David Hilbert wanted to establish the consistency of mathematics, and in particular Peano Arithmetic, using finitary methods. This program is widely thought to have completely failed, due to Kurt Godel's incompleteness theorems. However, as far as I know, Hilbert never gave a formal definition of "finitary methods". Has any mathematician ever gave a rigorous definition of "finitary methods"? I wonder if this has ever been formalized in the mathematical literature.
2026-03-26 20:40:07.1774557607
What does "by finitary methods" actually mean, in regards to Hilbert's proof of consistency program?
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There is now a large and interesting literature on this topic on the extent of finitistic methods, too much to review here.
One headline: William Tate, for one, has argued in detail that Hilbert's finitary methods are (or rather, should have been) what are capturable in the formal theory of Primitive Recursive Arithmetic.
But this isn't universally agreed. For example, Charles Parsons has argued that finitistic arithmetic doesn't reach as far.
For much more on this and related debates, you could usefully see for a start the article on Hilbert's Program in the Stanford Encyclopaedia, and some of the references there.