FOM: second-order logic is a myth

Vladimir Sazonov sazonov at
Sun Feb 28 07:11:52 EST 1999

Stephen G Simpson wrote:
> Me:
>  > I actually favor the `Platonist' or realist view in many contexts.
> Sazonov 25 Feb 1999 01:30:41:
>  > In which contexts? Are there contexts in which you do not favor
>  > this view?
> When I'm wearing my `naive working mathematician' hat, I usually take
> the Platonist view, because at this point in history most
> mathematicians think and talk as if actual infinities exist, the
> powerset operation is well defined, etc.  When I'm wearing my
> `f.o.m. professional' hat, I take a much more skeptical viewpoint.

I consider this as a normal position of non-Platonist, and
this is actually my own position, too. My problem with many
of your postings was that your views on this subject are,
nevertheless, often insufficiently clear. There is always
feeling that you are "living" in a ZFC universe, even as
`f.o.m. professional'.

I believe that in the discussion on V_2 it is this "ideological" 
question which is the stumbling-block. But you wrote: 

> The remarks in my original `second-order logic is a myth'
> posting (22 Feb 1999 16:16:41) were certainly not intended as an
> attack on the standard `Platonist' or realist view of sets, `the
> intended interpretation' of second-order quantifiers in terms of
> arbitrary subsets of the domain of individuals, etc. 

For a Platonist the "full" second-order logic is, by some 
miracle, quite real thing, not a myth. Here, as in the case of 
a religion, it is very difficult to convince somebody change 
his/her belief. I think that scientific beliefs, unlike religious, 
(say, in actual infinity) should be easily changeable, when 
necessary or just as an exercise. 

> Me:
>  > the study of V_2 as a whole is fruitless and hopelessly
>  > intractable.
> Here I was speaking as both a working mathematician (`the study of V_2
> is hopelessly intractable') and an f.o.m. professional (`the study of
> V_2 as a whole is fruitless').
> Sazonov:
>  > Does not this (i.e. "fruitless and hopelessly intractable")
>  > actually mean that the set of second-order validities V_2 is
>  > meaningless (if considered outside ZFC or any other
>  > [first-order] formal system which describes in its own way
>  > the set of "all" subsets of arbitrary set)?
> No, that's not what I was trying to say, and I don't even agree with
> that.

With what? With "V_2 is meaningless"?. (I.e. it is meaningful.) 
Or with "actually mean"? If you want, I can replace this by 
"resemble" or "seem to lead to".

> Some things are fruitful to study even if they are meaningless
> outside ZFC or a similar framework.

Sorry, I again hardly understand you: is V_2 meaningful or
meaningless (i.e. only a "myth")?.

> For example, infinite cardinal
> arithmetic has been fruitfully studied by Shelah, and he published a
> thick book on it.

Now I am in complete confusion. Do you say that Shelah made 
fruitful study of something meaningless (cardinal arithmetic??) 
outside ZFC or a similar framework?  It seems you consider that 
for me anything infinite is meaningless. My pesonal scientific 
interests need not coincide with what I consider meaningful.

By the way, I think the following is an interesting test 
question for Platinists/non-Platonists: 

	Is "full" second-order logic (for *many*-place 
	predicates, say, of 10 arguments) relativised to the 
	two-element set {0,1} also a myth (as well as the 
	ordinary non-restricted second-order logic)? 

Here I stress on "full", and also by logic I mean a *system of 
reasoning* on the predicates over {0,1}, not a (related) notion 
formalized, say, in ZFC or PA to prove some technical 
completeness-incompleteness, definability-non-definability, 
complexity theoretic etc.  (meta)theorems. 

Vladimir Sazonov

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