[FOM] Infinity and the "Noble Lie"
joeshipman at aol.com
Wed Dec 14 13:15:14 EST 2005
>is not clear to me that the Eda proof does not use notions of
>infinity. Specifically, can the PNT be proven in second-order
>arithmetic minus the Successor Axiom? (The Successor Axiom, by
>saying that every natural number has a successor, gives the natural
>numbers its infiniteness.)
The successor axiom is not what I regard as a use of "actual infinity".
What I care about is whether, when the proof is formalized in ZFC, the
ZFC "Axiom of Infinity" must be involved. If not, that means that the
theorem (if it can be stated in the language of arithmetic) can
actually be proved in Peano arithmetic, and Peano arithmetic is a
system about finite objects only. (The "potential infinity" involved
doesn't concern me.)
There is a strong isomorphism between Peano arithmetic and ZFC minus
the axiom of infinity. Define a bijection between the natural numbers
and the hereditarily finite sets as follows: if n is a natural number,
expressed as a sum of distinct 2-powers 2^i_1 + 2^i_2 + ... + 2^i_k,
then f(n) is the set whose elements are f(i_1), f(i_2),...,f(i_k).
Conversely, if X is a set, then g(X) is the sum over elements y of X of
2^g(y), so fg and gf are identity functions.
Since exponentiation is definable in PA, you can define each system's
basic relations and functions in the other system, and prove the
appropriate axioms (you may have to add the *negation* of the axiom of
infinity to get this equivalence).
There is no proof of Con(PA) in ZFC that does not use the axiom of
infinity; and if you believe that consistent theories have models then
believing Con(PA) really is the same thing as believing in an actual
infinity. But a finitist can say that he believes no contradiction can
be found in PA while denying that PA has a model -- he thinks the
proof of Godel's Completeness Theorem is, not wrong, but just
meaningless, because it speaks about infinite objects.
This is a subtle point. Godel's Completeness Theorem can be proved in
WKL0, which is conservative over Peano Arithmetic, so any consequence
of Godel's Completeness Theorem that speaks about integers only must be
accepted by a finitist who accepts only Peano Arithmetic. But the
finitist can reject the infinite model that Godel's Completeness
Theorem says exists, without being forced to deny Con(PA), because the
equivalence of consistency with having-a-model presumes the Axiom of
Infinity, even though the FINITARY consequences of the Completeness
Theorem don't depend on the Axiom of Infinity.
Regarding your other point -- can you provide an example of a statement
which can be proven in ZFC, and cannot be proven without the Axiom of
Infinity, but which (in the presence of the other axioms) does NOT
imply the Axiom of Infinity?
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