Re: Peano's Axioms

I'm surprised no-one else has answered this, but here goes...

Why the following is not mentioned as an axiom among Peano's Axioms.

Ax( x is number -> Ey( y=S(x) ) ).

It is not included nowadays, for the same reason that one of the
original
"axioms" of group theory, (closure under multiplication), is not.

They are automatic consequences of the use of the function
symbol or an operation symbol... EVERYTHING is assumed to have
a value when acted on by a (logical) function. Being a logical
function
is ontologically prior to and more basic than being a mathematical
function, these latter DO have to be proved functions.
(By axiom or otherwise.)

It's a bit of a toss-up which approach one takes.

Because its wrong.
Zero also belongs to N.

It is not wrong, but the above remarks show it is trivially true.
I suspect the latter poster got his successors and predecessors mixed.

------------------------------------------------------------------------
Bill Taylor
W.Taylor@xxxxxxxxxxxxxxxxxxxxx
------------------------------------------------------------------------
Successor-preserved properties hold universally if initially.
------------------------------------------------------------------------

.

Quantcast