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Re: Antifoundation axiom
Posted:
Mar 9, 2013 4:55 PM


On Mar 6, 7:59 am, Zuhair <zaljo...@gmail.com> wrote: > The following theory Violates foundation in preference to somewhat > plausible > axioms. > > ZF  foundation > + > For all x. Exist H(x) > + > for all x. x subnumerous to H(x) > + > AntiFoundation Axiom: An infinite Dedekindian finite set exists. > > Where H(x) is the set of all sets hereditarily subnumerous to x. > > Zuhair
You can't really have axioms for set theory. Axioms are a way to formalize our understanding of a subject e.g. number theory or geometry. They are so well developed and understood that we try to codify our great understanding with a formal system.
But how well do we understand sets? There are a dozen versions of set theory. You end up with conflicting theorems from one system to another!! Can you imagine 2 different versions of Geometry or Number Theory? At most one would be right.
You really have to treat sets as something we do NOT understand and so the next step is to firm up our limited understanding. That means we examine the issues and consider the options. The different set theories would be possibilities to examine. With each author declaring their version of set theory authentic  WITHOUT ANY RESPONSIBILITY TO JUSTIFY IT  you simply have conflict and the resulting chaos of misunderstandings that Wiki describes.
What are the fundamental questions?
Can a set contain itself? (Using Recursion Theory as a guide, the answer is yes. I posted on FOM a constructible X such that X={X} based on a proof in Recursion Theory.)
Do you consider relationships besides sets (relations)? Right now we have:
A. Everything is a set. B. x ~e x is not a set.
Direct inconsistency!
This is exactly like when you (Zuhair) attempt to formalize the Liar Paradox without first deciding the nature of the resolution. You fumble around with " 'This is false.' is both true and false." while I point out the analogies that show that it is neither.
Mathematics is supposed to be about the truth. With this freeforall at declaring what set theory is  rather than looking at the whole problem and considering each possibility  it is only about authors jockeying for power and influence, promoting themselves rather than giving an honest appraisal of other approaches.
Agree?
CB



