Axiom talk:Axiom of Continuity

The reason I quoted p. 198 was because I had to decide whether to use "consider an infinite line" or "consider the infinite straight line". Ultimately it probably doesn't matter, as we haven't defined either yet. --GFauxPas 05:46, 27 January 2012 (EST)

No worries, I'll just remove the p. 198 reference. --prime mover 05:55, 27 January 2012 (EST)

I'm working towards the view that the place in the citation where the concept is initially defined, that's the only instance you use, otherwise every single mention of every single instance of that concept will need to be referenced. Also, I am not usually in favour of giving an actual page number as they may vary between editions (although with direct citations to an article this is less of a problem), and also it goes against good software design practice to link to a presentational attribute (in this case a page number) to reference a content attribute (in this case an axiom number). This whole section will come up for review when I'm back from Gloucester tomorrow. --prime mover 05:55, 27 January 2012 (EST)

Dedekind's Axiom

It's certainly similar to Dedekind's Theorem, but is this exactly the the same thing as [1]? If so, I should put it on the page, and create a redirect. Givant says:

We turn finally to Ax. 11. It is a modified form of the well-known Continuity Axiom which, in its application to the theory of real numbers, was first stated in Dedekind [2]. The modification consists in the removal of some additional conditions often imposed on X and Y such as the condition that X ∪Y is a straight line. This both simplifies and generalizes the formulation of the axiom, and in consequence facilitates its use. Ax. 11 appears to be simpler than all other statements securing the continuity of geometric spaces which can be found in the literature.

He doesn't give wolfram's definition though. --GFauxPas 07:59, 27 January 2012 (EST)

I think it is not the same, the statement on this page is more general (as Givant mentions). It does not require the partitioning of $\R$ into $X$ and $Y$. In any case, by imposing more on $X$ and $Y$, the definition of Wolfram may be recovered. In any case, this just applies to the second-order axiom. The first-order schema is of course something different altogether. --Lord_Farin 08:22, 27 January 2012 (EST)

Perhaps Dedekind's axiom is different from dedekind's theorem? --GFauxPas 08:26, 27 January 2012 (EST)

Yes, but they both partition what is to be thought of as $\R$, resp. is $\R$. This is different from what this axiom does. In any case, Dedekind's Axiom is an abstraction from the result for $\R$ (Dedekind's Theorem) to what may become new models of geometry (or just geometry in a different language). This is eventually also the purpose of the Axiom of Continuity. But the paths these two considered axioms take differ. --Lord_Farin 08:38, 27 January 2012 (EST)

They're completely separate things, from what I understand. The Axioms of Tarski are concerned solely with geometry, with no reference to real numbers (of whatever dimension) at all. OTOH Dedekind's Axiom and Dedekind's Theorem are an attempt to demonstrate (by an appeal to an intuitive conception of a "real number line") that real numbers exhibit "continuity" and can be viewed as a bridge between algebra (defining relations between numbers) and geometry (concerned with relations between objects in "space"). --prime mover 19:22, 27 January 2012 (EST)