On 4/13/2013 11:19 AM, Nam Nguyen wrote: > On 12/04/2013 6:59 PM, fom wrote:
>> >> With the first two definitions technically being >> axioms because the symbols may not be eliminated >> through substitution. That is why the signature >> is formally introduced initially as >> >> <<M, |M|>, <c, 2>, <e, 2>> >> >> Obviously, I cannot specify an infinite domain. > > Exactly, fom. You, I, et al. don't seem to have a disagreement here. > > What we seem to have is a difference in understanding in where we > _can_ go from here, i.e., from one "cannot specify an infinite > domain"! >
> My presentation over the years is that it does _not_ matter > what, say, Nam, fom, Frederick, Peter, ... would do to > "specify an infinite domain", including IP (Induction Principle), > a cost will be exacted on the ability to claim we know, verify, > or otherwise prove, in FOL level or in metalogic level. > > The opponents of the presentation seem to believe that with IP > we could go as far as proving/disproving anything assertion, > except it would be just a matter of time. Which sounds like > Hilbert's false paradigm of a different kind. > > That's the difference on the two sides. >
The *paradigm* of first-order logic involves an ontology and a metaphysics with which you disagree. This is why I say that you are probably not using first-order logic in what you are trying to do. This is why I interpret what I have already seen as trying to take advantage of partiality. Partiality is not part of the *paradigm* of first-order logic.
That is not to say that what you are doing would not be interesting. It is to say that it is hard for people to have you say that you are using a certain set of principles but then also make statements incoherent with that set of principles.
First-order logic is not really about "truth verification".
Although "truth" is part of the jargon, it is, at this point, almost meaningless to talk of it as such.
In fact, I recently ran across an article where the model theorist Wilfred Hodges commented on the difference between Tarski's 1933 paper and his 1956 paper. In the article, Hodges condoned the greater abstraction that was concomitant with the generality of application of mathematical systems. The problem, of course, is that model theory professes to be about "truth" and "satisfaction". If model theory is to be about something else, then it should be using phrases like
"... is applicable"
"... is true"
"... is justifiably applicable by ..."
"... is a consequence of ..."
In the modern debate between "ideal language theory" and "natural language theory" what is at issue is the notion of meaning. The "ideal language theory" bases the meaning of language symbols solely on semantic truth assignments. The "natural language theory" bases the meaning of language symbols on pragmatic considerations that inform the content by which a truth assignment may occur.
In other words, the only real meaning of "truth" is that it facilitates the representation of details for the transformation
uninterpreted syntax -> interpreted syntax
Just yesterday I summarized how I now understand these matters in a correspondence:
So, mathematics has reached this curious place where the meaning of statements is given by semantic truth conditions based upon an indeterminable ontology presumed through purported denotation in relation to the descriptive introduction of names eliminable through representation within the uninterpreted syntax by means of Goedel arithmetization.
But, then, my views are non-standard. I am certain that everyone else is "actually" speaking about "actual truth" and that I am simply condemned to my own "private language".
You think I do not understand how you are trying to say something that may be relevant. I am trying to get you to understand that it will be difficult to succeed if you do not try to figure out what paradigm of logic you may be trying to apply. Once again, the textbooks are bad about all of these things. And, again, I direct you to