|
| |
|
|
|
MORE DETAIL ON FUNDAMENTAL... OVERSIGHTS
|
SECTIONS
DEFINITION
of “PARADOX”... AND HIDDEN “RULES OF DEFERENCE”
DEFINITION of “INCONSISTENCY”
DEFINITION
of “DEFINITION”
FINITE ARITHMETIC -> TRANSFINITE SET THEORY
RULES OF DEFERENCE?! WHAT LIES HIDDEN BY THESE... OVERSIGHTS?!
|
|
DEFINITION of “PARADOX”...
AND ITS HIDDEN “RULES OF DEFERENCE”
“Paradox”
no longer has a formal mathematical meaning, Cantor’s
Paradox to the contrary notwithstanding. For millennia, before Cantor, “paradox”
was synonymous with “inconsistency”
(see
formal definition, below), both
terms at that time being handled quite informally by today’s standards. But, here at
the beginning of the 21st Century, paradox in a mathematical theory is (still, since
Cantor) considered
acceptable whereas formal inconsistency is not. Examples of standard results from
set theory that are considered to be paradoxical but not to make set
theory inconsistent are:
-
a0 + 1 = a0
-
that there seem to be the
same number of even numbers as numbers since it seems that they can be paired
n <->
2n
-
“paradoxical
measure” is considered to be a legitimate area of study in mathematics
(measure theory, but the paradox derives from set theory)
-
Cantor held that
a0
(the cardinality of the set of all natural numbers Á 4 {1,2,3...})
is the largest cardinality that you can possibly get by the continued
process of adding 1s; but Cantor also had the
“relatively
paradoxical”
concept of transfinite ordinals, and there adding “1s” to the first
ordinal (“1st”) eventually gets you not only to the first transfinite ordinal
ω,
but then to ω
+ 1 (which is > ω),
and then eventually to the transfinite ordinal Ω (which is >>>
ω)
which has cardinality
 >>> a0
-
by the
standard theorem concerning the cardinality of set exponentiation,
 <<<
(since the cardinality of the first
term is countably infinite, i.e.
= a0, and a0 is very much less than
)
-
if, using the standard axioms and rules of inference of set theory, one
subtracts the elements of Á 4 {1,2,3...}
1 at a time, the cardinality remains a0 , even when all the elements have
been removed and all that remains is the empty set (hint: finite
induction)
This brings up a touchy subject, one that
can perhaps best be communicated satirically. To the standard axioms and
rules of inference of mathematical theory we must add the hitherto
unrecognized...
These “RULES OF DEFERENCE”
turn what would otherwise be “inconsistency”
(very bad, and worse, formally unacceptable to mathematicians) into
“paradox” (not only acceptable to mathematicians, but even romantic). The first result above — a0 + 1 = a0
—
is “paradoxical” since it seems to imply a mathematically unfortunate
consequence, i.e. that 1 = 0, which we would get if we subtracted a0
from both sides of the equation. But set theory dances around the fact that this
is formal inconsistency by saying “DON’T!” subtract that
a0
from both sides despite the fact that transfinite arithmetic is derived
from finite arithmetic and should share the same rules of inference.
(Believe it or not, mathematicians will actually say that we must not do
it because we will get an inconsistency; it somehow
remains mere “paradox” as
long as we “DON’T!”)
And “DON’T!”
try to relate
a0 + 1 = a0
to the fact that if we set subtract {1,2,3...}
from e.g. {0,1,2,3...} we get {0} and not the empty set. In fact, set
theory has evolved a whole pantheon of “DON’T!”s
that lie totally outside the official formal axioms and rules of inference — or
any other formal part — of set theory, there
to keep things from slipping into... “inconsistency”.
More on these, later.
The second result is paradoxical for much the same reason, e.g. if we set
subtract {2,4,6...} from {1,2,3...}, instead of the empty set we get {1,3,5...},
which is far from empty.
But the sets {2,4,6...} and {1,2,3...} seem to have the same number of
elements if we go by the n <->
2n mapping, so subtracting them should... uhhh... well, anyway,
“paradox”,
but not formal inconsistency since we “DON’T!”
The other results follow along those same lines.
Modern mathematics, not just set theory, pretty
much depends on putting what is actually fatal inconsistency off to mere “paradox”.
And to a theory’s
rules of inference we tacitly add “RULES OF DEFERENCE”, the ever-growing lists
of “DON’T!”s.
(See
RULES OF DEFERENCE?! WHAT LIES HIDDEN BY THESE... OVERSIGHTS?!,
below.)
|
|
SECTIONS
DEFINITION
of “PARADOX”... AND HIDDEN “RULES OF DEFERENCE”
DEFINITION of “INCONSISTENCY”
DEFINITION
of “DEFINITION”
FINITE ARITHMETIC -> TRANSFINITE SET THEORY
RULES OF DEFERENCE?! WHAT LIES HIDDEN BY THESE... OVERSIGHTS?!
|
|
DEFINITION of “INCONSISTENCY”
First let's go back and look at a
standard definition of a “theory”.
