Does physicalism imply everything can be defined at least to a certain extent?
Two senses of definition are 'real' and 'nominal'.
'Real' definition describes the essential, fundamental feature(s) of a thing. If everything is physical, then a 'real' definition could (in principle) be given of every individual (state, object or event), every class and every attribute. This is clearly so since the essential, fundamental feature of every individual (state, object or event), every class and every attribute would be that it is physical.
One snag is that if everything is physical, then a 'real' definition would have itself to be physical. I have difficulties conceiving what a physical definition of a physical individual, class or attribute would be.
Perhaps you could get round the problem by refining 'everything' : you could have 1st-order things such as physical individuals, classes and attributes, and 2nd-order things such as concepts and signs in terms of which 1st-order things can be defined. But some analytic work needs to be done on 'everything'.
'Nominal' definition is the analysis of one concept or sign in terms of another concept or sign. (Pluralise if need be as above : 'concepts or signs'.) Thus 'A prime number is a positive integer that is divisible only by itself and 1.' If - IF - everything can be nominally defined under physicalism, it can be nominally defined under any other theory too - spiritualism, panpsychism, animism and so on and on. Why not ? This gives no punch to the claim that under physicalism everything can be defined, since in this respect physicalism is no different from any other theory.
I don't know about "physicalism" as far as its metaphysical denotations/connotations go, but as far as physics goes the answer is "no", i.e., not everything can be defined.
That's erroneously contradicted by https://books.google.com/books?id=pQXSBwAAQBAJ&pg=PA54 where Peres states his "G* Strong superposition principle. Any orthogonal basis represents a realizable maximal test." It's realizable that's the key word here.
There are an uncountably infinite number of possible orthogonal bases (an elementary fact from functional analysis), but only a countably infinite number of constructible test apparatus (there are several references about that, but I'm not offhand finding them). And that boils down to the existence of states that can neither be prepared nor tested for (contrary to Peres). And, like I parenthetically mentioned, I know I've read several such references, and will edit this to cite them if I come across them.
I'd typically leave it at that, but since this is philosophy rather than physics, I like to more (much more) broadly (and entertainingly) think of the above idea along the lines (perhaps in the spirit) suggested by Jean Giraudoux in The Madwoman of Chaillot (the first sentence is what's directly relevant, the rest icing on the cake),
COUNTESS. [...] are you so stupid as to think that just because we're alone here, there's nobody else in the room? Do you consider us so boring or so respulsive that of all the millions of beings, imaginary or otherwise, who are prowling about in space looking for a little company, there's not one who might possibly enjoy spending a moment with us? On the contrary, my dear -- my house is full of guests, always. They know that here, at least is one place in the universe where they can come when they're lonely and be sure of a welcome and a pleasant hour. And for my part, I'm delighted to have them.
Physicalism is restricted to a finite number of particles (at least in the accessible universe) and hence to a finite number of definitions. If there are infinitely many ideas, for instance numbers, then most of them cannot be defined.
If we consider what the mathematicians of the 19th century could not yet know, and what those of the 20th century seem to have pushed out of their minds, we recognize: The universe contains less than 10^80 protons and certainly less than 10^100 particles which can store bits. Therefore, an upper limit of the number of definitions is 10^100. The supremum is certainly less.