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Transcript
1
Embeddings
This is part of a longer thesis advancing a refutation of strong AI. To download the thesis visit: Poincare’s thesis
For an introduction to the work as a whole visit: Introduction to Poincare’s thesis by Peter Fekete
1.1 The story so far
Arithmetic is based on the set
 0,1,2,3, ... , which is an unbounded domain of potentially infinite
elements; it is equipped with a unique principle of reasoning, complete induction, that enables one to
infer from finite premises to a conclusion about the set as a whole.
In this way the collection
 0,1,2,3, ... is “completed”, or, “the boundary is attained”, because we have successfully inferred
from any to all. An analytic logic cannot be defined directly over
for it forms a chain and the
lattice that it generates is just another chain; furthermore, because
has no upper bound, the
resultant lattice has no maximal point that can serve as 1 in an analytic logic. Arithmetic is not prima
facie a form of analytic logic.
If arithmetic is a form of analytic logic after all, then the set
 0,1,2,3, ... must be
embedded within a structure over which an analytic logic can be constructed. With this in mind, firstorder set theory has been developed.
In this theory a collection of entities known as ordinals is
defined; these include transfinite ordinals. The ordinal that most concerns us is  , which represents
the notion of an actual, or completed infinity with the same members as
, but now conceived as
bounded into a set, which is a definite, determinate multiplicity;  is said to be the first ordinal that
follows in the sequence 0, 1, 2, 3, ... and to be the first ordinal of infinite size, or cardinality. An
analytic logic can be defined over  as follows: imagine that the extended real line, 0,1 , is
partitioned into  segments that we shall call atoms. Although these atoms may be numbered and we
may count them from 0 to 1 this primitive relation is ignored when we construct the lattice over it, for
otherwise we shall not obtain an analytic lattice as required; the lattice that arises is the Cantor set,
2  0,1 . In order to assess the claim that arithmetic is analytic, we must consider the possibility

that all the theorems of arithmetic, including all those that depend on the natural order of
and
principle of complete induction, may be derived from the analytic properties of this structure.
The Cantor set is subdivided into “regions” that correspond to its ideals and filters. One of
these is the proper ideal of all finite subsets of  . Under the Boolean Prime Ideal theorem (Boolean
representation theorem), which in turn rests upon the Axiom of Choice, this ideal can be extended to a
maximal ideal  with maximal element  [Chap. 5 / 7.42]; likewise, there is a prime filter (ultrafilter)
 with minimal element  . Under the assumption of the Axiom of Choice,  and  become sets and
we have     . The distance of  from 1 is 1 unit in the metric, and  from 0 is also 1 unit. Hence,
the Axiom of Choice effectively transforms  into a co-atom (prime) and  into an atom of the
complete one-point compactification,



