Question

This is an excellent and interesting question! You are asking whether the 2-step iteration of a normal measure μ μ on a measurable cardinal κ κ is uniquely factored by the steps of the iteration itself.

(I think I have just been exchanging email with you--orperhaps with one of your colleagues, with a differentname?--about the one-dimensional version of this question.)

My answer is that No, you haven't quite got all the possibilities.

The reason is that actually, there answer is a commuting square ofelementary embeddings for the 2-step iteration, as followsYes.The ultrapower

Let me denote κ₀ just by κ and j₀₂ is just the ultrapower of V by the measure j. Sinceμx μ, and it ₁ is a standardmodel-theoretic fact that the ultrapower of any structureby a product measure μ x in M₁, it has theform j₀₁(m)(κ), where m =(ν_α | α < κ). Since youhave said that μ₁ is to first take not inran(j₀₁), we may choose theultrapower by ν, ν_α to be all different, and then take the ultrapower by μ.The lower half of the square differentfrom μ₀. In this case, there is a partitionof κ as the iteration you have defineddisjoint union of X_α,with the diagonal being j_{02Xα in να and nonein μ0. The upper half of the square is Let x = (Xα | α< κ). Note that j02κ is also equal to the compositionj01(Xα) for any α <κ, and similarly κ1 is not inj(Xα). But κ is inj01(x)(β) for some β <κ1, since this is a partition ofκ1. This Apply j12 to conclude thatκ1 is not obviousin j(x)(β) for this β.Thus, but itcan be verified by proving there is some β in the model-theoretic fact interval [κ,κ1) having the form β =j(f)(κ1) for the function f that Imentionedpicks theindex. ThusFrom this, it follows from normality ofμ0 that we can take i write κ =j(g)(κ1) for some function g, since anyβ < κ1 generates κ viaj01:V-->M. In my favored terminology, the seedκ1 generates κ via j and k = in factgenerates all β in [κ,κ1) viaj01restricted to M.}

Similarly, suppose that δ is in the interval[κ₁,and this will map elementarilyinto j(κ)). We know δ =j₁₂(f)(κ₁) for some function fon κ₁ in M₂ with ₁. We also know f =j_{0201(F)(κ) for some F in V. Thus, δ =k.ij(F)(κ, κ1). But in thiscaseIn Y, we let(α,β) be the smallest pair with δ =j(F)(α,β). It cannot be that both are belowκ1, since this would be insideran(j12) and so the least pair must have N β= Mκ1. Thus, δ generatesκ1, which we already observed generates}

To summarize, every ordinal in the interval[κ₁,j(κ)) generatesκ₁, which generates all the ordinalsβ in [κ,κ₁), any of whichgenerate κ and all the other such β.

This is enough to answer your question. The k " N in yourquestion is just j₀₁ " an arbitrary elementary substructure ofM_{12 containing ran(j), so suppose we have Yelementary in M2 and ran(j) subset Y.}

This The case Y= ran(j) is not equal to any one of the sets on your listcases. It isOtherwise, Y has somethingnot j₀₂ " Vin ran(j). Every object in M₂ has formj(h)(κ,κ₁) for some function h, since it includesby looking at the smallest pair of ordinals to generate agiven object with j(h), we see that there must be ordinalsbelow j_{01(κ). It is not j12 "Mκ) in Y. If Y contains any ordinal δ inthe interval [κ1 since it does not include κ. And} ,j(κ)), then it iscontain both κ and κ₁, since weobserved that any such δ generates these ordinals. Inthis case, Y = M₂, since it is not transitivethose two ordinalsgenerate everything.

A much So we assume that Y contains no suchδ. In this last case, Y must contain some ordinalβ in the interval [κ,κ₁). Sinceany such β generates κ, Y contains all suchordinals. It follows that ran(j₁₂) subset Y andin fact = Y, since if Y contained anything more interesting follow-up it wouldhave to have an additional ordinal δ in[κ₁,j(κ)).

So we've seen that your three cases are the onlypossibilities. And like your previous question, there is whether thefactorings arising in noneed to assume that Y or N is somehow internally definable.

By the commuting square are exhaustiveway, and I will give this some was a problem that I had solved many yearsago for my dissertation, although perhaps other people hadalso thought about it. I think it may be thecase was interested in understandingwhich pairs of ordinals (α,β) generate productmeasures via an embedding j, and this question is very muchrelated to thatif you add .

(Click the edit history to see my extra previous answer, which was just about the case when μ₁ is in the range of j₀₁, then a case for which the list answer is exhaustive...no.)