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:
“Children are potentially free and their life directly embodies nothing save potential freedom. Consequently they are not things and cannot be the property either of their parents or others.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“Democracy is timelessly human, and timelessness always implies a certain amount of potential youthfulness.”
—Thomas Mann (1875–1955)
“Truth exists. The sole purpose of this proposition is to assert the existence of truth against imbeciles and sceptics.”
—Edward Herbert Of Cherbury, Lord (1583–1648)