
Re: An observation on definability and description
Posted:
Mar 16, 2013 8:15 PM


On 03/16/2013 04:40 PM, Dan wrote: > On Mar 16, 9:01 pm, Frederick Williams <freddywilli...@btinternet.com> > wrote: >> Dan wrote: >> >>> On Mar 16, 4:29 pm, fom <fomJ...@nyms.net> wrote: >>>> On 3/16/2013 3:43 AM, William Elliot wrote: >> >>>>> On Sat, 16 Mar 2013, fom wrote: >> >>>>>> Many of my posts refer to description theory and definability. >> >>>>> If it takes the axiom of choice to prove it's existence, is it definable? >> >>>> There is an important aspect to Robinson's >>>> remarks from "On Constrained Denotation". >> >>>> If the diagonal of a model domain is "constructed" >>>> of "specified" in relation to descriptions, then >>>> the axiom of choice  in the guise of "the Cartesian >>>> product of nonempty sets is nonempty"  is not >>>> an optional axiom for any reason. It is necessary >>>> to interpret the sign of equality. >> >>>> So, there is a sense of "definable in principle" >>>> that should applies to any language act asserting >>>> singular reference. >> >>>> This is why I tried to point out the distinction >>>> between the "extralogical" presupposition and >>>> the notion of "definability in a model". I cannot >>>> see how any extralogical use of singular reference >>>> does not implicitly invoke the axiom of choice. >> >>>> That is, in my opinion, the sense of Robinson's >>>> remarks. >> >>> I hold the view that the Axiom of Choice allows us to reach a >>> seemingly paradoxical position where we can 'define' a set without >>> defining any of its 'individual elements' per se , as an individual >>> element . >> >>> To be more precise : >>> The axiom of choice allows us to say that if a set S , as some >>> collection of sets , is 'definite' , then the set C of 'choice >>> functions' on S is definite . Whether any member E of C is definite is >>> another matter . >>> That's why in ZFC you can think of the 'totality' of wellorderings of >>> R as definite , without being able to define any well ordering of R . >> >>> 'definiteness ' is a weak form of constructivism criteria . I've come >>> to the (as I imagine unpopular) opinion that for a mathematical theory >>> to be sound and 'unambiguous' in interpretation , all elements we >>> should ever refer to in the theory must be 'definite' . >> >> What do you mean by 'definite'? >> >>> For set theory that means giving up AC , along with an 'absolute' >>> notion of uncountability >> >> 'Absolute' is a technical term in set theory. Are you using it in that >> sense or some other? >> >> ('Definite', used to be a technical term in set theory, but no matter.) >> >>> (in a more extreme form , giving up all >>> uncountable ordinals ) . >> >>  >> When a true genius appears in the world, you may know him by >> this sign, that the dunces are all in confederacy against him. >> Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting > > No... not exactly technical ... more along the lines of > 'incorporating' Skolem's paradox ... or having some sort of modality > for 'countability' . > > Assume I were to give you the set of all sequences of integers you > could ever think of (by 'think' of a sequence, I mean give a unique > unambiguous (finite) definition , though not necessarily within any > specific formalism) . > > Is there any integer sequence not within this set? > By assuming that the totality of integer sequences you could ever > think about is countable ,therefore the existence of a bijection > between the natural numbers and this set , you can apply Cantor's > diagonal argument . > But this rests of several shaky assumptions , and , relativizing the > totality of sequences you can think about, you obtain a sequence you > can't think about . > A more apt question would be 'can you think of a bijection between N > and all the integer sequences you could ever think about?' > to which the answer is 'no' . > By going with Parmenides's identity of Thought and Being , at least in > Mathematics , you would say that the set of > 'all integer sequences you can think about' coincides with the set of > 'all integer sequences' . As we have seen , it being countable can be > used to construct a 'nonthinkable' integer sequence , therefore it is > uncountable . > By the same argument , countable ordinals should coincide with > 'thinkable ordinals' . Therefore , the totality of thinkable/ > countable ordinals , should we ever consider it , would form the > first uncountable/unthinkable ordinal. The continuum hypothesis , in > this light, asks of the existence of a bijection between the > 'thinkable sequences' (the set of all countable integer sequences is > R) , > and the 'thinkable ordinals ' (the set of all countable ordinals forms > the first set with cardinality greater than N) .
For the coutable ordinals, sometimes there's a recursive, computable relation R on N , meaning phi(a,b) = 1 if a R b phi(a,b) = 0 if not(a R b), with phi: NxN > {0, 1} computable,
and we can show (in some formal theory, say) that phi is welldefined, with range in {0, 1} and that phi (through R) produces a wellordering of N, the natural numbers.
There are only countably many computable functions. So, at some point in omega_1, we run out of computable functions to represent wellorderings on N, I think.
After that, it must get more complicated.
Maybe all the countable ordinals are thinkable, but I'm not really convinced ...
dave
> Obviously , we can't think of such a bijection , and neither its > existence nor its nonexistence leads us to contradiction . > It asks us to relate two 'potential' totalities that can only be > terminated/completed after 'thinking has stopped' , essentially , a > meaningless question . > > >
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