Re: Galileo's Paradox

Eckard Blumschein wrote:
On 11/29/2006 7:36 PM, Tony Orlow wrote:
Eckard Blumschein wrote:
On 11/29/2006 3:58 PM, Tony Orlow wrote:
where one set contains all the elements of another, plus more, it can rightfully be considered a larger set.
All of oo?


Yes. All of the naturals are integers. Only half of all the integers are naturals.

All of the points in (0,1] are in (0,2], but only half of all of the points in (0,2] are in (0,1].

You are equating two quite different notions: smaller and half as large.

Smaller means less large, meaning less than 1 times as large, as in a fraction in [0,1), such as 1/2. What is different about smaller and half as large?

In case of two finite heaps of size a and b of numbers, a=b/2 implies a<b.
Generalize where possible. Why is this not true in the infinite case?

In case of a=oo and b=oo, we may have a=b/2 while a is not smaller than
b but simply not comparable: oo = oo/2.

So, you adhere, then, to the tenets of imaginary alephs and the creed of transfinitology? What you say is true for the standard generalization from the finite using the *existence* of a bijection, called cardinality, but to claim that there is no valid justification for seeking a formulaic method that produces more intuitive results is an empty statement. To say that there are not twice as infinitely many point in (0,2] and in (0,1] is equivalent to claiming that there are more points in one or the other of (0,1] and (1,2], if there is any correlation at all between this infinite count, and measure, which there clearly can be.

Tony
.

Relevant Pages

  • Re: Galileos Paradox and the Project of the Reals
    ... the positive integers which I defined the other day. ... A finite real, then, may be defined as any finite natural, or any number between any two finite naturals on the real line, by subdivion of the unit interval. ... We can also construct a linear enumeration of the reals using powers as I suggested with the H-riffic numbers. ...
    (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)
  • 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, ... number of reals on the line. ... to the subsequent logic that claims such a set cannot have infinite values. ... standard orderings, since sets in general don't come with little tags ...
    (sci.math)
  • Re: infinity
    ... >>> surjection into a proper subset is a generally good definition. ... >> about the surjection definition of infinite, ... All finite naturals are finite, ... Constant relationships such as this can be taken to infinity, ...
    (sci.math)