Re: Well Ordering the Reals

Tony Orlow wrote:
Robert Low said: 
If you keep the axiom of infinity, you have limit ordinals,
since the axiom of infinity is saying precisely that one
particular limit ordinal (omega) exists.
What it says, according to MathWorld, is that there exists a set that includes the null set, and for every element in the set, the successor is in the set, defining the Von Neuman ordinals. 

No, this does *not* define the von Neumann ordinals. It says that
one particular class of von Neumann ordinals is a set.

What causes an inductive definition of a set to imply the existence of elements outside the set? Basically, the idea is that the size is always greater than the largest value,, and so the size of the set of all finites is larger than all finites. But, that is easily remedied by starting with {1} instead of {0}, and having the size EQUAL to the largest value. It's a simple matter of definition and an error of 1 that requires limit ordinals, as far as I can see. Limit ordinals are a cute trick but not a good soluton. 

Sorry, but I'd struggle even to make up a story that makes sense
using those words. Trying to extract sense from them in that order
is completely beyond me. You seem to think that if we left 0 out
of the set of ordinals then there would be no limit ordinals, but
I can't imagine why you would think that. There would still be
the set {1,2,3,....} and it would still be a limit ordinal, not
a successor, and there still wouldn't be an element of the set
which was the number of elements of it.
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >> If you keep the axiom of infinity, you have limit ordinals, ... > of all finites is larger than all finites. ... Limit ordinals are a cute trick but not a good ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... Tony Orlow wrote: ... >> of all finites is larger than all finites. ... Limit ordinals are a cute trick but not a good ... Which appears to contradict your first paragraph above, ...
    (sci.math)
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