Quentin Anciaux wrote:
> Also... a list consisting of "A exists" and "A does not exists" is
> consistent to you ?
No, that would be inconsistent.
>
> Could I "infer" A exsits or A does not exists from this list ? If I
> takes the states separately, there is no contradiction... but If I
> take the states as following each other (in any order) then there must
> be a rule that ties those states together... and how could it be a
> rule if it change at every steps ?
I was thinking of lists like "A exists at t." and "A does not exist at t+1."; so
it is explicit that the propositions in the list do not directly contradict each
other. In our models of the universe we rely on various regularities which are
subsumed under "the laws of physics" to compare propositions that refer to
different spacetime events. But if we're going to contemplate all
mathematically consistent universes and try to derive the "laws of physics"
then we have only the laws of logic to relate one proposition to another. I
think they are to weak rule out completely arbitrary universes like my
list-universe. Maybe by "mathematically consistent" you mean more than just
free of logical contradiction; maybe you mean "including ZF" or ZFC - that would
capture a lot of mathematics.
Brent
>
> 2008/11/9 Brent Meeker <meekerdb.domain.name.hidden>:
>> Quentin Anciaux wrote:
>>> 2008/11/9 Brent Meeker <meekerdb.domain.name.hidden>:
>>>> A. Wolf wrote:
>>>>>> I can if there's no rule of inference. Perhaps that's crux. You are requiring
>>>>>> that a "mathematical structure" be a set of axioms *plus* the usual rules of
>>>>>> inference for "and", "or", "every", "any",...and maybe the axiom of choice too.
>>>>> Rules of inference can be derived from the axioms...it sounds circular
>>>>> but in ZFC all you need are nine axioms and two undefinables (which
>>>>> are set, and the binary relation of membership). You write the axioms
>>>>> using the language of predicate calculus, but that's just a
>>>>> convenience to be able to refer to them.
>>>>>
>>>>>> Well not entirely by itself - one still needs the rules of inference to get to
>>>>>> Russell's paradox.
>>>>> Not true! The paradox arises from the axioms alone (and it isn't a
>>>>> true paradox, either, in that it doesn't cause a contradiction among
>>>>> the axioms...it simply reveals that certain sets do not exist). The
>>>>> set of all sets cannot exist because it would contradict the Axiom of
>>>>> Extensionality, which says that each set is determined by its elements
>>>>> (something can't both be in a set and not in the same set, in other
>>>>> words).
>>>> I thought you were citing it as an example of a contradiction - but we digress.
>>>>
>>>> What is your objection to the existence of list-universes? Are they not
>>>> internally consistent "mathematical" structures? Are you claiming that whatever
>>>> the list is, rules of inference can be derived (using what process?) and thence
>>>> they will be found to be inconsistent?
>>>>
>>>> Brent
>>> Well I reverse the question... Do you think you can still be
>>> consistent without being consistent ?
>>>
>>> If there is no rules of inference or in other words, no rules that
>>> ties states... How do you define consistency ?
>> A set of propositions is consistent if it is impossible to infer contradiction.
>>
>> Brent
>>
>
>
>
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Received on Sat Nov 08 2008 - 21:42:06 PST