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It does.
Section 4 in the article linked in the grandparent discusses higher-order logics and how they can all be simulated by and reduced to second-order logic.
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.
In response to
Godel's Completeness and Incompleteness Theorems
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
In response to
Second-Order Logic: The Controversy
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!
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I fail to see how this is evidence of Nonsecondorderizability of some possible sentences.
There is no known trick to encode all sentences expressible in higher-order logics into first-order logic, but there is such a trick to encode all sentences expressible in higher-order logics in second-order logic.
The trick in question is described in the SEP article. Doesn't that suffice as a reference and starting point for studying the notion that second-order logic can encode higher-order logics?
i misread it XD trhanks for your help