For the Neo-Fregeans, logicism is first and foremost a means to meet Benacerraf's challenge. The contention is that Hume's Principle provides us with an attractive semantic and epistemological theory, which avoids both extreme Platonism and fictionalism. This answer does not extend, however, to earlier versions of logicism - the ones defended by the historical Frege and Russell, which do not use any abstraction principle. From the neo-logicist perspective, the old versions of logicism no longer constitute credible philosophies of mathematics.
In this paper, I suggest that the central position occupied today by the Benaceraff's dilemma blinds us to the possibility of other forms of philosophical agenda, which the ancient logicists attempted to fulfill. Focusing on geometry and the theory of reals, I show that, beside the unification and reduction of all mathematics to logic, another issue was at stake in The Principles as in Principia: how to carve mathematics at its joint? Russell wanted to arbitrate between the various conceptions of mathematical architecture, and found a rational way of doing this. If both mathematics and logic have changed since Russell's time, there is reason to believe that the architectonic issue is still alive today.
This paper surveys the model-theoretic aspects of two topics initiated by Kazuyuki Tanaka related to the Weak König Lemma: self-embedding, and conservativity for sentences of the form ∀X ∃!Yθ(X,Y), where θ(X,Y) is an arithmetical formula. It includes a few recent developments and proof sketches.
In the conference of Computability Theory and Foundations of Mathematics 2015, we had special sessions on Professor Kazuyuki Tanaka's work in honor of his 60th birthday. It was a great honor for me to give a talk about determinacy of infinite games in that session. In this paper, accordance with works by Professor Tanaka on determinacy, we introduce a collection of related researches.
Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way, and provides a fundamental tool in the study of computing theory and computability theory. In search of complete lattices on which the Laswon topology is Hausdorff, Gierz, Lawson and Stralka introduced in  quasicontinuous lattices, which inherit many good properties of domains. Gierz, et al. pointed in  that Rudin's Lemma for finding a “cross-section” of certain descending family of sets plays a central role in the development of the whole theory of quasicontinuous lattices. In this paper, we study Rudin's Lemma from reverse mathematics point of view and prove that the Rudin's Lemma is equivalent to ACA0 over RCA0.