Re: Motion P1788/M0013.04 - Comparisons - Overflow / Infinity
Ian
On 20 Sep 2010, at 23:22, Ian McIntosh wrote:
> You're right, as I said Overflow is an infinite set just as Infinity is an infinite set, so [a, Overflow] represents an infinite family of intervals just as [a, Infinity] represents an infinite family of intervals. How does one evaluate
> [1, Infinity] \subseteq [1, Infinity] ? Which Infinity is larger? It's the same problem, one which I did not claim to solve.
NO! You are falling into two traps at once.
First, our interval model says intervals are subsets of the reals R; our notation says [1, Infinity] is just shorthand for {x in R | 1 <= x < oo}. Hence it is ONE set, not an infinite family of sets like [1, Overflow]. And standard set theory unambiguously says [1, Infinity] \subseteq [1, Infinity] is true.
Second, even if we used an interval model on the extended reals R*, e.g. csets, then Infinity is a "single member" of R* just as much as 3.75 is. Not a placeholder for lots of fuzzy things. So [1, Infinity] \subseteq [1, Infinity] is true there also, because ([1, Infinity] in that model) is just ([1, Infinity] in our model) with one extra point added.
That is what mathematicians have understood by R*, and the points -oo and +oo in it, since Lebesgue's "Sur une généralisation de l'intégrale définie", Comptes Rendus, 29 April 1901, and probably earlier; and it has stood the test of time.
It is also pretty clearly how the authors of 754, both the 1985 and the 2008 version, understand infinity.
John