# On Formally Undecidable Propositions of Principia Mathematica and Related Systems

@inproceedings{Gdel1962OnFU, title={On Formally Undecidable Propositions of Principia Mathematica and Related Systems}, author={Kurt G{\"o}del and Bernard Meltzer and Richard Schlegel}, year={1962} }

Gödel’s famous proof [2, 1] is highly interesting, but may be hard to understand. Some of this difficulty is due to the fact that the notation used by Gödel has been largely replaced by other notation. Some of this difficulty is due to the fact that while Gödel’s formulations are concise, they sometimes require the readers to make up their own interpretations for formulae, or to keep definitions in mind that may not seem mnemonic to them. This document is a translation of a large part of Gödel… Expand

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#### References

SHOWING 1-10 OF 11 REFERENCES

57 And of course the domain of the definition must always be the whole domain of individuals

- 57 And of course the domain of the definition must always be the whole domain of individuals

58 Variables of the third kind may therefore occur at all empty places instead of individual variables, e.g. y = φ (x), F(x, φ (y)), G [ ψ (x, φ (y)), x] etc

- 58 Variables of the third kind may therefore occur at all empty places instead of individual variables, e.g. y = φ (x), F(x, φ (y)), G [ ψ (x, φ (y)), x] etc

But for every formula in which the sign = occurs, there exists a formula without this sign

Variables of the third kind may therefore occur at all empty places instead of individual variables, e.g. y = φ (x), F(x, φ (y)), G [ ψ (x, φ (y)), x] etc

XXXVII, 2, I have shown of every formula of the restricted predicate calculus that it is either demonstrable as universally valid or else that a counter-example exists

s) represents any complex of the variables x 1 , x 2

x m ) {Φ k

Φ n , such that In = 4', All (X) = X+ I and for every 4:,k (I < k _ n) either 1. (x 2

- Since F is recursive, there is a recursive function Φ (x) such that F(x) ~ [Φ (x) = 0], and for Φ there is a series of functions Φ 1

Ψ n be the functions presumed to exist, which yield a correct proposition when substituted for φ 1 , φ 2 , . . . φ n in (E x 0 ) C. Let its domain of individuals be I

- Proof: Let Ψ