LoPSE

If you look at Bob Hale and Crispin Wright's Logicism in the twenty-first century, in this book (esp. pp. 168-169) (also: at Frege's works themselves and at other explications of Frege), the way things are supposed to have gone is this: Frege used Hume's Principle (see my previous post) to derive second-order Peano Arithmetic. Yet, he was unhappy with the explanatory role of Hume's Principle: "we can never—to take a crude example—decide by means of our definitions whether any concept has the number JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is a number or not." So, he introduced extensions by means of Basic Law V (the extension of F is the same as the extension of G iff exactly the same objects are Fs and Gs), and defined numbers in terms of extensions. While people usually think that Frege moved to BLV to avoid the Ceasar problem, I haven't found anywhere, neither in Frege, nor in secondary literature, a clear...

Hume's Principle (HP), as used nowadays, states that the number of Fs is the same as the number of Gs iff there is a 1-1 correspondence between Fs and Gs. While this sounds pretty obvious, with second-order logic in the background you can use this to derive second-order Peano Arithmetic (PA). (all this is well known, just like the role of HP in a fairly fashionable stream in phil of math called neologicism - check out this  or this  if you haven't heard of this stuff). Anyway, the person who really used HP to obtain PA was Frege, so just in case you wondered why the principle is called Hume's Principle, I dug up the passage where Hume formulates it (Treatise 1.3.1). It's only moderately interesting, but if you're geeky enough to be still reading this, you might be just geeky enough to be interested in the quote: We might proceed, after the same manner, in fixing the proportions of quantity or number, and might at one view observe a superiority or inferiority b...

The paper on an arithmetization of Yablo's paradox with provability instead of truth, written jointly with Cezary Cieśliński is now available (open access) in its final form published in the Journal of Philosophical Logic . Abstract:  We investigate what happens when ‘truth’ is replaced with ‘provability’ in Yablo’s paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Gödel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo paradox. We also look at a formulation which employs Rosser’s provability predicate.

I've just finished reading the book . Here are some remarks (copy-pasted from LaTeX, apologies for resulting infelicities). Piotr Łukowski's Paradoxes is an abridged, revised and translated version of a book written in Polish (the English version has 194 pages and the original habilitation is 535 pages long). It divides into four parts: Paradoxes of Wrong Intuition, Paradoxes of Ambiguity, Paradoxes of Self-Reference  and Ontological Paradoxes . The first part deals with a selection of pseudo-paradoxical arguments which Łukowski  attempts to explain away by indicating why the intuitions involved in them are mistaken.  The second part is devoted to pseudo-paradoxical arguments which can be accounted for by pointing out ambiguities. The third part discusses the classics: some variants of the Liar paradox, and Richard's, Berry's and Grelling antinomies, among others. The part devoted to ontological paradoxes revolves around vagueness, change and identity through time...