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The Hilbert program, seeking to secure the consistency of higher mathematics by finitary reasoning about the formal system underlying it, was dashed by Gödel’s incompleteness theorems, which show that no consistent formal system can prove even its own consistency, let alone the consistency of a higher system. We shall describe several proofs of the first incompleteness theorem, via the halting problem, via self-reference, and via definability. After this, we’ll discuss the Rosser variation, the second incompleteness theorem, and Tarski on the non-definability of truth. Ultimately, one is led to the inherent hierarchy of consistency strength underlying all mathematical theories.

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In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibl چه حکایت از فراقت که نداشتم ولیکن...

In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run.

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Wait a minute

I will definitely do it

Just wait a minute

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The technological singularity (also, simply, the singularity)[1] is a hypothetical future point in time at which technological growth becomes uncontrollable and irreversible, resulting in unfathomable چه حکایت از فراقت که نداشتم ولیکن...

In philosophy, the matter of personal identity[1] deals with such questions as, "What makes it true that a person at one time is the same thing as a person at another time?" or "What kinds of things a چه حکایت از فراقت که نداشتم ولیکن...

In the metaphysics of identity, the ship of Theseus is a thought experiment that raises the question of whether an object that has had all of its components replaced remains fundamentally the same obj چه حکایت از فراقت که نداشتم ولیکن...

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