Posts tagged with transitivity

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• @isomorphisms: Higgs Boson Particle to be observed on/before 31 Dec 2012 = 68.0% @Intrade bit.ly/MP3Eo0
• @isomorphisms: Higgs 2013 = 80%, Higgs 2014 = 85%, Higgs 2015 = 80.1%. Oops #EMH #arb
• There's \$2 three years from now sitting on the table for whoever wants to pick it up.

## the Begats

The word `<` is normally defined to mean `less than` in some quantifiable sense. For example, considering the set `{3,6,1441}`, one could say that `3<6<1441`.

But in the abstract language of partially ordered sets, < is reinterpreted many ways — to mean `proper subset of ⊂`  (`contained by`), `divides`, “is hotter than" or … any transitive relation — even `begat`.

Consider the set

• `{Cain, Enoch, Irad, Mehujael, Methusael, Lamech₁, Jabal, Jubal, Tubal-cain, Naamah} ∪ {Adam, Abel, Seth} ∪ {Seth, Enosh, Kenan, Mahalalel, Jared, Enoch₂, Methusaleh, Lamech₂, Noah} ∪ {Noah, Shem, Ham, Japheth}`.

Transitivity means that it’s impossible for Adam < … < Adam < … (where Adam refers to the same man, not to another person also named “Adam”. We can call him Adam₀ if it’s a problem).

Then the fourth and fifth chapters of Genesis yield the following relations among the members of that set.

• `Cain < Enoch₁ < Irad < Mehujael < Methusael < Lamech₁ < Jubal`
• `Lamech₁ < Jabal`
• `Lamech₁ < Tubal-cain`
• `Lamech₁ < Naamah`
• `Adam < Cain`
• `Adam < Abel`
• `Adam < Seth`
• `Adam < Seth < Enosh < Kenan < Mahalalel < Jared < Enoch₂ < Methuselah < Lamech₂ < Noah`
• `Noah < Shem`
• `Noah < Ham`
• `Noah < Japheth`

It’s not like we reduce `Enoch₂` to “the thing between Jared and Methuselah” — there is other information attached to `Enoch₂` such as that he walked with God and was no more (whereas all the others were noted to have died). Likewise to say that `18` is the integer between `17` and `19` isn’t to ignore the fact that `6` divides `18` or that it represents a legal bright-line in some countries.

Nor do we assume that every pair `(a,b)` from that set should be comparable. (In a totally ordered set either `a<b` or `b<a, ∀a,b`.) But in case of the begats:

• Mehujael Noah
• Shem Tubal-cain
• Jared Jabal

You can put more than one order on a set. For example, regarding salsas from the set {mild, medium, hot}:

• mild < medium < hot, where < means the predicate “is less spicy than”
• medium hot mild, where x≻y means the predicate “She prefers x to y”.

One could launch off into defining ≻ᵢ's for every  of the population. (This is the kind of thing one needs to do in talking about Arrow’s Theorem.) That’s when notation begins to suck.