[FOM] f.o.m. documentary 2

Christopher Menzel cmenzel at tamu.edu
Sun Feb 19 09:28:19 EST 2012

Am Feb 18, 2012 um 8:15 PM schrieb Michael Lee Finney:
> Perhaps the term "resolved" is not quite correct, but neither Cantor's
> pardaox nor Russell's paradox can be properly handled within the
> system [of ZF set theory]. The system has simply been made small 
> enough that, hopefully, they do not occur.

I think this misrepresents the situation rather badly. It is true that Zermelo himself considered his initial 1908 axiomatization to be a pragmatic solution to the problem of the paradoxes. But the development of the iterative/cumulative conception of set culminating in Zermelo's own 1930 paper "Über Grenzzahlen und Mengenbereiche" yielded a clear and intuitive picture of the natural models of ZF that provided a compelling explanation of the paradoxes and a satisfying justification of the restriction of the Naive Comprehension schema found in the Separation schema: Sets are collections that are formed at some stage of the cumulative hierarchy; some conditions on sets (notably, the conditions "x ∉ x" and "x is a cardinal" that generate the Russell and (so-called) Cantor paradoxes) pick out objects of arbitrarily high rank which, therefore, never jointly form a set at any stage of the hierarchy.

> 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.

That is far from obvious to me. In ZF, you can (arguably) offer a very satisfying explanation for why certain conditions simply do not specify a logical object of the requisite sort. That seems to me to count as a pretty reasonable way to resolve the problem without having to acknowledge some sort of shadowy denotation for every conceivable term.

> 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.

That they *can* be revised is of course an interesting and important project that a lot of people have been pursuing for quite some time.  That they *must* be is not quite so clear.

> 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.

Seems to me these properties, and stronger ones still, notably, decidability, are of immense practical importance for, notably, automated reasoning, where a great deal of research is still working within the framework of classical logic — notably, many description logics, which are so fundamental to recent work in the development of the semantic web. But maybe I don't know what sort of practical usage you have in mind.

Chris Menzel

More information about the FOM mailing list