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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.
On 23 July 2010, [12] LessWrong user Roko posted a thought experiment to the site, titled "Solutions to the Altruist's burden: the Quantum Billionaire Trick". [13] [1] [14] A follow-up to Roko's previous posts, it stated that an otherwise benevolent AI system that arises in the future might pre-commit to punish all those who heard of the AI before it came to existence, but failed to work ...
(In particular, the sentence explicitly specifies its domain of discourse to be the natural numbers, not, for example, the real numbers.) This particular example is true, because 5 is a natural number, and when we substitute 5 for n , we produce the true statement 5 × 5 = 25 {\displaystyle 5\times 5=25} .
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]
Word2vec is a group of related models that are used to produce word embeddings.These models are shallow, two-layer neural networks that are trained to reconstruct linguistic contexts of words.
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. [1] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!"
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.