2^Z < Z^!  namely subsets of Z as fixedpoint
-sets of permutations of Z, where seqdim(Z!) = 2, with generators
{a,b} = {+1, swap01} see /ism.htm ...

Ciao, Nico Benschop -- http://piazza.iae.nl/benschop/finite.thm
http://piazza.iae.nl/benschop/cantor.thm
http://piazza.iae.nl/benschop/ism.thm
----------------------------------------------------------------------

Subject: Re: a .9999rep = 1 debate, can someone comment please?
Date: Tue, 07 Nov 2000
From: Nico Benschop
Org.: Amspade Research (Digital)
Newsgrp: sci.math
-----
Guillermo Phillips wrote:
>
> Steve,
>
> Yes I do indeed mean an infinite number of 9s and 0s.
>
> I think there is some mileage in representing numbers this way :
>
> 1.  It 'feels' like a natural extension to the way numbers are
> ordinarily represented: if you can have an infinite number of places
> to the right, then there should correspondingly be an infinite number
> of places to the left of the point.
>
> 2.  Doing arithmetic this way causes no logical inconsistencies
> (as far as arithmetic is concerned - try it).
>
> 3.  There are two 'pleasing' things which come out of this notation
> a.  If we define ....999.000... = -1 then ....999.999.... must be zero.
> Dividing this number by 9 we would get .....111.111.... which must
> also be zero. In fact any infinitely repeating finite sequence of
> numbers (from the left of the point to the right) would represent zero.
> This allows you to find the negative of any rational number which
> contains a repeating sequence eg -(1/7) = ....142857142857.000000....
>
> b.  If we represent things in binary, then ....11111.0000.... = -1.
> This turns out to be just twos complement arithmetic using an infinite
> number of bits.  This nice thing about it is that all you would have
> to do to find the negative is to flip all 0s to 1s and vice versa :
> ie 1 = NOT(-1)  OR  ...0000.1111....= NOT(....1111.0000....).
> This works for any number - which is neat.

Indeed.

> So it may seem ridiculous to have an infinitely large number represent
> a negative finite number.  But by the same token, isn't just as
> ridiculous to represent a positive finite number, by the addition of
> an ever decreasing number of smaller numbers?
>
> You can't prove it as such, fundamentally you need a definition
> somewhere. -- Guillermo
-------------

> "Steve Lord"  wrote in message
> news:Pine.OSF.4.21.0011061038040.31349-100000@mathlab.WPI.EDU...
> > On Sun, 5 Nov 2000, Guillermo Phillips wrote:
> >
> > > Can anyone prove that "....99999.00000....." equals -1?
> >
> > Define "....99999.00000....."  If you're going to say
> > "An infinite number of 9's before the decimal place and an infinite
> > number of 0's after", then you have to ask yourself: does this make
> > sense?
> >
> > > But try adding 1.  So isn't this the same question - ie one of
> > > definition?
> >
> > It's a question of "Does ....99999.00000.... make sense?"  Since
> > you brought it up, _assuming that ....99999.00000.... is defined_,
> > what is 1 + ....99999.00000.....?   -- Steve L
--------

What you're up against, Guillermo, is: does it "make sense" to extend
integers to inf. # msd (more significant digits). The extension to the
right is no problem (re 'convergence'). What you suggest (by symmetry)
has been done by Hensel early last century: the p-adics, which however
are NOT integers, but _residues_  mod p^k (prime p, lim k --> inf).

That somehow caught-on because it was backed-up by a lot of theory,
although of limited use because it is in a way 'asymptotic' by nature,
but still interesting, of course. Notice: in p-adic number theory the
most significant digit place has been moved out of sight, to the
infinite left. But don't think these are integers (the 'smallest'
unit being 1 all right;-) They are & remain RESIDUES (= rest mod p^inf).
But they do satisfy interesting arithmetic closure properties.

So you need to counter the 'lack-of-convergence' (in fact 'divergence')
by another basic concept, like 'closure' or something, and back it up by
interesting algebraic properties (which you partially did) _and_ come
with a useful context for 'application' (although p-adics seem to lack
this IMHO). Otherwise: an uphill battle against 'business as usual';-)

NFB -- http://www.iae.nl/users/benschop/campaign.htm (Hensel lift & FLT)
(Fermat and the cubic roots of 1 mod p^k, prime p=1 mod 6)

Charles Taylor (McGill Univ) in "Sources of the Self":
"We have to fight uphill to rediscover the obvious".
-- http://www.iae.nl/users/benschop/simple.htm --

Re: 1/0 . . . (sci.math 10jan2001)

