Re: infinity

Ross A. Finlayson said:
Have you heard of the counterexample in real analysis where there
exists real positive non-zero, say, iota, less than any other positive
real? Now you have.

No, I had my fingers in my ears and was humming, so it never happened. ;)


With that kind of notion of a nilpotent infinitesimal, nested intervals
does not hold. Also, integration at least looks a lot more intuitive.

There is something to be said for the notion, but you shouldn't rule out
further subdivision of infinitesimals either.


In a circular form of argument, in well-ordering the reals there is a
degenerate non-empty interval, or not.

I think that notion can work, if the continuity of the real line can be
accepted to be a matter of the scale or unit of measurement. It seems clear to
me that it can.


The natural naturals, are, how you say, as from Peano characterization,
incompletely described, as Paris and Kirby's observation is that some
true statements about the natural integers imply there being infinite
integers. I say characterization instead of axiomatization, for, many
platonists agree that Peano's assertions about the natural numbers do
not affect them. Instead it is definition, and in this and several
related cases there is more to those numbers than those incomplete, and
arguably inconsistent, axiomatizations.

For instance, the fact that the Peano axioms don't address the quantitative
nature of the naturals at all, and therefore can't incorporate the concept of
measure, but only count and order.


Nonstandard you might think it is. There is no universe in ZF.
Quantify over sets, in a set theory.

Sounds second-order to me, Ross. Maybe that's acceptable. I am thinking that we
really actually need to retool the logical proof aparatus itself from the
ground up by defining it more quantitatively. I'm working on that. maybe I'll
post a thread on it.


Ross



--
Smiles,

Tony
.

Relevant Pages

  • Re: abundance of irrationals!)
    ... All I know is that what we know about infinite ... > the sets I call finite have larges members. ... The set of all finite naturals is not infinite, ... Sets defined by mapping functions from the naturals to the reals which have ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... >> less than sqrthas no largest member. ... > The set of all finite naturals is not infinite, ... >> I WILL claim that your incomplete definition of cardinality ... > Sets defined by mapping functions from the naturals to the reals ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... the sets I call finite have larges members. ... I WILL claim that your incomplete definition of cardinality ... >> the naturals ... > Yes a function from the naturals to the reals, ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... don't have a definition for an arbitrary set of its "standard ordering" ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... you had said that the existence ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ...
    (sci.math)