[FOM] 481:Complementation and Incompleteness
Michael Lee Finney
michael.finney at metachaos.net
Sat Feb 18 14:15:02 EST 2012
I cannot direct you to any web sites, though you can probably find
some via a web search.
However, there is a great deal of literature. For instance, recently
there is Ross Brady's book Universal Logic which is an excellent
attempt to solve the problem using depth relevant logic and naive set
theory. There is Peter Aczel's book "Non-Well-Founded Sets" and
Vicious Circles by Jon Barwise and Lawrence Moss. As well as Anderson
and Belnap's "Entailment I & II", the "Frontiers of Paraconsistent
Logic" and many more.
Category theory has the problem that it is based on functions which
only exist inside of limited sets. Therefore, trying to apply category
theory to larger areas has foundational issues. Even if you define
functions using classes, you only move the problem back a step. There
are some attempts to use Category theory as the foundations instead of
Set Theory, but so far as I know they have either not been succesful
or very awkward. I haven't gotten into category theory much, so I
can't give you any references here. I only care about it relative to
foundational issues.
For the insufficiency of classical logic, there are the books "The
Principles of Mathematics Revisited" by Jaakko HIntikka and
"Dependence Logic" by Jouko Vaananen. There are quite a few papers,
but the basic idea is that first order logic is insufficient to
characterize common mathematical structures (such as a finite set).
At least some second order logic is required. Like ZFC, stratifying
logic into "orders" is another band-aid to avoid the paradoxes.
In all of these areas there is are many, many papers -- some of which
I have and I am quite certain many moref which I don't since I am not
in the academic environment. However, the problems with logic have
been recognized for a very long time. That was where some of the
earliest logic variations such as modal logic and intuitionistic logic
originated in the early to mid 1900's.
Some of the material is available on the web, but more and more the
academic publishers are making it extremely difficult to obtain copies
of papers if you are not a member of an academic institution which
subscribes to everything. At 10$ a paper, it gets extremely expenseive
very rapidly. Likewise for subscribing to journals. Access to local
university libraries can help, but not really solve the problem.
Michael Lee Finney
c> Dear Michael Lee Finney,
c> Knowing something of the "incorrectness and/or
c> "insufficiencies" of both first-order logic and ZF, I would like to
c> find out a little more about your FOM post regarding them. I
c> realize that my question covers a great deal of territory and is
c> thus not easy to answer in much detail, but I wonder whether you
c> could make some brief comments and/or direct me to some sites
c> pertaining to your FOM remarks. I am not asking you to spend much
c> time on my query, but anything helpful you may say will be highly
c> appreciated.
c> Thanks.
c> Charlie Silver
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