Question

maths can never come to an end...... Paragraph likhna hai

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Answer 1

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Will mathematics be “completed” at some point? Will there be a time when there is nothing more to add to the body of mathematics and research has exhausted? That Godel's incompleteness theorem undoubtedly proves that mathematics will never, in the sense that you're asking, be "complete."

I wanted to dive a little deeper there because I feel that a lot of the answers to this question are somewhat philosophical and relatively wishy washy, but Godel's theorem actually offers a very logical, rigorous, and "mathy" way to answer this question which I was surprised that no one other than Marisa had mentioned.

Basically, Godel proved mathematically that it is impossible to know everything there is to know about math.

That, at least for me when I first learned it, was kind of a mind-blown moment. Both the meaning of this proof (we'll never know everything!) and the fact that the proof even exists (you can rigorously prove that we will never know everything?) were somewhat unsettling. Remember that this is not philosophy, we're starting with numbers here. Godel proved this using pretty much just arithmetic. It is Fact.

And yet, the proof is undeniably complicated (taking up over 20 pages of solid text), because, well, he was super rigorous, as math is.

In a slightly more precise and complicated statement, Godel's proof states that if a system has enough complexity to be able to express the rules of arithmetic, then the system itself cannot be both complete and consistent.

The fact that arithmetic is needed is a somewhat important distinction. Not all systems are complex enough to prove arithmetic, and those systems could happily be both complete and consistent. However, considering that Mathematics includes both these systems and systems that include arithmetic, it follows that we'll never prove everything about Mathematics.

But what is this concept of consistency which I keep talking about? Though completeness is easier to grasp, it's also worth going over really quickly.

Completeness basically means that within a system of rules and axioms, you can prove all possible Truths within that system. That is, if something is true, there is a way to prove it by starting with the axioms and manipulating them using the rules of the system (the process of proof).

Consistency basically means that given the rules and axioms of a system, you cannot arrive, via the process of proof, to a contradiction. That being, I can't prove that 1 = 2 or something crazy like that .

The implications of this are pretty cool and extend to a lot of things beyond mathematics. It basically says that there are things out there that are true and you'll never be able to prove them to be true. It's almost offers peace of mind when contemplating the vast knowledge of the universe, and can even justify believing in something yet unprovable by mere faith while still claiming to be a strongly logical individual (Note, this does not mean that you can continue believing in something that has been proven false because of faith, that would make you a very illogical individual).

$\textit{Hope this helps you maths might never end but this is an open ended Question}$

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Answer 2

Step-by-step explanation:

I have answers but given after sometime. I'm sorry for ur answering !!