In this post I want to look at an issue I notice from was in doubt when looking at the comment of F. G. Dorais in the post In model theory, does compactness easily imply completeness? F. G. Dorais remark was:
Blockquote ...The first, which comes through rather clearly, is that Model Theory could ultimately be done without any formal syntax and deduction rules...
An alternative is Fraisse
I think F. G. Dorais was talking about Fraisse's development of model theory via back and forth. It is, however, not clear to me that this is more free from syntax and deduction rule than the traditional one in a meaningful way.
I think Fraissen view does show that Model Theory could be done without a specific choice of syntax. But I don't it seems unreasonable to think that the traditional model theorists would be naive enough to believe that a specific choice of syntax does matter.
The main question I want to ask:
Is there any difference between Fraissean point of view to traditional point of view BEYOND switching from formal language (syntax) syntax and deduction rule to its informal languagecounter part?
It appears is not immediately clear to me that the Fraissean definition just change the language from the standard formal one to an informal onedid anything more than doing so.
It don't see that
If this is the case then there is any mathematical significance switching from formal language to informal languagenothing genuine new about Fraissean point of view than the traditional one. For example, there seems to be no fundamental difference between writing $ N \vDash S0+S0=SS0 $ and speaking out loud "in $N$, one plus one is equal to two". In both cases, we use language which are ultimately meaningless. There is no more meaning in the utterance of "one" than writing $S0$. (If a parrot shout “one plus one is equal to two”, his statement would have no meaning).
In both cases meaning is given by the interpretations. The only difference is the interpretation in the case of "one" is more familiar than in the case of $S0$. This difference has nothing to deal with the subject matter of mathematics. Likewise, there is no meaningful difference between " there exist .. in M" and $M \vDash \exists ...$.