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## clique principle – Independence via forcing vs common collapses

Are there recognized statements in $V_{ω+ω}$ impartial via forcing after $mathrm{Col}(ω,<κ_1)*mathrm{Col}(κ_1,<κ_2)*mathrm{Col}(κ_2,<κ_3)*…$ the place $κ_1<κ_2<κ_3<…$ are supercompact?

If no, what if we commence with $mathrm{Col}(ω_1,<κ_1)$ as a substitute? If sure, can we purchase an independence in $V_{ω+2}$?

*Notes:*

- level to that the forcing is an iteration quite than a product.
*Extension:*It is unclear methods to greatest proceed previous $ω_ω$, however the easiest altenative is to disregard eccentric cardinals, and collapse above the common cardinals, utilizing Easton uphold for the iteration.- There are different collapsing notions in addition to $mathrm{Col}$, and I’ll settle for an retort for the symmetric common collapse (it’s unaffected; not optimistic whether it is equal right here since after collapsing $κ_n$, the subsequent step absorbs $mathrm{Add}(κ_n,1)$).

**Motivation**

The clique summary universe $V$ can breathe ‘scrambled’ by forcing. sizable cardinal axioms proffer residue, however largely solely on the scale of the sizable cardinals. common collapses can typically unscramble the consequences of forcing, maybe giving us a require approximation of the correct $V$, and a unaffected canonical nonrestrictive principle of uncountable units, with out the pervasive incompleteness of ZFC.

The above iteration corresponds to first maximizing the reals as much as a sufficiently closed and symmetric level, then continuing with units of reals, units of units of reals, and so forth; and it’s the easiest forcing implementing this maximization thought. After $mathrm{Col}(ω,<textual content{inaccessible})$, because of residue, $mathrm{HOD}(ℝ)$ units of reals are measurable, maintain Baire property, and consummate subset property. And $mathrm{Col}(ω_n,<textual content{inaccessible})$ provides inescapable residue properties for $mathrm{OD}(mathrm{Ord}^{ω_n})$ subsets of $P(ω_n)$.

I selected supercompactness as it’s the level the place the complexity of the sizable cardinal construction above $κ$ matches the complexity of $V$ above $κ$. A feeble extender mannequin for supercompactness should approximate $V$ (In search of Ultimate-L by Hugh Woodin). moreover, there are indications (modulo iterability) that sizable cardinal axioms beneath supercompactness maintain canonical fashions with all reals $Δ^2_2$ in a countable ordinal, and presumably too feeble to ‘purchase’ third bid arithmetic. too, with simply inaccessibles, I believe the assorted sq. ideas will breathe impartial after the collapses airplane if supercompacts animate in $V$.

There are inescapable similarities between supercompactness and residue properties we would like $ω_n$ to carry; and consistency proofs of some forcing axioms (conjectured to breathe equiconsistent with supercompact) stop up collapsing a supercompact to breathe $ω_2$. As for beginning with $mathrm{Col}(ω_1,<κ_1)$, the chance is that sizable cardinal axioms themselves present sufficient closure for countable units, with unaffected forcing independences restricted to the uncountable.

It is just too workable that some forcings go away traces that survive the common collapses. A stationary subset of $κ_n$ will stay stationary within the extension because of $κ_ntext{-c.c}$ of the collapse via $κ_n$, and $κ_n$-closure of the relief, however with the supercompactness, I assassinate not know whether or not we are able to masterstroke this.

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