  0,1,2, ... ,      .
1.2 First-order theories in general
Analytic logic subdivides broadly into two parts: 1.
Pure analytic logic, which comprises the propositional and predicate calculus.
Also, the predicate calculus with identity may be included in this category.
2.
First-order theories in which a theory about some specific class of objects is
said to be embedded in first-order logic; first-order logic is said to provide the
underlying logic of the theory.
Examples of this are first-order Peano
arithmetic, first-order set theory and the first-order theory of rings.
First-order theories are defined by the addition of axioms to those of first-order logic. The axioms
usually form a list that can be mechanically generated (recursively enumerated). It is also claimed that
all mathematics can be written and conducted in first-order set theory, so that any other first-order
theory could be regarded as a species of first-order set theory.
First-order theories fall into two
categories: 1.
Those theories that are “complete”, or if they are not complete then could be
completed.
2.
Those theories that are “essentially incomplete”. 1
The questions that arise are as follows: 1.
How is the underlying lattice modified or altered when the proper axioms of a
particular first-order theory are added?
In particular, are new lattice points
added when this happens? How, in terms of the lattice to which the logic is
related, are the models of first-order theories related?
2.
Why are some first-order theories complete and others incomplete, and what, in
terms of the lattice, does this distinction entail?
The axioms that define propositional logic collectively may be taken as a name of 1 in the
correspondent lattice. [4.10.3] Their role is to define the structure of the lattice and hence they may
be conjoined to any lattice point   1   . It is a theorem that these axioms are realised in a lattice of
just two elements; the prime indecomposable algebra 2  0,1 .
This is effectively a proof of the
completeness [Chap.4 Sec.6] of the axioms, because it demonstrates that every lattice of which 2 is a
factor is a lattice in which those axioms are realised. Since 2 is a factor of every (Boolean) lattice
1
For technical explanation of this term see Monk [1976] chapters 13 to 16, and Tarski, Mostowski and Robinson
[1953].
whatsoever, every lattice inherits those axioms: completeness indicates properties that are inherited
by all lattices.
A formal predicate logic could be defined over a finite lattice, but in this case those lattice
points corresponding to quantifiers could all be eliminated in favour of finite joins or meets (finite
lattice points) so such a logic would be merely a propositional logic masquerading in disguise.
Therefore, it is natural to consider that a formal predicate logic is one where there is at least a
countably infinite collection of lattice points. In the Gödel-Henkin completeness theorem [4.1 above]
the property of completeness is attached to all predicate logics; this theorem explicitly attaches this
property to every denumerable lattice, which is the minimal lattice to which a non-trivial predicate
logic applies. The proof of the theorem works by constructing a model – that is a lattice – out of the
very terms of the language out of which the logic is itself constructed: a denumerable lattice takes the
place of 2 in the predicate version of the completeness theorem, and acts as the minimum model.
This model is not irreducible in the same sense in which the Boolean algebra 2 is irreducible, but it is a
kind of factor nonetheless.
The effect of the addition of axioms  to those of predicate logic is a combination of one or
more of the following: -
1.3 The addition of axioms
1.
The axioms may force the lattice to contain a minimum number of lattice points;
in other words, to affect its cardinality.
2.
The axioms may define a lattice point within a lattice. A lattice point defines (a)
an ideal or down set – every lattice point that falls in this down set is an
instance of  . If  is such a lattice point then      . All lattice points in the
ideal correspond to models of  .
Also we have (b) a filter   ... of
consequences of  - every member of the filter represents a structure that is in
some sense a dilution of the structure defined by  .
3.
The axioms may define second-order relations on the lattice. For example, the
axioms of equality

define relations on the filters contained in the lattice.
[5.1.7]
1.4 Example, the axioms of number theory
In addition to those axioms that govern equality, the axioms of formal number theory
include the following: -
S1
S 2
S 3
S 4
x  y  x  z  y  z 
x  y  x  y
0  x
x  y  x  y
These four axioms cannot be satisfied in a finite domain. They firstly imply that every
element x of the domain has a successor x  ; that is,   x  y  y  x  2, and this is
impossible in a finite domain without circles, which are ruled out by 0  x  .
2
  x  y  y  x    x  y  y  x    y  y  a  1  a  0  a  0  0
This
illustrates principle 1 above: The axioms may force the lattice to contain a minimum
number of lattice points – to establish its minimum cardinality.
The possibility of adding more axioms to an existing theory creates a picture of the lattice being
embedded in another larger lattice, and of these axioms defining both an ideal and a filter in that
larger structure. Every Boolean lattice is capable of being embedded in a yet larger lattice, and the
collection all Boolean lattices forms an unbounded collection – in other words, a proper class. Once
we progress from first-order logic to a theory embedded in first-order logic we progress to a model in
which the theory corresponds to a lattice point not identical to 1, and hence defining both a filter and
an ideal in the lattice. All lattice points that fall within the ideal correspond to models of the axioms.
Every lattice point is the disjunction (join) of other lattice points. The lattice may or may not be
atomic.
1.5 Example of embedding a theory within the lattice
Consider the theory of
2

