Definition talk:Minimally Inductive Set

To be useful, this definition requires the use of the Axiom of Infinity - otherwise it is not well-defined.

The definition:


 * $\omega = \{ x : ( x \cup \{ x \} ) \subseteq K_{II} \}$

Where $K_{II}$ is the collection of all ordinals that are not limit ordinals does not require the axiom of infinity. Furthermore, it is the definition given in Takeuti. Andrew Salmon 20:12, 9 February 2012 (EST)


 * The way it's defined on the page is easier to understand than your definition, as I have to figure out what a collection is and what limit ordinals are to understand yours. --GFauxPas 20:31, 9 February 2012 (EST)


 * Hmm...I guess this ties back into the philosophy of the wiki in general. If we can define something in a way that requires fewer axioms, but is more difficult to understand for someone inexperienced in the subject, which definitions do we give.  Do we give the fewer-axioms definition, or the easier-to-understand definition.  What if (let's suppose) this definition's use requires the existence of an inaccessible cardinal? Andrew Salmon 20:56, 9 February 2012 (EST)


 * How about both definitions? --GFauxPas 21:18, 9 February 2012 (EST)


 * That sounds best...and we need to prove as a theorem that the two are equivalent. Andrew Salmon 23:16, 9 February 2012 (EST)

Why do you need to define this without the Axiom of Infinity? Mister Effin Naive here wants to know: since AoI is one of the basic ZF axioms that we adopt before starting these definitions, what's the problem with taking it on?

You also got to admit that before you can use this definition, you also got to define an ordinal, whereas the original simple definition doesn't need it.

But the problem is bigger than this, as follows.

Takeuti's approach is from a completely different direction. I worry that if we merely attach Takeuti's definitional approach onto the bottom of the pages which have been generated according to the Halmos / ZF approach, there is going to be a serious problem with tracking the argument through. Such-and-such a step depends on a particular definition / statement which has been created according to ZF - but then it's been changed to match the Takeuti approach (or that there's a second section to the page from which that does not follow).

We have got round this in the case of geometry by setting up a completely new axiom schema (see Tarski's Axioms) for which the equivalence to Euclid's defs is an ongoing exercise to be demonstrated. I wonder whether the same needs to be done here. Otherwise there is no clear set of axioms from which the entire website can be said to derive its proofs and then we'll have to rename the site VagueNotionOhYouKnowWhatIMeanItsObviousInnitWiki. --prime mover 01:27, 10 February 2012 (EST)


 * If Takeuti doesn't use ZF, then a similar approach to Tarski's geometry is required. I'm not sure about it at the moment. --Lord_Farin 03:17, 10 February 2012 (EST)