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Science Forum Index » Logic Forum » Additional Axioms for (1st order) Formal Systems.
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| namducnguyen |
 Posted: Tue Jan 13, 2004 12:08 am |
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Hi all,
Sometimes last year, I posted a question on whether or not there could
always
be additional axioms to any 1st order theory that's strong enough to
carry out the
concept of infinity. Still haven't found answer for the question, but I
now think
there is a possible hope for the quest, if we restrict our attention to
the 1st
order theory of N, instead of a more general formal system. The hope is
that if
we could always add axioms to N, then perhaps a "parallel" mechanism
might be
possible for a more general formal system [capable of portraying
infinity]. In
this post, however, we'd focus our attention solely to N, the (1st
order) theory
of the natural numbers.
First of all, let's assume the convention that the letter N could denote
either
the theory of the natural numbers, or a model of the theory. Secondly,
any natural
number as well as N, as a model , are (ZF) sets. Thirdly, we'd assume
the normal
Peano axioms that N is a set of natural numbers iff:
pa1. There exist an 1-1 function f from N onto N.
pa2. There exist an element 0 in N that has no inverse image w.r.t. the
function f.
pa3. Induction axiom: if there is a subset S of N such that if 0 is in
S and
(n \in S) => (f(n) \in S), then S = N.
Fourthly, given a general non empty set N, if a function f in N
satisfies pa1 - pa3,
then f is said to be an NF (Natural Function) function. And finally, by
N1 we mean a
formal system with only pa1, N2 with pa1 plus pa2, and N3 is just N.
Now, there seems to be a conflict in our objective to state however many
axioms
we've added to N, we could always add more:
a) on the one hand, axioms in this context are *non-logical* and,
together with the
non-logical symbols, these axioms will determine the theory
question. In other
words, if the objective of a formal system F is to enable us to say
something
about certain "reality", then we have to *clearly state* the
non-logical axioms.
For example, it's sort of silly to state that ZFC is ZF plus one more
additional
axiom - without precisely stating what that axiom [AC in this case] is.
b) on the other hand, our attempt to show *additional axioms are always
possible*
would mean that when all said and done, there would be *infinite*
numbers of
axioms to content with. How on earth could we possibly "clearly
state" infinitely
many number of axioms?
Given the difficult conflict above, in this post, we'll consider the
following
compromise: there is not much we could do about the infinite numbers of
axioms
in b), therefore in a), we would "relax" the requirement of clearly
stating an
additional axiom. Instead, we'll construct additional axioms that are only
"carrier" axioms: their sole purpose is to "allude" to the existence of the
"intended" axioms and in this way we don't have to explicitly state them
[the intended axioms].
Additional Axiom (Meta) Theorem
===============================
Starting from N, at least one more additional axiom can be added to the
current
formal system; and the adding process can be repeated indefinitely.
Proof - Construction of new axiom(s):
1) Let N3 = N, the normal 1st order theory of natural numbers.
2) Let N4 be N3 plus the following axiom:
There exists an equivalence relation R of NF's in N (as a model) and
a particular
NF function, say f1, such that
~(f \in R[f1])
with R[f1] being an equivalence class of R containing f1.
3) Let N5 be N4 plus the following axiom:
There exists an equivalence relation R' - not necessarily different
from R - and
2 NF functions f'1 and f'2 such that:
~(f \in (R'[f'1] U R'[f'2]))
4) For a general formal system Nk then (where k >= 4), we can add the
following axiom:
There exist an R of NF's and n n NF functions f1, f2, ..., fn such that:
~(f \in (R[f1] U R[f2] U ...U R[fn])) (4.1)
[as many n as it takes to repeat the adding process from N3 to the
current Nk.
Please note also that w.r.t. to the current system Nk, 'n' is a
constant, not a variable.]
Motivation for the above axiom-construction
==================================
The general form (4.1) of an additional axiom basically axiomatizes
that there exist an additional
NF function f of different "type": different type than those types that
have been used so far.
[Here a "type" is basically an equivalence class R[fi] of NF functions,
as expressed in (4.1)].
However odd the whole idea of constructing additional axioms as
"carrier" axioms may sound,
there is something related to N that actually suggests the idea. The
difference between N1 and N2
is pa2, which suggests the NF function f between N1 and N2 has undergone
a "type speciation": from
a general 1-1 function to only 1-1 that has 1 element without inverse
image. From N2 to N3 = N, the
speciation has gone an additional step further, namely f now must also
be inductive. The expression
(4.1) is then nothing more than an implied further type speciation on
the NF f.
Many thanks in advance for any correction or comments on the whole idea.
---Nam |
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