THEORY
— A “theory” consists of a set of the axioms (the primitive
assumptions, standard by definition),
a set of rules of inference (again, standard by definition) , and all
the “theorems” (propositions) that can possibly be derived from
them.
We
need to emphasize this last point in the definition of a “theory” because it is an essential part of what
mathematicians overlook when dealing with questions of the inconsistency of
set theory. A theorem is still formally a theorem even if
mathematicians have failed to recognize it or discover its proof, even
if it contradicts a theorem already proven.
An
example: if in a finite arithmetic theory (there are several variants, and
many more possible) we have a theorem that a
= b, and we have the (finite) number c, then a
- c = b - c is also a theorem, since we must
derive every possible theorem that the rules of inference allow, and in finite
arithmetic there is a rule of inference that allows adding or subtracting
equal quantities from both sides of an equation (and still getting an
equality).
It
is actually somewhat easier to define “inconsistency” before
“consistency”, and then, if needed, define the latter in terms of the
former.
INCONSISTENCY — A
theory is “inconsistent” if it has as theorems both a proposition/theorem
and its negation, i.e., for emphasis, if it is even possible to
derive both a proposition/theorem and its negation/contradiction from the
axioms using the standard rules of inference.
We
also need to emphasize with regard to derivation:
In all mathematics, the elements of a
theory are
ostensibly “spacetime independent”. I.e. any axiom, rule of
inference, or theorem can be used at any “place-time” in a derivation. Set theory
does not stick to this formal requirement, and mathematics does not have adequate
arrangements for mathematicians not to stick to it. More on this, later.
We
should also note that there are 2 common ways for a theory to be
inconsistent:
The combination of axioms gives us an
inconsistency (under the rules of inference).
A single axiom can be inconsistent by itself in
what it assumes, making the combination inconsistent (again, under the
rules of inference).
This latter shows up in problems
associated with the standard Axiom of Infinity.
This
is historical fact. Note the... oversight with regard to the standard definition of
inconsistency:
If a theory is inconsistent, then one
must be able — theoretically — to validly derive (both sides
of) a contradiction, and
any statement to the effect that “the assumption that you can validly perform that
derivation is proven false (merely) by the fact that you arrive at a
contradiction” is per force a mathematically incompetent statement
to make under any circumstances whatsoever.
It would
be essential to question/determine whether it was derivable from only the standard
axioms using only the standard rules of inference, but — historical
fact — these imminent mathematicians did not do so, perhaps because the
derivations were so straightforward. The assumption they actually seem to
hold as most fundamental is that
set theory must be consistent, and therefore any
derivation/proof to the contrary is invalid.
|
|
SECTIONS
DEFINITION
of “PARADOX”... AND HIDDEN “RULES OF DEFERENCE”
DEFINITION of “INCONSISTENCY”
DEFINITION
of “DEFINITION”
FINITE ARITHMETIC -> TRANSFINITE SET THEORY
RULES OF DEFERENCE?! WHAT LIES HIDDEN BY THESE... OVERSIGHTS?!
|
|
DEFINITION
of “DEFINITION”
It
is standardly agreed in mathematics that when a definition is used in a
proof, that, in order for the proof to be valid, it is necessary that the
proof must remain essentially unchanged semantically when the original definition is
substituted for the defined entity. E.g. if we define the set of all
natural numbers, Á,
as Á 4 {1,2,3...},
in any proof where we use the symbol Á
we must be able to substitute not only {1,2,3...}, but
its original meaning-definition and so on, until we get to the
primitive entities whose existence is postulated in the standard axioms.
And when we do so, the essential meaning-semantics of the proof and its stages of
derivation must remain unchanged, even though the form(s) may be quite
different. (E.g., the proofs tend to get a bit longer.)
Historical fact — this “simultaneous existence” is not a “to the effect”
paraphrase; it is a direct quote, from a context like that paraphrased above.
Note the logical similarity to:
An
obvious problem with this new “simultaneous existence” is that if there is
a fatal flaw in the original construction, e.g. with regard to later usage, it
may remain invisible for quite some time... and has.
|
|
SECTIONS
DEFINITION
of “PARADOX”... AND HIDDEN “RULES OF DEFERENCE”
DEFINITION of “INCONSISTENCY”
DEFINITION
of “DEFINITION”
FINITE ARITHMETIC -> TRANSFINITE SET THEORY
RULES OF DEFERENCE?! WHAT LIES HIDDEN BY THESE... OVERSIGHTS?!
|
|
FINITE ARITHMETIC -> TRANSFINITE SET THEORY
Set theory is taken so much
for granted that it is no longer a field of specialization in mathematics.
