Re: infinity

David wrote:

So is w the smallest infinite natural?
If so, is w-1 or w/2 defined? If so, are they finite or infinite?
What is w+1 or 2w? How do you compare (order) infinite naturals?
------------
I hope you've already read my article on my web site about " Inductive
Infinite Numbers"
If you haven't, then see: http://zaljohar.tripod.com/infinite.txt

Now back to your question, I defined natural number as that number
which can be DIRECTLY defined in terms of the meaning of one and
summation and predecessor, A number like -1 for example is not a
natural number becuase we should know the converse of summation
operator which is "subtraction" in order to define it, so -1 is
indirectly traced to the meaning of one and summation and predecessor ,
while natural numbers should be directly traced to these.

Now w defined in my article comes directly from the meaning of one and
summation and predecessor, it doesn't require the knowledge of
subtraction to define it.

Now w-1 is of coarse smaller than w but it cannot be defined without
subtraction operator
exactly like -1 above, that's why w-1 is an infinite Integer but not an
infinite natural,

About w/2<w. w/2 is an infinite rational number , and you know
rational numbers are not natural numbers.

About the third question of yours"What is w+1 or 2w? How do you
compare (order) infinite naturals?

I already answered that in my article so please read it , but in breaf
it is like below:

w+1,w+2,w+3,................,2w,2w +1, 2w +2 ,2w +3,.............,3w,3w
+1,.....,4w,...,5w,

....................................................., w^2,w^2 +1, w^2
+2,............., w^2 +w,....,w^2 +2w

w^2 +3w,...........
,2w^2,...,3w^2,........,w^3,...,w^4,....,w^5,..................,w^w ,
w^w +1,.....

.........endlessely.

In ascending order.

About the descending order from w (The Infinite integers)

w,w-1,w-2,w-3,........,3,2,1,w-w=0,-1,-2,-3,.....,-w,-w-1,-w-2 ,...(
same as above but in negative)

It can easily reasoned.

so what is the problem?

Please read the article I refered to.

Zuhair

.

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: infinity
    ... So is w the smallest infinite natural? ... natural number becuase we should know the converse of summation ... Now w-1 is of coarse smaller than w but it cannot be defined without ... compare infinite naturals? ...
    (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)