Hilbert's program then and now

In Dale Jacquette (ed.), Philosophy of Logic. Amsterdam: North Holland. pp. 411–447 (2007)
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Abstract
Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s
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Godel's Functional Interpretation.Avigad, J. & Feferman, S.

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Takeuti's Well-Ordering Proof: Finitistically Fine?Darnell, Eamon & Thomas-Bolduc, Aaron
Proof Theory.Avigad, Jeremy

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