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in this theory. A ˄ denotes the set of all theorems
that can be proved from A in the proof. Finally, we set
ʱ
φ
to indicate that
is provable from
TRUE A / X
={ ʱ L A : I A |= ʱ }
where X is a placeholder index enabling reference to modifications in the underlying
language
L A and axiom system A . For instance, writing TRUE A / I will denote the set
of all first-order arithmetical formulae true on the standard first-order interpretation
I A
of arithmetic.
3.2.3 Context for the Theorem
Our context is set out diagrammatically in Fig. 3.1 . The reader will be able to make
sense of much of this figure given his/her assimilation of the review provided in
Sect. 3.2.2 . Here's the remaining explanation that is needed:
First, a circle C inside another circle C indicates a proper subset relation; that is,
C . The contents of each circle is just a set of formulae.
Four axiom systems of arithmetic appear in the diagram. At the inner core, in the
smallest circle, the system EA , “elementary arithmetic,” appears. More precisely,
every theorem (consequence) of the axioms EA is what composes the innermost
circle. Smith [ 18 ] refers to this axiom system as “Baby Arithmetic,” and I follow
suit, and so deploy ' BA ,' as can be seen the diagram. No one could take BA seriously
as an axiom system that captures what even moderately intelligent pre-teen humans
know about arithmetic. For example, while any true instance of an equation of the
form n
C
+
m
=
k is deducible from BA , and likewise any true instance of an equation
of the form n
k as well, BA is severely limited, since for instance it doesn't
even allow formulae and deduction with the quantifiers
×
m
=
. There is thus little
point here in saying anything further about BA , since even moderately intelligent
schoolchildren know truths of arithmetic that make use of variables and (implicit)
quantifiers (e.g.,
and
x
(
x
×
1
=
x
)
, which such children would recognize as x
×
1
=
x ).
In other words, an alien-fair
T gi would have to include questions like the following,
which aren't in the innermost circle.
Q
x
(
x
×
1
=
x
)
?
In Fig. 3.1 , I indicate that an agent in the innermost circle understands everything
within this circle. I do this by depicting the “face” of the agent inside this circle. But
notice that there is an arrow flowing from this picture of the agent that leaves the
innermost circle and travels to the immediate superset, that is, to the next circle. And
notice that this arrow has a check on it. What this says is that any agent of moderate
intelligence who has reached pre-college development will not only understand BA ,
but also, if they have an understanding of basic arithmetic, Q , or—as it is sometimes
known— Robinson Arithmetic .
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