Re: Non-standard arithmetic

On Jun 11, 5:36 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
apoorv says...



On Jun 10, 8:31=A0pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
Those particular axioms don't really define a nonstandard
model in any real sense. The point of nonstandard models
is that the usual rules of arithmetic apply to the nonstandard
elements, as well.
That is the issue. Non Standard models of PA do not distiguish between
the standard (finite)elements and nonstandard (infinite) elements.

That's sort of the *point* of nonstandard analysis, is that
you can work with infinite and infinitesimal numbers and
(under certain circumstances) use the same laws of arithmetic
as ordinary numbers. This allows you to replace the use
of limits in many applications by the use of infinitesimals.

For example, the usual definition of what it means for
f(x) to be continuous at x=0:

forall epsilon > 0, exists y > 0, forall z > 0,
if z < y then |f(x)-f(z)| < epsilon

The definition in terms of infinitesimals is much
simpler, and possibly more intuitive:

f(x) is continuous at x=0 if, whenever y is an
infinitesimal, then f(y) - f(0) is an infinitesimal.

The usual definition of the derivative of f(x) at x=0 is

df/dx = that k such that forall epsilon > 0, exists y > 0,
forall z > 0, if z < y, then |(f(z) - f(0))/z - k| < epsilon.

The definition in terms of infinitesimals is:

df/dx = that standard real k such that if z is an infintesimal,
then (f(z) - f(0))/z - k is infinitesimal.

It would be more reassuring to have a theory in which such a
distinction existed.

There *is* a distinction in nonstandard analysis, which is
that standard naturals are called "standard" while nonstandard
ones are called nonstandard (or hyperfinite). It's just
that if you have a statement of arithmetic that does not
use the term standard (or related concepts such as hyperfinite,
infinitesimal, etc.) then anything that is true of standard
numbers is true of nonstandard numbers.
The intended interpretation of the Peano Axioms is that the variables
range over standard naturals and the Axiom of Induction is stated with
that interpretation in mind.

If we know, a priori , that the axioms could have a model
with non-standard naturals, then the axiom of induction
,as usually stated , runs counter to all intuition -- every property
of all standard naturals automatically carries over to all non-
standard naturlals. We would perhaps have stated the Axiom of
Induction as:
[P(0) & An (P(n)->P(S(n)))]->[n is standard -> P(n)]
( with additional axioms to define/distinguish standard and non-
standard naturals).




The axiom Sc=c and therefore x+c=c or x*c=c
capture the  intuitive notion of infinity-that is, infinity augmented
by anything is infinity.

But why should there be only one infinity? The point of nonstandard
analysis is that "infinite" should intuitively be understood as
"very very big". There can be more than one very very big number.

In nonstandard analysis, you do have the corresponding
facts:

1. If x is infinite, then so is Sx.
2. If x is infinite, and y is positive, then so is x+y.
3. If x is infinite, and y is positive, then so is x*y.

What you are talking about is not nonstandard analysis, but
is an extended notion of arithmetic.

To me, the interesting feature of the axioms stated in the opening
post is that the theory incorprates addition and
multiplication and is omega inconsitent .
-apoorv


.

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