Re: Calculus XOR Probability

Tony Orlow wrote:
Yes, and if the functions f anf g are defined over all the reals, then they can
be considered to be defined in the limit as well, and formulaic expressions of
oo can therefore be compared and ordered in this way. That's what's
interesting. If 2*x<x^2 for all x>2, and oo>2, then 2*oo<oo^2.


David R Tribble said:
That's a neat trick, considering that oo is not a member of the reals.
Apparently, defining a simple relation between the members of a
set (such as defining 2x<x^2 for all x>2 in R) can call into existence
new values in the set that previously did not exist in the set before
the relation was defined.


Tony Orlow wrote:
If we are talking about the hyperreals, then that's not happening. You can
insist there are no infinite values on the real line, but that's just refusal
to consider a new idea. Values aren't "called into existence". If those values
exist, then the formulaic expression applies to them.


Matt Gutting said:
Yes, but first, if you're not "calling the values into existence", you have to
prove that they exist.


Tony Orlow wrote:
No, I can make a statement that depends on whether they exist or not. (E oo A
neN oo>n) ^ (E meN A n>m f(n)>g(n)) -> f(oo)>g(oo).

So instead of proving that oo exists in the system of standard
arithmetic, you instead have added an axiom that defines oo.
The result, of course, is no longer standard arithmetic, but a system
based on the set R U {oo}, a.k.a. the real projective line.
(See http://en.wikipedia.org/wiki/Real_projective_line)


So, IF an infinite value
exists, and f(n)>g(n) for all n greater than some m, then we can conclude that
for this infinite value oo, f(oo)>g(oo). Tada! Infinite induction.

And here is your axiom that defines what you call "infinite induction".

Can we assume that these are the first two axioms of the T-arithmetic
system?

Of course, you're still missing axioms (or theorems) that define
your "standard unit infinity" Big'Un (is it the same as oo?),
Little'Un, and the basic arithmetic operations that apply to them.

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... Daryl McCullough said: ... > Tony Orlow says... ... >>the axiom of infinity as it stands, and perhaps suggest replacing it with the ... >>something that doesn't call the finite set infinite. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... Tony Orlow wrote: ... If each "increment" has an infinite number of others following it, as is the case, it does no matter how many or how few it has preceding it. ... One first gets the set of naturals via the inductive axiom, one then shows that the set produced by that axiom satisfies our mathematical definition of an infinite set. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... Tony Orlow wrote: ... I can create a system that is an extension of the reals by defining ... Without this axiom, ... He claims that the Peano axioms allow for infinite numbers, ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... >> Tony Orlow wrote: ... > of an infinite set, ina stepwise manner, forever. ... induction, that in the infinite for however finitely many there's ... axioms of ZF besides regularity are theorems of the null axiom theory's ...
    (sci.math)
  • Re: A puzzle for Cantorists
    ... Any universe of sets used as a model of ZF ... For example, I "believe" in the Axiom of Choice, ... comprehensive line of argument about why and how infinite sets are ... the hierarchy" is not mathematics. ...
    (sci.math)