Re: infinity
From: Roger Beresford (mail_at_beresford22.freeserve.co.uk)
Date: 07/25/04
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Date: 25 Jul 2004 15:36:53 -0700
"Andrew B. Park" <novakyu@yahoo.com> wrote in message news:<cd49f7
Snip
> BTW, I would prefer to think of infinity rather in terms of "limit"
> than as just a concept. Treating it as a limit makes it easier to grasp
> it conceptually, I think.
No-one appears to have approached this "rationally" (pun intended),
though failing eyesight precludes my studying the thread in detail. I
offer an engineer's rational exposition:-
(1) The Natural Numbers 'N (short ASCII for Double-struck Capital N)
can be created from set theory by labelling the empty set with 0
(nought), the set containing the empty set with 1 (unity) etc. 'N has
(trichotomous) order, a<b, a=b, or a>b; this allows the binary
functions {Min[a,b]=Max[b,a]=a if a>=b, else =b} to be defined.
'N also has the following non-invertible emergent properties, members
of 'N giving members of 'N:-
Successor a+1=n (next)
Addition, a+b=s (sum)
Multiplication a*b=t (times)
Powers a^b=p (power)
Symmetric Difference d, Min[a,b]+d =Max[a,b].
Division, roots, negation and subtraction are not applicable to 'N.
(2) Sets of elements can be free {x,y,w,..}, ordered {a,b,..} (with
lexicographic or other order), or "indexed" {n2,n5,n1..}. Indexed sets
have some properties that survive re-ordering.
(3) I postpone the development of negation & the (signed) integers 'Z
(and, later, the reals 'R) because traditional negation is a special
n=2 case of equivalence relations on n-element indexed sets.
(4) Following Landau. An equivalence relation on ordered pairs,
{a,b}~{c,d} iff a*d=b*c, gives the unsigned rationals 'Q+, with
division as the inverse of multiplication. Using "zero" and "one" to
distinguish 'Q+ fom 'N, zero is {0,a}~{0,b} and one is {a,a}~{b,b}.
(5) Attempting to define Infinity "oo" as {a,0}~(b,0} leads to a
Goedelian Bifurcation. We cannot have both 0~{0,0} and oo~{0,0}, so
two incompatible systems must be developed. The first case has new
axioms 0~{0,0} & {a>0,0} is undefined but this appears to be a dead
end; the second has new axioms oo~{a>0,0} & {0,0} is undefined. This
introduces a tower of infinities, ooo~{oo,0}, etc. Mathematics is not
a democracy, but the second aalternative appears to be popular because
it "goes somewhere".
(6) The situation changes when we tackle division algebras for "vecs"
{a1*d1, a2*d2,...} (generalized vectors, indexed sets of signed
coefficients ai and basis dimensions d1). They have (left & right)
multiplicative inverses Ai, with (*B)*Ai=B. I call these algebras
"Hoops" because they are shape-conserving rings or loops. The shape is
the list of the factors of the multiplication table determinant.
Division by zero (ultra violet divergence to physicists) occurs when
one or more of these factors is zero, because the factors are the
denominators for partial fractions defining multiplicative inverses.
http://library.Wolfram.com/infocenter/Mathsource/4894 (which is
intended to be comprehensible to non-Mathematica users, using the free
MathReader that is downloadable from the same page) develops Hoops,
showing that that they are all "collapsed" from Moufang Loops via
generalized negation. Real 'R, Complex 'C, & Quaternion 'H algebras
are collapsed from the C2, C4, and Quaternion groups; Octonions 'O
from the 16-element Octonion Moufang loop. "Ordinary negation" is used
in these (and many other) cases, so Reals 'R are an equivalence
relation A~{a,b}~{c,d} iff a+c=b+d, and -A~{b,a}~{d,c}, 0~{a,a}. 'R,
'C, 'H, & 'O are the only "real algebras without divisors of zero"
because they alone conserve the sums of squares of their elements. The
inverses are only infinite in the trivial real case {0,0,..}. (Note
that these algebras have division by zero in the 'C, 'H, & 'O fields.)
(7) Many other algebras (Clifford, Study, Davenport, Terplex,
Pauli-sigma and Dirac-gamma algebras, etc.) are hoops; exterior
algebras & Lie algebras are hoops with restrictions. Most conserve
more than one function; the inverses split into partial fractions with
each function being a denominator. This allows the infinities to be
"factored out". Division by zero can then be replaced by working in a
constrained (lower symmetry) sub-algebra where one or more functions
are constrained to be zero. The original is a super-symmetric version
of the sub-algebra.
(8) Hence, in many algebras, "anything/0" implies that the
calculations should be "renormalized" into a sub-symmetric
sub-algebra.
Roger Beresford.
"One should always generalize." (C. G. Jacobi.)
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