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The same theory describes every real closed field, not just the real numbers. [6] However, there are other number systems that are not accurately described by these axioms; in particular, the theory defined in the same way for integers instead of real numbers is undecidable , even for existential sentences ( Diophantine equations ) by ...
The theory of real closed fields is the theory in which the primitive operations are multiplication and addition; this implies that, in this theory, the only numbers that can be defined are the real algebraic numbers. As proven by Tarski, this theory is decidable; see Tarski–Seidenberg theorem and Quantifier elimination.
While a monkey is used as a mechanism for the thought experiment, it would be unlikely to ever write Hamlet, according to researchers.. The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, including the complete works of William Shakespeare.
Artificial intelligence (AI), in its broadest sense, is intelligence exhibited by machines, particularly computer systems.It is a field of research in computer science that develops and studies methods and software that enable machines to perceive their environment and use learning and intelligence to take actions that maximize their chances of achieving defined goals. [1]
AI Chatbots in their own words: ChatGPT: OpenAI - GPT-3.5 (GPT-4 available) "I'm an AI language model created by OpenAI called GPT-3.5, designed to understand and generate human-like text based on ...
is a sentence. This sentence means that for every y, there is an x such that =. This sentence is true for positive real numbers, false for real numbers, and true for complex numbers. However, the formula (=) is not a sentence because of the presence of the free variable y.
Elon Musk has said he believes AI is “one of the biggest threats” to humanity, and that the UK’s AI Safety Summit was “timely” given the scale of the threat.
First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all men are mortal", in first-order logic one can have expressions in the form "for all x, if x is a man, then x is mortal"; where "for all x" is a quantifier, x is a variable, and "...