WHY DO WE LEARN SETS IN MATH ? PROPERLY ANSWER FOR BRAINLIEST
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Sets came to solve a similar problem sets are collections of mathematics objects which themselves are mathematical objects... This is of course doesn't mean that we should learn set theory just for that purpose.. I hope it's the brainiest answers
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Set theory is important because it is a theory of integers, models of axiom systems, infinite ordinals, and real numbers, all in one unified structure. This allows it to serve as a foundation for all of mathematics, anything you talk about in mathematics can be formalized in set theory naturally and easily, and studying set theory allows you to prove theorems about mathematics itself. The formulation of set theory in the late 19th century motivated the metamathematics of the 20th century, with all the astonishing results about provability.
It is an extremely important subject, and I am not going to do it justice in this answer. I would recommend to read Paul Cohen's book "Set Theory and the Continuum Hypothesis", together with some historical work from the late 19th century or early 20th century, like Frege and Cantor, to see where the ideas are coming from, and further work from more recent authors, like Saharon Shelah, who is a big name with big theorems and big books.
I will give an answer that focuses on the first three things, integers, ordinals, and models, because I personally think it is good to conceptually separate out the real numbers, as they are described in set theory. The real numbers are important for usual day-to-day mathematics, but in set theory, they can be a headache, because they are a different kind of infinity than the integers, ordinals, and logical models.
It is an extremely important subject, and I am not going to do it justice in this answer. I would recommend to read Paul Cohen's book "Set Theory and the Continuum Hypothesis", together with some historical work from the late 19th century or early 20th century, like Frege and Cantor, to see where the ideas are coming from, and further work from more recent authors, like Saharon Shelah, who is a big name with big theorems and big books.
I will give an answer that focuses on the first three things, integers, ordinals, and models, because I personally think it is good to conceptually separate out the real numbers, as they are described in set theory. The real numbers are important for usual day-to-day mathematics, but in set theory, they can be a headache, because they are a different kind of infinity than the integers, ordinals, and logical models.
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