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Topic: An observation on definability and description
Replies: 12   Last Post: Mar 16, 2013 9:56 PM

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David Bernier

Posts: 3,540
Registered: 12/13/04
Re: An observation on definability and description
Posted: Mar 16, 2013 8:15 PM
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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 non-empty sets is non-empty" -- 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 "extra-logical" presupposition and
>>>> the notion of "definability in a model". I cannot
>>>> see how any extra-logical 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 well-orderings 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 'non-thinkable' 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 well-defined,
with range in {0, 1}
and that phi (through R) produces a well-ordering 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 well-orderings on N,
I think.

After that, it must get more complicated.

Maybe all the countable ordinals are thinkable,
but I'm not really convinced ...


> Obviously , we can't think of such a bijection , and neither its
> existence nor its non-existence 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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