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:
“To throw obstacles in the way of a complete education is like putting out the eyes; to deny the rights of property is like cutting off the hands. To refuse political equality is like robbing the ostracized of all self-respect, of credit in the market place, of recompense in the world of work, of a voice in choosing those who make and administer the law, a choice in the jury before whom they are tried, and in the judge who decides their punishment.”
—Elizabeth Cady Stanton (1815–1902)
“The pure work implies the disappearance of the poet as speaker, who hands over to the words.”
—Stéphane Mallarmé (1842–1898)
“Sport in the sense of a mass-spectacle, with death to add to the underlying excitement, comes into existence when a population has been drilled and regimented and depressed to such an extent that it needs at least a vicarious participation in difficult feats of strength or skill or heroism in order to sustain its waning life-sense.”
—Lewis Mumford (1895–1990)