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)

I agree with PM that Assuming the AoI isn't a problem, as ZF(C) is the default. But it doesn't necessarily follow that we don't want equivalent definitions. But this assumes the definitions really are equivalent, and I have no idea whether that's true. If Takeuti uses ZF, then just adding "this definition assumes the existence of ordinals" or whatever would be good enough. In any event, I'm not sure something has to be provably extant to be defined. We can define a proper class even if they don't exist, can't we? --GFauxPas 06:34, 10 February 2012 (EST)

Your last comment highlights the different philosophical schools of thought between the intuitionist and classical schools of mathematics. Intuitionists do not accept the existence of any object they can not construct or demonstrate a means of construction of. If something is proven not to be false, an intuitionist would still not accept that it is true.

For my own part, I am prepared to accept the existence of classes, if there is an axiom system that declares that existence. But it is a big mistake to just tack on a load of class-theoretical definitions to the existing set-theoretical ones until we have made the effort to put in place that axiomatic structure. If we then find we need two completely separate threads from there: one for set theory and one for the equivalent results in class theory, then so be it, and we can bridge that come when we cross to it. --prime mover 10:36, 10 February 2012 (EST)

Just a comment on Euclid, I thought that his geometry as usually presented was incomplete because it was unable to prove Pasch's axiom. Takeuti does use ZFC, but he doesn't introduce the Axiom of Infinity until after he introduces $\omega$. I thought it was still significant that we can construct the natural numbers in a finitistic (if that's a word) universe. Andrew Salmon 10:12, 10 February 2012 (EST)

Maybe it is, maybe it isn't. It may not use AoI but it does use that $K_{II}$ (whose abbreviation is even more opaque than its definition) which needs to be demonstrated that this also arises from an axiomatic framework that may or may not be ZFC. Although the way it uses the language of classes suggests that it may be Bernays-von Neumann, which still has not been documented rigorously on here because of a lack of sufficient source works defining it (I have Bernays's work and also Hamilton's Logic for Mathematicians, that's about it). It ultimately boils down to: this definition is worth adding to ProofWiki but it feels as though it's been parachuted in from thin air and has no (on PW anyway) axiomatic backing on any level.

And I'd rather we don't get sidetracked here by an in-depth discussion on the difference between Euclidean and Tarskian geometry - this was raised solely as an example of how it is envisaged this site should handle alternative axiom schemas. --prime mover 10:36, 10 February 2012 (EST)

As far as $K_{II}$, it is a proper class, and there you could think of $A \subseteq K_{II}$ as saying "all elements of $A$ are not limit ordinals. With proper classes, the only thing we need to be careful of is only that we shouldn't assume that something holds for a proper class just because it holds for all $x$ or that something holds for some $x$ just because we can construct a class that has the property. Andrew Salmon 11:35, 10 February 2012 (EST)

Well okay, but why is it called "$K_{II}$"? Its very name suggests it is but one of a series of carefully named objects, which means there's a system here crying out to be defined. Until this has been done (or at least planned and talked about in a proper page, not just buried under heaps of crap on a talk page) there is no place for it in ProofWiki. --prime mover 13:19, 10 February 2012 (EST)

Yes...there are two. $K_I$ Takeuti uses to denote the set of all limit ordinals. $K_{II}$ is simply $(\On \setminus K_I )$.

Continuing...

Without AoI, there is no such thing as a minimal infinite successor set, so from the perspective of that particular term, there is no reason to avoid it. However, it does make good sense to define the notions of ordinal and finite ordinal/von Neumann integer in a way that is independent of that. There are various ways to do this. If an ordinal is defined as a transitive set which is strictly well-ordered by the epsilon relation, then a finite ordinal is:

1. An ordinal which is strictly well-ordered by the inverse of the epsilon relation.

2. $\varnothing$ or a successor ordinal whose elements are all successor ordinals.

3. An ordinal whose successor is a set of successor ordinals.

There are probably other ways to put it.

This allows one to speak sensibly of the class of natural numbers without accepting AoI, whereas the (apparerently non-standard) definition in S & F leads in that case to the useless idea of every set being a natural number. --Dfeuer (talk) 18:22, 4 April 2013 (UTC)