Longest element of a Coxeter group

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29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

Not to be confused with Silver Machines.

In set theory, Silver machines are devices used for bypassing the use of fine structure in proofs of statements holding in L. They were invented by set theorist Jack Silver as a means of proving global square holds in the constructible universe.

Preliminaries

An ordinal α is *definable from a class of ordinals X if and only if there is a formula ϕ(μ0,μ1,,μn) and β1,,βn,γX such that α is the unique ordinal for which Lγϕ(α,β1,,βn) where for all α we define α to be the name for α within Lγ.

A structure X,<,(hi)i<ω is eligible if and only if:

  1. XOn.
  2. < is the ordering on On restricted to X.
  3. i,hi is a partial function from Xk(i) to X, for some integer k(i).

If N=X,<,(hi)i<ω is an eligible structure then Nλ is defined to be as before but with all occurrences of X replaced with Xλ.

Let N1,N2 be two eligible structures which have the same function k. Then we say N1N2 if iω and x1,,xk(i)X1 we have:

hi1(x1,,xk(i))hi2(x1,,xk(i))

Silver machine

A Silver machine is an eligible structure of the form M=On,<,(hi)i<ω which satisfies the following conditions:

Condensation principle. If NMλ then there is an α such that NMα.

Finiteness principle. For each λ there is a finite set Hλ such that for any set Aλ+1 we have

Mλ+1[A]Mλ[(Aλ)H]{λ}

Skolem property. If α is *definable from the set XOn, then αM[X]; moreover there is an ordinal λ<[sup(X)α]+, uniformly Σ1 definable from X{α}, such that αMλ[X].

References

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Please note that errors have been found in some results in this book concerning Kripke Platek set theory.