
Re: A canonical form for small ordinals
Posted:
Jan 18, 2014 6:51 PM


On 1/18/2014 3:48 PM, Ross A. Finlayson wrote: > On 1/17/2014 3:27 AM, quasi wrote: >> quasi wrote: >>> quasi wrote: >>>> William Elliot wrote: >>>>> Paul wrote: >>>>>> >>>>>> I saw it asserted without proof that all ordinals, alpha, >>>>>> less than epsilon_0, can be uniquely expressed as >>>>>> >>>>>> (omega^beta)*(gamma + 1) >>>>>> >>>>>> for some gamma and some beta such that beta < alpha, >>>>>> where omega is the smallest infinite ordinal. This isn't >>>>>> clear to me. Could anyone help or give a reference? >>>> >>>> Note that alpha = 0 must be excluded since the ordinal 0 has >>>> no such representation. >>>> >>>>> Cantor's normal form. >>>> >>>> Assuming 0 < alpha < epsilon_0, then as suggested by >>>> William Elliot, Cantor's normal form clinches it. >>>> >>>> Proposition: >>>> >>>> For all ordinals alpha > 0 there are exist unique ordinals >>>> beta and gamma such that (omega^beta)*(gamma + 1). >>> >>> I meant: such that alpha = (omega^beta)*(gamma + 1). >>> >>>> Existence: >>>> >>>> Let beta be the least exponent in the Cantor normal form for >>>> alpha. Then the normal form factors as >>>> >>>> (omega^beta)*(gamma + 1) >>>> >>>> for some gamma. >> >> Clarifying remark: >> >> The least exponent beta may occur more than once in the normal >> form, but since the normal form has only finitely many terms, >> it will still be the case that once omega^beta is factored out >> (on the left) the remaining factor will not be a limit ordinal, >> and hence can be expressed in the form gamma + 1, for some >> ordinal gamma. >> >>>> Uniqueness: >>>> >>>> Suppose alpha has two representations: >>>> >>>> alpha = (omega^beta1)*(gamma1 + 1) >>>> >>>> alpha = (omega^beta2)*(gamma2 + 1) >>>> >>> >From the uniqueness of the normal form for alpha, together with >>>> the uniqueness of the normal form for gamma1 + 1 and gamma2 + 1, >>>> it follows that beta1 = beta2, and based on that, the uniqueness >>>> of the normal form for alpha then yields gamma1 = gamma2. >>>> >>>> This completes the proof of the proposition. >>>> >>>> Now suppose 0 < alpha < epsilon_0. >>>> >>>> Applying the proposition, there are unique ordinals beta and >>>> gamma such that alpha = (omega^beta)*(gamma + 1). >>>> >>>> Then >>>> >>>> alpha = (omega^beta)*(gamma + 1) >>>> >>>> => omega^beta <= alpha >>>> >>>> But the above, together with alpha < epsilon_0, yields >>>> >>>> beta < alpha >>>> >>>> as required. >>>> >>>> Bottom line: >>>> >>>> Sometimes William Elliot is right on target, and this was >>>> one of those times. >> >> quasi >> > > And, then back to Von Neumann? This is up to "epsilon_zero", > "epsilon_nought" as it were, changing the order of exponentiation > in the form, here is then defined, if so "defined" in the > different, anyways I don't care as it is still induction through > definable ordinals. > > Then, quasi, I'd as rather it be more directly productive in the > constructive, here for me that conclusion is still quite proper. > > > Interesting stuff, that, building for example all products spaces, > here with the general interest in a result like this. > > William, Elliott, quasi, or Dr. Quasi or whatever it is these days, > please explain what you see as the significance of the result you > have proven. > >
Rather that's "Elliot".

