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In fact, an interpretation of the Euclidean axioms was found where all the Euclidean
axioms were true apart from the fifth axiom which was false. This discovery was the
first meta-theorem of geometry, proving the mathematical impossibility of deducing
the fifth axiom from the other ones.
Let us consider an example of FOL theory. We use: i) three variables x
,
,
z rang-
ing over the individuals of some biological population with sexual reproduction;
ii) four predicate symbols G
y
F , that is, symbols of relations over the domain
consisting of our individuals, such that:
,
A
,
M
,
G
(
x
,
y
)
means x generated y ,
A
(
x
,
y
)
means x is an ancestor of y ,
M
(
x
)
means xisamale ,
F
(
x
)
means x is a female .
By using logical symbols, predicate symbols, variables, and parentheses we can
put the following sentences in formulae expressing usual facts about sexual
reproduction:
•∀
x
(
y
(
G
(
y
,
x
)
M
(
y
)))
(Everybody has a father)
•∀
x
(
y
(
G
(
y
,
x
)
F
(
y
)))
(Everybody has a mother)
•∀
x
(
M
(
x
)
F
(
x
))
(Everybody is male or female)
•∀
x
(
y
(
G
(
x
,
y
)
A
(
x
,
y
)))
(Parents are ancestors)
•∀
x
(
y
(
z
(
A
(
x
,
y
)
G
(
y
,
z
))
A
(
x
,
z
)))
(The ancestors of parents are ancestors)
•∀
x
( ¬ (
A
(
x
,
x
)))
(Nobody is self-ancestor).
These sentences constitute the axioms of a theory, consisting of all the logical conse-
quences of them. Can we interpret them in a different domain, with different mean-
ings for predicate symbols, in such a way that they turn to be true also in this new
interpretation? With a detailed analysis, we can discover that these axioms cannot
be fulfilled by a real biological population, consisting of a finite number of indi-
viduals. On the other hand, we can interpret coherently these formulae on natural
numbers. However, in which sense can we prove that the father's unicity is not a
logical consequence of the given axioms? And, in which sense the non-existence of
a common ancestor for all individuals,
, is a logical consequence of
them? Could we find an algorithm generating all the logical consequences of these
axioms? What does it mean
¬∃
x
(
yA
(
x
,
y
))
? Can this axiom be added to the
theory, while preserving its consistency (avoiding a contradiction)?
Peano's axioms of natural numbers are given by the following sentences ( P is
any predicate).
y
x
(
G
(
y
,
x
)
M
(
y
))
0 is a number.
Each number has a successor.
0 is not successor of any number.
Distinct numbers have distinct successors.
If P
(
0
)
and, for every number n, the implication P
(
n
)
P
(
n
+
1
)
holds, then, for
every n, proposition P
(
n
)
holds.
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