Disjunction Property Implies Existence Property
In 1974 Harvey Friedman proved that in any recursively enumerable extension of intuitionistic arithmetic, the converse of the above also holds. Throughout the following discussion assume is such an extension of arithmetic. Harvey Friedman's proof is quite intricate, making of self-referential sentences in a similar way to the one used in the proof of the incompleteness theorem. They key step is to find a bound on the existential quantifier because bounded existential formula can be written as a disjunction,
- .
However, given an arbitrary existential formula, can we hope to replace it by a bounded one?
Read more about this topic: Disjunction And Existence Properties
Famous quotes containing the words property, implies and/or existence:
“No man acquires property without acquiring with it a little arithmetic, also.”
—Ralph Waldo Emerson (1803–1882)
“Ideas improve. The meaning of words participates in the improvement. Plagiarism is necessary. Progress implies it. It embraces an author’s phrase, makes use of his expressions, erases a false idea, and replaces it with the right idea.”
—Guy Debord (b. 1931)
“The settlement of America had its origins in the unsettlement of Europe. America came into existence when the European was already so distant from the ancient ideas and ways of his birthplace that the whole span of the Atlantic did not widen the gulf.”
—Lewis Mumford (1895–1990)