Re: Can ZFC prove Addition is Associative?



"G. Frege" wrote:

On Mon, 12 Feb 2007 16:53:41 +0100, G. Frege <nomail@invalid> wrote:


For simplification our discussion let's drop negation and just stick
to the "implicational part" of propositional logic, ok?

So Frege's system consisted of the following axioms (for the
"implicational part" of propositional logic):

B1 p -> (q -> p)
B2 (p -> (q -> r)) -> ((p -> q) -> (p -> r))
B3 (p -> (q -> r)) -> (q -> (p -> r))

Instead of removing B3 as deducible from B1 and B2, we
weaken B2:

C1 p -> (q -> p)
C2 (q -> r) -> ((p -> q) -> (p -> r))
C3 (p -> (q -> r)) -> (q -> (p -> r))

but this is no longer complete [...]. So we strengthen the system
by adding Hilbert's axiom [ (p -> (p -> q)) -> (p -> q) ]:

D1 p -> (q -> p)
D2 (q -> r) -> ((p -> q) -> (p -> r))
D3 (p -> (p -> q)) -> (p -> q)
D4 (p -> (q -> r)) -> (q -> (p -> r))

Right.

Just stumbled over the following historical fact:

In his article Logische Grundlagen der Mathematik [Logical Foundations
of Mathematics], 1922, D. Hilbert introduces an axiom system (as a
logical framework for mathematics). Again I'll just consider the
"implicational part" of propositional logic:

"I. Axioms for Implication

A -> (B -> A)

A -> (A -> B) -> (A -> B)

(A -> (B -> C)) -> (B -> (A -> C))

(B -> C) -> ((A -> B) -> (A -> C))"

This is exactly the system D1-D4.


Well, well. :-)

--
hz
.

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