[FOM] uniqueness of FOL

[Charlie]

	   Not everyone loves Compactness.   Frankly, its acceptability seems predicated
on all the "nice theorems" it makes available.  However, consider this interpretation:
All infinite information can be reduced to some finite amount of information with no
loss at all.  <-- This interpretation makes compactness seem unacceptable (of course,
many logicians will squawk at this, since it would deprive them of plenty of adorable 
theorems).

On Sep 5, 2013, at 8:09 PM, Noah David Schweber <schweber at berkeley.edu> wrote:

> (Apologies if this is a duplicate; I had some trouble trying to send this the first time.)
> 
> I actually think we might be closer to formulating (and proving!) the unique role of first-order logic than Prof. Friedman indicates.
> 
> Lindstrom proved (1969; http://onlinelibrary.wiley.com/doi/10.1111/j.1755-2567.1969.tb00356.x/abstract;jsessionid=899FEBB4283B1F501009D02212BEC7C4.d01t03) that first-order logic is the maximal "regular logic" satisfying 
> 
> (i) both Downward Lowenheim-Skolem and Compactness;
> 
> (ii) both Downward Lowenheim-Skolem and Recursive Enumerability.
> 
> If we take "usable" to mean "proofs are finite and verifiable," then this is almost the desired result; the only issue is the role that the downward Lowenheim-Skolem property plays.
> 
> Personally, I also take the Lowenheim-Skolem property as a requirement of a usable logic; but I understand that the argument there is slightly weaker.
> 
> (Some papers by Shelah might be helpful for visualizing compact logics stronger than FOL; see, e.g.,http://shelah.logic.at/files/375.ps,www.jstor.org/stable/1997362, or http://link.springer.com/article/10.1007%2FBF02011631)
> 
> On the other hand, Lindstrom's Theorem seems to me a perfect formalization of the following:
> 
> (*) First-order logic is the strongest usable logic for studying countable structures.
> 
> (Maybe one might argue that the statement above should be "a strongest," not "the strongest;" but I tend to feel that logics not containing first-order logic are automatically of less interest from the point of view of logic-for-foundations.)
> 
> Don't get me wrong; I love second-order, infinitary, and other weird logics. But I wouldn't view any of these as "usable" unless they are interpreted in such a way as to be just (many-sorted) FOL.
> 
> 
> On Wed, Sep 4, 2013 at 9:09 PM, <meskew at math.uci.edu> wrote:
> In a message dated August 25, 2013, Harvey Friedman wrote the following:
> 
> "For any of the usual classical foundational purposes, you need to be able
> to get down to finite representations that are completely non problematic....
> 
> "Furthermore, first order logic is apparently the unique vehicle for such
> foundational purposes. (I'm not talking about arbitrary interesting
> foundational purposes). However, we still do not know quite how to
> formulate this properly in order to establish that first order logic is in
> fact the unique vehicle for such foundational purposes."
> 
> I would be interested to hear Dr. Friedman or others elaborate on why it
> appears that FOL is uniquely suited for classical foundational purposes.
> I agree that it is well-suited, and that several other logics are not as
> well-suited because they don't "get down to finite representations that
> are completely non-problematic."  But what is the evidence that FOL is
> uniquely suited for these purposes?  As Dr. Friedman said, we cannot
> firmly establish this at present.  But if this thesis is "apparent," is
> the evidence of an empirical nature, an intuitive nature, a philosophical
> nature, or does it take the form of theorems about logics in general that
> seem to support this broad thesis?
> 
> Thanks,
> Monroe
> 
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