Re: 1/0 -- Boole: "The Laws of Thought" (1854) has nothing on 1/0 Author: Nico Benschop Date: 10 jan 2001 Forum: sci.math -------- "Clifford J. Nelson" wrote: > > Danny Kelly wrote: > > > > I have noticed something kind of interesting. > > > > The square root of -1 is obviously not a real number (ie, not a member > > of the real number system), but it is a complex number. Further, > > considering it to "exist" has helped further mathematics and many > > sciences in general. You would probably say that in much of today's > > math, it is indispensable; we couldn't do without it. > > > > But what about 1/0. We say it is "undefined". > > I think that most people, when they say that, mean that it is not a > > number. It is obviously not an element of the real number system > > (because (x/0) * x can never equal 1 (for any x)). > > > > ---My question is, WHY IS THERE NOT STUDY INTO A NUMBER SYSTEM THAT > > HAS AS AN ELEMENT THE NUMBER 1/0 ? There is only one possibilty > > that I have come up with as a possible answer, that it has somehow > > been "determined" (or more accurately, "guessed, with some suggestive > > evidence") that it would not be as useful as sqrt(-1). > > > > Yours truly, Sam Kelly > > Sorry if this has been posted already. > George Boole interpreted 1/0 in his only book "The Laws of Thought" > in 1851. -- Cliff Nelson ^^^^ that was 1854, ...with an earlier monograph 1848 on binary logic interpreted as an idempotent (binary) image {0,1) of integer arithmetic. Moreover, I don't think he interpreted 1/0 as anything useful;-) For him 'logic' was the calculus of binary properties {x,y} of objects, interpreted as idempotent arithmtic with: commutation law x.y = y.x, and idempotent: x.x = x --> x(x-1)=0 --> x \in {0,1} x+x = x (saturation for all x <> 0: mapped to 1) In such arithmetic context division by 0 is definitely tabu !-) --NFB -- http://www.iae.nl/users/benschop/work.htm

Subject: Re: 1/0            -- "On infinity in arithmetic"
Date: Tue, 09 Jan 2001
From: Nico Benschop
Org: Amspade Research - Digital
Newsgrp: sci.math
--------
"David W. Cantrell" wrote:
>
> In article ,
>   nirh1@netvision.net.il (Nir) wrote:
> > There is a simple reason why that wouldn't be useful: if you want to
> > introduce a number which equals "1/0" and preserve the field structure
> > of the real numbers at the same time, then you have to consider a
> > field in which the number 0 has an inverse, but there is just one such
> > field and it consists of one element, nameky the number 0 itself, so
> > it isn't very useful, is it ?
>
> There are very useful systems which are not fields! For example, think
> of the systems used in cardinal arithmetic, ordinal arithmetic,
> interval arithmetic, and floating-point arithmetic.
> If you're going to have 1/0 defined in a useful system, it won't be a
> field! --   David Cantrell

In the usual arithmtic closures with 1 as multiplicative (.) neutral
element (1.a = a.1 = a),  and 0 as additive neutral (0+a = a+0 = a),
the '0' under (.) functions as an "ideal" or "sink" (0.a = a.0 = 0)

The "inverse" a/b of b/a is normaly defined under (.) with (a/b).(b/a)=1
Then 1/0 should yield as inverse 0/1=0 with product (1/0).0 = 1,
which contradicts a.0 = 0 for all 'a'.
So 1/0 is not in the normal arithmetic closure system S.

In fact, under associative rules  for (.) the two commuting idempotents
1.1=1 and 0.0=0 with 0.1 = 1.0 = 0, form an ordered pair: 1 .ge. 0,
which relation .ge. (greater or equal, >/ ) is transitive, anti-symmetric
and reflexive (thus an ordering relation), with converse 0 \< 1.

Yet it might be nice to have an element 'w' (for reverse 'new';-)
that elevates the sink '0' under (.) _and_ under (+),  sothat:

w.a = a.w = w [for all 'a' in system S(+,.) including a=0]

and:   w+a = a+w = w [ ,, ] ,  and also:   w.w = w+w = w

Like 0,  w is an ideal of S(.) but 'dominating', since 0.w = w.0 = w
( 'w' is a "stronger" sink than '0' ;-)

Use "\<" for ordering .le. (lower or equal) then:

S(.) has three idempotents: {w,0,1} ordered w \< 0 \< 1,

while S(+) has two   idempotents: {w,0}   ordered w \< 0.       --NFB

PS:
a matter of 'taste' to think of w as 'infinite' (or 1/0, or omega ?-)

