If it could be shown that one of Gentzen's consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert's program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen's second proof can be finitistically justified. In particular, the focus is on Takeuti's purportedly finitistically acceptable proof of the well-ordering of ordinal notations in Cantor normal form. The paper begins with a historically informed discussion of finitism and its limits, before introducing Gentzen and Takeuti's respective proofs. The rest of the paper is dedicated to investigating the finitistic acceptability of Takeuti's proof, including a small but important fix to that proof. That discussion strongly suggests that there is a philosophically interesting finitist standpoint that Takeuti's proof, and therefore Gentzen's proof, conforms to.

The question of the analyticity of Hume's Principle (HP) is central to the neo-logicist project. We take on this question with respect to Frege's definition of analyticity, which entails that a sentence cannot be analytic if it can be consistently denied within the sphere of a special science. We show that HP can be denied within non-standard analysis and argue that if HP is taken to depend on Frege's definition of number, it isn't analytic, and if HP is taken to be primitive there is only a very narrow range of circumstances where it might be taken to be analytic.The latter discussion also sheds some light on the connections between the Bad Company and Caesar objections.

G. Genzten’s 1938 proof of the consistency of pure arithmetic was hailed as a success for finitism and constructivism, but his proof requires induction along ordinal notations in Cantor normal form up to the first epsilon number, ε0. This left the task of giving a finitisically acceptable proof of the well-ordering of those ordinal notations, without which Gentzen’s proof could hardly be seen as a success for finitism. In his seminal book Proof Theory G. Takeuti provides such a proof. After a brief philosophical introduction, we provide a reconstruction of Takeuti’s proof including corrections, comments, re-organization and notational adjustments for the sake of clarity. The result is a much longer, but much more tractable proof of the well-ordering of ordinal notations in Cantor normal form less than ε0, that nevertheless follows Takeuti’s strategy closely. We end with some more general comments about that proof strategy and the notion of accessibility more generally.

The importance of Georg Cantor's religious convictions is often neglected in discussions of his mathematics and metaphysics. Herein I argue, pace Jané (1995), that due to the importance of Christianity to Cantor, he would have never thought of absolutely infinite collections/inconsistent multiplicities, as being merely potential, or as being purely mathematical entities. I begin by considering and rejecting two arguments due to Ignacio Jané based on letters to Hilbert and the generating principles for ordinals, respectively, showing that my reading of Cantor is consistent with that evidence. I then argue that evidence from Cantor's later writings shows that he was still very religious later in his career, and thus would not have given up on the reality of the absolute, as that would imply an imperfection on the part of God.The theological acceptance of his set theory was very important to Cantor. Despite this, the influence of theology on his conception of absolutely infinite collections, or inconsistent multiplicities, is often ignored in contemporary literature. 1 I will be arguing that due in part to his religious convictions, and despite an apparent tension between his earlier and later writings, Cantor would never have considered inconsistent multiplicities (similar to what we now call proper classes) as completed in a mathematical sense, though they are completed in Intellectus Divino.Before delving into the issue of the actuality or otherwise of certain infinite collections, it will be informative to give an explanation of Cantor's terminology, as well a sketch of Cantor's relationship with religion and religious figures. Such will comprise the first part of this paper, after which I will argue that although there is tension between how Cantor discusses the absolute infinite before roughly 1896, and inconsistent multiplicities thereafter, due to his continuing and even strengthening religious convictions, Cantor would have maintained his earlier position that inconsistent multiplicities are not mathematically extant, but also not merely potential. I achieve this aim by first pointing out that the evidence taken by Jané (1995) to show

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