We should emphasize that this is transfinite
set theory. Finite set theory is taken even more for granted, if that is
possible. And the origins of transfinite set theory are the victims of
serious... oversights.
Transfinite set theory
is formally derived from finite arithmetic (the number 1 and its
successors, created by cumulatively adding 1), but it is derived in such a
way that one cannot substitute the original definitions for the newly
defined entities, i.e. it is defined in such a way that the old axioms and rules of
inference must be abandoned if one is to maintain even a semblance of
theoretical consistency.
Obvious examples:
-
Set theory abandons the standard
rule of inference of finite arithmetic that says that
“equal quantities may be subtracted from both sides of an equation”
when it arrives at the transfinite arithmetic equation
a0 + 1 = a0
.
-
Set theory abandons the natural
relationship between set subtraction and arithmetic that existed in finite
arithmetic; e.g. even though one may set subtract the set {1,2,3...} from
{0,1,2,3...} and get {0}, one may not relate that to a falsification of
a0 + 1 = a0
since {1,2,3...} and {0,1,2,3...} are formally considered to have the same
number of elements (their cardinality).
-
Set theory ignores the
remainder of 1 when integer dividing 1 by any transfinite number, even though it
acknowledges the quotient of 0.
This last has to do with the
fact that Cantor and every leading set theorist up to the beginning of the
21st Century have felt the need to deny the natural existence of
infinitesimals, which we might as well call
“transfinitesimals”, and implicitly
affirm that the Archimedean Axiom cannot be extended to transfinites.
Set theory abandons (some
of) its origins
for the reasons described above, that if it did not do so, set theory
would obviously be inconsistent, since “1 = 0”
would be
considered too paradoxical to be mere paradox. Unfortunately, as we
also saw above, this ploy does not work on several grounds, and set theory
remains inconsistent even though we have disguised its inconsistency with
an agreement not to apply standard rules of inference in
standard ways.
Set theory’s excuse has chutzpah:
-
We
“DON’T!”
don’t subtract a0 from both sides of a0 + 1 = a0
because, if we did, set theory would be inconsistent!
(Therefore we
“DON’T!”
allow it, even though subtracting equal quantities from both sides of
an equation is allowed in the finite arithmetic which is the foundation of
set theory and its transfinite arithmetic)
We...
“oversight” the
fundamental formal definition of theoretical
inconsistency, that if it is even possible to derive a
contradiction then the theory is inconsistent. Transfinite
arithmetic now has its own “simultaneous
existence” and does not need to refer (completely) to its finite origins; indeed
it must
not do so, because if it does, that will “make” set theory
“inconsistent”. As a reminder...
If
mathematics — set theory in particular — is going to insist on proceeding as it has for well over a
century, it-we must make formal acknowledgement of the
formal role
of “RULES
OF DEFERENCE”,
the “DON’T!”s
that are so essential, but are not “spacetime/place-time independent” as is formally required
in current mathematics.
|
|
SECTIONS
DEFINITION
of “PARADOX”... AND HIDDEN “RULES OF DEFERENCE”
DEFINITION of “INCONSISTENCY”
DEFINITION
of “DEFINITION”
FINITE ARITHMETIC -> TRANSFINITE SET THEORY
RULES OF DEFERENCE?!
WHAT LIES HIDDEN BY THESE... OVERSIGHTS?!
|
|
RULES OF DEFERENCE?!
WHAT ELSE LIES HIDDEN BY THESE... OVERSIGHTS?!
Unwritten,
even unspoken as such, the hidden “Rules
of Deference”,
the “DON’T!”s
that are invisibly ubiquitous and so essential to current standard formal
theory, but formally disallowed by that same theory, are
waiting to be acknowledged by the mathematical community.
The hint of all that lies
hidden, of what is in store for mathematics, can be summed up as (repeated
here for emphasis):
Think of all the places
where derivations are rejected because of...
“lack of convergence”, “result undefined”, “contradiction”, etc. Now,
think: should (a consistent) theory have produced “convergence”, a
“defined result”, whatever...?
Set theory... well, new directions will be
needed. But we can all take heart: many, many, many papers will
need to be written to transition away from the standard inconsistency of
19th-20th Century set theory. It may even be that “inconsistency” can be
co-opted in the new theory — the new theories — that will evolve. This article should have given
even an unsympathetic reader at least pause with regard to the possibility
of inconsistency in today’s standard set theory. The
PAIAS articles on
Set Theory and its inconsistency follow up on the above and show how standard
inconsistencies derive from the fundamental result in set theory that
a0 + 1 = a0
. There are also further problems with the standard Axiom of Infinity.
|
|
