From: Brent Meeker <meekerdb.domain.name.hidden>
Date: Sat, 08 Nov 2008 17:39:36 -0800

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
>>> 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

--~--~---------~--~----~------------~-------~--~----~
You received this message because you are subscribed to the Google Groups "Everything List" group.
To post to this group, send email to everything-list.domain.name.hidden
To unsubscribe from this group, send email to everything-list+unsubscribe.domain.name.hidden
For more options, visit this group at http://groups.google.com/group/everything-list?hl=en
-~----------~----~----~----~------~----~------~--~---
Received on Sat Nov 08 2008 - 20:39:52 PST

This archive was generated by hypermail 2.3.0 : Fri Feb 16 2018 - 13:20:15 PST