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i misread it XD trhanks for your help
In recent years second-order logic has made something of a recovery, buoyed by George Boolos' interpretation of second-order quantification as plural quantification over the same domain of objects as first-order quantification (Boolos 1984). Boolos furthermore points to the claimed nonfirstorderizability of sentences such as "Some critics admire only each other" and "Some of Fianchetto's men went into the warehouse unaccompanied by anyone else" which he argues can only be expressed by the full force of second-order quantification. However, generalized quantification and partially ordered, or branching, quantification may suffice to express a certain class of purportedly nonfirstorderizable sentences as well and it does not appeal to second-order quantification.
an unstated assumption in Godels Incompleteness Theorem
Exceuse me I h ave come up with a possible way around Godels theorem.
A crucial fact in the theorem is that the theory T (any extension of PA) can encode recursively "x proves y".
but we well know that there are many fast growing functions that can't be proved total in PA.. thus...
if we define a theory of mathematics which has a very complicated (algorithm complexity)defintiion of "x proves y" (in ZFC meta-theory for example), so fast growing that it can't be define in T.
then T may be a theory containing arithmetic for which godels theorem does not apply.. may even a consistent theory T exists than can prove its own consistency!
doesnt answer my question
hey that was really interesting about whether or not "all properties" in second order logic falls victim to the same weirdness as the idea of a natural number does in first order logic. I never thought of that before.
oh yeah do you have a reference for "there's tricks for making second-order logic encode any proposition in third-order logic and so on."? I would like to study that too, even though you've already given me a lot of things to read about already!