[FOM] Infinity and the "Noble Lie"

[joeshipman@aol.com]

Boucher:

>Still it
>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.)
>

I reply:

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?

-- JS