Statement of The Theorem
Tennenbaum's theorem states that no countable nonstandard model of PA is recursive. Moreover, neither the addition nor the multiplication of such a model can be recursive. The strategy for proving Tennenbaum's Theorem is to assume that there exists a recursive non-standard model of PA and then show that this leads to a contradiction. This may be achieved by constructing a non-recursive set S and then showing - under the assumption that such a recursive non-standard model exists - that S is recursive, leading to a contradiction. Such a non-recursive set S must involve non-standard elements and be constructible for any non-standard model of PA
Read more about this topic: Tennenbaum's Theorem
Famous quotes containing the words statement of, statement and/or theorem:
“The honor my country shall never be stained by an apology from me for the statement of truth and the performance of duty; nor can I give any explanation of my official acts except such as is due to integrity and justice and consistent with the principles on which our institutions have been framed.”
—Andrew Jackson (1767–1845)
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)