Wonderful video and question if you have a moment: Can any statement then be considered an Axiom? Since no proof is necessary? Also - I read somewhere that “2+2=4” is an axiom and “3 is a natural number” is an axiom. Are we sure those are technically axioms though?!
@MatheMagician
10 ай бұрын
Hi Mathcuriousity, thank you for your question. Technically you could choose any set of (consistent) statements as your axioms and develop a corresponding "mathematics" accordingly. However, in mathematics nowadays a certain set is chosen and commonly accepted. (and even there one particular axiom - the axiom of choice - is under debate). In particular this set is as small as possible; a statement like "3 is natural number" follows from definitions, which means that it is superfluous as axiom, which is why it is not considered an axiom. Hope this helps, good luck!
@MathCuriousity
10 ай бұрын
@@MatheMagician Hey thanks for writing back! Ok so thank you for confirming my suspicion that this other KZitem video was wrong: it tried to say “3 is a natural number” is an axiom. Now I understand how in peanos axioms for natural numbers that “0 is the first number” is an axiom? Maybe? , but to say “3 is a natural number” is more like a proposition in that it follows from the axioms right?
@veilofmayaa
4 жыл бұрын
Excellent explanation. Clear and to the point. Thank you.
@naveensundar4765
4 жыл бұрын
What is a lemma?
@MatheMagician
4 жыл бұрын
Hi Naveen, thanks for your question. When the proof of a theorem is very large, it is sometimes split in smaller steps called lemmas. So in order to prove the theorem, you prove the lemmas first and then you use the lemmas to prove the theorem. Hope this helps, good luck!
@naveensundar4765
4 жыл бұрын
@@MatheMagician thanks!
@adosar7261
4 жыл бұрын
When we define something e.g. a function should we also prove that a function exists?
@MatheMagician
4 жыл бұрын
In principle yes, however, in many practical cases you may be able to use existing theorems etc. It depends how fundamental you want to be.
@adosar7261
4 жыл бұрын
@@MatheMagician Thanks for the answer. For example when we define determinant we have already shown that there exists a unique number that can be obtained by multiplying and adding the entries of the matrix. Can we do the same with function? Can we prove that there exists a mapping and we call that mapping function? In other words is every definition we make just an abbrevation for entities that can be proved to exist by the axioms?
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