2
. This is a particular ring. The members of this ring are


equivalence classes 0  0,2,4,6,... and 1  1,3,5,7, ... . Its model is the set 0, 1
conjoined with the ring axioms. This model is already based on a prior partition of the
space
into these two equivalence classes. Then 0  and 1 stand for two mutually
exclusive atoms, so 0  1  0 and 0  1  1 . (This is the Boolean algebra 2.) Let us
attempt embed this theory within the lattice defined by base partition of 0,1 into
parts.
Then
0,1 , 0,3 , 2,1 , 2,4

all particular models of the theory – they are
examples from an infinite list of sets. The model as a whole is any one of these, so we
attempt to form the disjunction: 0,1  0,3  2,1  2,4  ... , but this is not possible as
it stands because this disjunction gives the 0 of the lattice. We must treat each member
of the list
0,1 , 0,3 , 2,1 , 2,4 , ...
as a new atom, that is an ordered pair:
 0,1 .
Hence, if we start with a partition of the lattice in which the atoms were represented by
0 , 1 , 2 , ... , in order to interpret
2
in this framework, we must rewrite the atoms in
terms of more fundamental atoms:  0,1 ,  0,3 , ... and the model is the disjunction (join)
of all of these. Let  stand for this join of atoms. This lattice point,  , has at least two
different representations.
(1) As the join,    0,1 ,  0,3 , ...  ... of the atoms of the
lattice that constitute all its individual models; (2) As the meet of certain axioms, for
example, those governing the predicate calculus, those governing the ring theory and
one concerning the size of the domain of the ring.
It would be as well to pause to reflect at this point how profoundly our picture of the lattice has now
altered as a result of the change up from (a) first-order logic to general idea of (b) a theory embedded
in first-order logic.
The lattice corresponding to first-order logic is already a lattice based on a
potentially infinite partition, but the axioms of the theory govern the whole lattice and conjointly are a
name of 1 in the lattice. The lattice corresponding to a first-order theory must also be an infinite
lattice, but a larger one3 – the possibility of an actually infinite partition arises; the theory itself,
denoted in general by  , represents a lattice point somewhere in the middle of the lattice. This lattice
point is itself a join of infinite atoms4, so it does not belong to the ideal of all finite subsets of the
lattice and already represents a species of limit point in the lattice – that is, an infinite join. In order
to reach this lattice we have had to rewrite our atoms. Thus, even if a lattice is atomic it is always
possible to embed that lattice into another lattice with atoms that lie below those original atoms. A
finite illustration of this phenomenon would be simply what happens when we step from the Boolean
algebra 22  1, 0,p, p  to the Boolean algebra 24  1, 0,p, p,q, q  ; we have doubled the number of
atoms, so that combinations that appear to be atoms in 22 become joins in 24 . (Compare p with
p  q .) What is an atom is relative to the partition of space – and when the partition is finer then the
atoms split. What this reveals is that what is atomic in a lattice is not absolute.
1.6 (+) Definition, floor, lowering the floor, ceiling
A given atomic lattice has a set of atoms which shall be called the floor of the lattice. We
progress up the lattice by means of the lattice algebra, or equivalently, by the logical
operations defined over the lattice. But now we discover that there is an alternative
route that takes us out the lattice to another lattice, a progression that is possible in the
proper class of all lattices; we can progress to different atoms.
This process shall be
called lowering the floor. For example, when we rewrite the atoms of the lattice from
0 , 1 , 2 , ...
to  0,1 ,  0,2 ,  0,3 , ... , 1,2 , ... we are lowering the floor of the lattice. We
lower the floor when we progress from a model of
to a model of
2
. The co-atoms of
a lattice constitute its ceiling and when we lower the floor of the lattice we also raise its
ceiling. Lowering the floor is a process of embedding a lattice within a larger lattice.
Boolean spaces
Boolean lattices
Duality
Raising the ceiling
Embedding of lattices
Refining the partition
Lowering the floor
This is part of a longer thesis advancing a refutation of strong AI. To download the thesis visit: Poincare’s thesis
3
I mean here “larger” in the sense of proper subset; the larger lattice may have the same cardinality as the smaller
one.
4
If the lattice is atomic; otherwise, it is generated from below by an infinite set.
For an introduction to the work as a whole visit: Introduction to Poincare’s thesis by Peter Fekete