[FOM] Provability of Consistency
Sam Sanders
sasander at me.com
Tue Mar 26 15:42:21 EDT 2019
Dear FOM,
> All mathematical statements in Anton Freund’s post are correct. However, it appears that this post misrepresents Hilbert’s program and hence misses the main point of the PoC paper.
The misrepresentation of Hilbert's intent that Sergei Artemov is talking about, would not be a standalone case:
The “Grundlagen” by Hilbert and Bernays (esp. Supplement IV) are cited as a motivation for/first instance of
the formalisation of mathematics in second-order arithmetic. However, Hilbert and Bernays allow for third-order
parameters in their system H, which of course yields a very different framework. The other systems in Supplement IV
are introduced, without much development. What is more, the main systems of Kohlenbach’s higher-order reverse
math can be written down in system H, and hence a lot of higher-order math can be done/reversed there.
Best,
Sam
PS: Those interested in the aforementioned differences: in second-order arithmetic, it is hard to
find natural examples of splittings: A <=> [B +C] and disjunctions: D <=> [E V F]. In third-order
arithmetic, these are a dime a dozen. More fundamental results concern compactness and
may be found here:
https://www.worldscientific.com/doi/abs/10.1142/S0219061319500016
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