These are (the basic) theorems.
It is the recovery theorems that seal the case.
All of them are theorems about first-order model theory.
But as we noted earlier, these theorems don’t extend to the PoI.
In Section 3.1, I discuss the theorems’ function in decision theory.
Sometimes quite fantastic conclusions are drawn from Gödel’s theorems.
He viewed (34) and (35) as the most surprising of his surprising theorems.
In addition, mathematical theorems are, at least in principle, all of hypothetical form.
We now describe the proof of the two theorems, formulating Gödel's results in Peano arithmetic.
A related question is just how much logic is needed to reproduce probabilistic existence theorems within a qualitative framework.
Besides their intrinsic interest, these theorems may be used as existence theorems in various combinatorial problems.
However, if mathematical theorems do not exactly hold in nature, these theorems at least serve with sufficient precision in practice.
It is demonstrable that the theorems of Ackermann set theory about well-founded sets are exactly the theorems of ZF (Lévy 1959; Reinhardt 1970).
In the case of other theorems, however, the negative results that are often shown by the limitative theorems of metamathematics may no longer hold.
We have Soundness and Completeness theorems for classical quantificational logic with identity as well as Compactness and Löwenheim-Skolem theorems.
This allows the anti-Platonic point that theorems about the beautiful and theorems in mathematics have nothing to do with one another, even if some theorems are beautiful.
In intuitionistic reverse mathematics one has a similar aim, but then with respect to intuitionistic theorems: working over a weak intuitionistic theory, axioms and theorems are compared to each other.
In contrast to other branches of mathematics, many of the problems and theorems of number theory can be understood by laypersons, although solutions to the problems and proofs of the theorems often require a sophisticated mathematical background.
Many of Mally’s surprising theorems are derivable in KD, but they have, as it were, lost their sting: those theorems lead to surprising consequences when combined with Mally’s Axiom I and his definition of f , but they are completely harmless without these postulates.
In order to appreciate the physics literature aimed at proving no-go results for time machines it is helpful to view these efforts as part of the broader project of proving chronology protections theorems, which in turn is part of a still larger project of proving cosmic censorship theorems.
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