Subject: Re: infinity - an ignorant view Date: Mon, 08 Oct 2001 From: Nico Benschop Org'n: Amspade Research (Digital) Newsgrp: sci.math glenn wrote: > > On 4 Oct 2001 erickniffin@my-deja.com (Eric Kniffin) wrote: > > But we can't reach the end of an infinite set, and > > there is no such thing as getting to the end of an > > infinite amount of time. > > Infinity is a quite easy idea to visualize. Much easier from > the combinatorial complexity of relatively big finite sets. > > On finite sets, counting is percieved as a finished deed. Instead, > on infinite sets counting is not a finished action. Hence, in > order to have something finished, even in these sets, we go up > one level higher, namely on maps. Mappings between infinite > sets are percieved as finished deeds. They are the substitute > of counting. > And while counting is something finished on a finite set, a map > from an infinite set to itself is also finished, since we have > the perfect knowledge of the undelying law which furnishes the > map. As a perfect example of this, take the map > f : N -> N, n |-> f(n) = n. > While counting of the set N = {0,1,2,...} is indeed endless, > the map f is a finished thing. Notice that f is nothing but a > "counting" of N. But this time, we are in a higher level than > mere coutning. We alledge for ourselves the perfect knowledge > of the "law" f. > In this sense, someone could say that the "theory of infinite" > is a glorification of the theory of maps. ...[*] > -- glenn Re[*]: Indeed, that's how I see 'infinity' as a useful concept, to be used in 'practice' like: - "Infinite precision" for the reals on interval [0,1] representented by strings over some natural base B of >1 nrs, - "Limit" of a converging series, such as Pi/4 = 1 - \sum 2/(16n^2-1), provided arithmetic rules apply 'consistently' to those limits;-) - "Asymptotic" behaviour: predictable and do-able, yet missing a finite end/result - like a deterministic algorithm on finite data. - "Infinite State Machine" ISM: A x Z --> Z with (finite) input alphabet A, but a countable two-sided infinite set of states (the integers Z) to model 'infinite memory' (-time;-) behaviour. It seems to me that most people do not have trouble with "time" to be infinite and being linearly ordered (past-present-future). Which is the intuition undelying Peano's discrete arithmetic model of the Naturals N(+) with one generator "1" under addition, starting anywhere, but usually at 0, roughly: N = 0{+1}* -- And again Cantor's R01 of reals on [0,1) in binary code: countable_length infinite strings to the right of the binary point with representation: R01 = B* = .{0,1}* - which again is a concept definied "up to iteration", just like N, BUT requiring 2 'generators'. Then "iteration" as model of a 1-dimensional ('linear') process without necessarily an end, is the clue of 'countable'. Which concept is taken for granted and often denoted by a star (..)*, and used in higher order definitions such as Cantor's powerset 2^N where now the stringlength is 'countable' but the resulting set not (by the diagonal argument, taking care of the 2-dimensional structure of any countable list of reals in binary code) with >1 generators, here 2, or any base b= |B| >1 for B*.
From: Nico Benschop Subj: Re: pi greek View: Complete Thread (17 articles) Newsgrp: sci.math (18dec1998) ----------------------------------------------------------------------- > Tangent60 wrote in message <19981217141536.04109.00000390@ng-fq1.aol.com>... > >> I'd like to know where I can find a table with pi greek > >> (3.14159265 ...) on the Internet. I need to know pi with > >> a precision of about some hundred digits. > > 3.14159265358979323846264338327950288419716939937510582097494459230781 > 64062862089986280348253421170679821480865132823066470938446 ----------------------------------------------------------------------- William L. Bahn wrote: > > But the question begs to be asked, " Why do you need pi with a !! > precision to a hundred digits?" The farthest objects we can see !! > are about 15 billion light years away. So we can calculate the !! > circumference of the resulting circle to within the diameter !! > of an electron with only 40 digits (give or take a couple). !! > What possible application do you have where the precision !! > and accuracy of all of the other values warrants using !! > the value of any constant to this level of precision? !! > > I guess I can see some applications where pi is being > used as a number sequence and not as an actual value. ----------------------------------------------------------------------- [** Kevin Brown wrote (..jun98): Leibniz noted that arctan(1)=pi/4 implies the remarkable relation: pi 1 1 1 1 1 ---- = 1 - --- + --- - --- + --- - --- + ... 4 3 5 7 9 11 It's interesting to consider what Leibniz (or Gregory) might have done with this result. **] NB (16jun98): ------------ So: pi/4 = 1 - (1/3 - 1/5) - (1/7 - 1/9) - (1/11 - 1/13) - (... = 1 - \sum { 1/(4n-1) - 1/(4n+1) } = 1 - \sum 2/(16.n^2 -1), sum over all n>0. Hence: (1 - pi/4)/2 = \sum 1/(16n^2 -1) = 1/3.5 + 1/7.9 + 1/11.13 + which is a fascinating simple series! I wonder what this yields for instance in base 4, or base 16 code. Has that ever been done? ---------------------------------------------------------------------- So, rather than decimals, pi in hexadecimal code (0..9A..F) might be more interesting (from a structure analysis point of view;-) Ciao, Nico Benschop -- http://www.iae.nl/users/benschop