[FOM] f.o.m. documentary 2

[Christopher Menzel]

Am Feb 22, 2012 um 11:12 AM schrieb Michael Lee Finney:
> First, I have to say that the very idea of the "cumulative heirarchy"
> that is created in "stages" gripes my gut.

Psychological aversion noted.

> As I see it, everything
> is there all at once, we merely discover the structure within the
> axioms. There is no "creating", no "stages", nothing to be "reached"
> or anything like that.

The metaphor of "forming" sets in successive stages that is often invoked in informal expositions of the cumulative hierarchy is just that, a metaphor; some people find it helpful in priming the necessary intuitions for approaching the actual mathematics. But in ZF proper, the metaphor is gone; there are indeed "stages", or "levels", but these are fixed mathematical objects of the form V_α = ∪{℘(V_β) | β < α}. The cumulative hierarchy is indeed "there all at once", just as you desire.

> Otherwise, if you have
> 
>   1. A quantified logic which allows incomplete and inconsistent
>   statememts without triviality,
> 
>   2. Allows arbritary "order" in the logic,
> 
>   3. A foundation which is simple, clean and provides for the
>   founding of both set theory and category theory in a natural
>   manner,
> 
>   4. All of the above is just as easy to use, in the same manner, as
>   the classical foundations (under classical conditions), allowing
>   all of conventional mathematics with effectively no changes,
> 
>   5. In addition to containing ZFC (as theorems), the set theory
>   allows naive comprehension, non-wellfounded sets, Russell's set
>   and other inconsistent structures,
> 
>   6. Does NOT satisfy completeness, etc..
> 
> Would you consisder the above to be a reasonable replacement for the
> current logical foundations?

Um, well, dunno, maybe. I was simply responding to your claim, as I understood it, that ZF (i) is just an ad hoc weakening of naive set theory that is "hopefully" consistent but in which (ii) the Russell and Cantor paradoxes are not "properly handled", which I understood to mean not properly explained. I argued that both claims are unwarranted. I argued nothing more about foundations generally.

> Godel's work essentially says
> that either our systems are intrisincally incomplete or there are
> inconsistencies in the system. To me, that argues that allowing the
> presence of inconsistency is absolutely necessary. Triviality under
> those conditions merely shows that the logical foundations are 
> wanting.

Not sure I'm getting you. You appear to be arguing that completeness might be achievable if we allow inconsistency and prevent triviality, i.e., explosion. But explosion is what gets you completeness in an inconsistent classical system; if you allow inconsistency but prevent explosion you have no guarantee that I can see that completeness is suddenly possible. So I'm not sure how this is supposed to show some sort of advantage for your suggested approach. But I'm pretty far out of my bailiwick here so will say no more.

> Difficulty in building automated tools is no reason to restrict the foundations.

But the issue isn't difficulty; it's impossibility. It is impossible to build a (first-order) system in which validity is decidable, let alone tractable. But my point, once again, wasn't one about foundations generally; it was, once again, simply a response to a claim you made that seems unwarranted, namely, that a logic with "desirable" properties (like decidability) "is a 'toy' and is incomplete for practical usage." The development of very non-toylike decidable logics for automated reasoning seems to be a pretty clear counterexample to this claim.

Chris Menzel

*****
> CM> 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.
> 
> CM> I think this misrepresents the situation rather badly. It is
> CM> true that Zermelo himself considered his initial 1908
> CM> axiomatization to be a pragmatic solution to the problem of the
> CM> paradoxes. But the development of the iterative/cumulative
> CM> conception of set culminating in Zermelo's own 1930 paper "Über
> CM> Grenzzahlen und Mengenbereiche" yielded a clear and intuitive
> CM> picture of the natural models of ZF that provided a compelling
> CM> explanation of the paradoxes and a satisfying justification of the
> CM> restriction of the Naive Comprehension schema found in the
> CM> Separation schema: Sets are collections that are formed at some
> CM> stage of the cumulative hierarchy; some conditions on sets
> CM> (notably, the conditions "x ? x" and "x is a cardinal" that
> CM> generate the Russell and (so-called) Cantor paradoxes) pick out
> CM> objects of arbitrarily high rank which, therefore, never jointly
> CM> 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.
> 
> CM> That is far from obvious to me. In ZF, you can (arguably)
> CM> offer a very satisfying explanation for why certain conditions
> CM> simply do not specify a logical object of the requisite sort. That
> CM> seems to me to count as a pretty reasonable way to resolve the
> CM> problem without having to acknowledge some sort of shadowy
> CM> 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.
> 
> 
> CM> That they *can* be revised is of course an interesting and
> CM> important project that a lot of people have been pursuing for
> CM> 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.
> 
> CM> Seems to me these properties, and stronger ones still,
> CM> notably, decidability, are of immense practical importance for,
> CM> notably, automated reasoning, where a great deal of research is
> CM> still working within the framework of classical logic — notably,
> CM> many description logics, which are so fundamental to recent work
> CM> in the development of the semantic web. But maybe I don't know
> CM> what sort of practical usage you have in mind.
> 
> CM> Chris Menzel