[FOM] f.o.m. documentary 2
Michael Lee Finney
michael.finney at metachaos.net
Sat Feb 18 14:15:09 EST 2012
Perhaps the term "resolved" is not quite correct, but neither Cantor's
pardaox nor Russell's paradox can be properly handled within the
system. The system has simply been made small enough that, hopefully,
they do not occur.
Just as with a set of linear equations, a set of logical equations can
be underdetermined, overdetermined or fully determined. So, every set of
logical equations must be undetermined, inconsistent, false or true.
Since every logical object is described with a set of logical
equations, all four possibilies exist. So, if your foundations cannot
handle incomplete and inconsistent propositions you are unable to
discuss every logical object. So, objects such as Cantor's set cannot
be discussed within your system. Refusing to talk about it is not
resolving the problem. The foundations must be revised to resolve the
problem which means that the objects can be discussed formmaly and
within the system without destroying the system.
For category theory, the basic problem is simple. Category theory
relies on functions. Functions are "small" sets. Therefore caregory theory
cannot be applied to large objects. Different "hacks" have been
created, usually along the line of extending the set / class heirarchy
and introducing "large" functions in that context. But that is a never
ending problem and doesn't really resolve the issue. What is necessary
here is different. Either functions must apply to every set so that
there is no concept of "small" vs "large" set or functions must
themselves be primitive objects in some manner (I have some ideas
there, but they are not ready for prime time). The latter is where
some attempts to found mathematics on Category Theory instead of Set
Theory originate.
These issues extend to the logic itself. Logic was not originally
first order, second order, etc.. It was broken up just like set theory
to avoid logical paradoxes.
When you tell me that a logic is formally "complete" or has
"desirable" upward or downward properties, my response is that it is a
"toy" and is incomplete for practical usage. Restricting your language
so you can't say the bad words is nothing more than hiding in the
closet. This is all that the foundations of the '30s acheived. It is
time to come out of the closet, or at least to peek out the door.
Michael Lee Finney
828-406-3064
michael.finney at metachaos.net
MM> Hello Michael
MM> When you say, within the context of referring to ZFC, that "the
MM> paradoxes are STILL not properly handled within the foundations", what
MM> exactly are you getting at here? Also, in what sense(s) to you believe
MM> that category theory has "outgrown" ZFC? Further explanation would be
MM> welcome.
MM> Many thanks
MM> Margaret
MM> On 14/02/2012 20:37, Michael Lee Finney wrote:
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