Induction and Well-Ordering

[buildlackey]

good job.  I like it when people who initially did not understand  something explain how they got it.  To me it is much more accessible than explanations from people who live and breath the stuff and go too fast.

[fausto_]

Excellent explanation! I love the passion that you have for the topic 😃

[cleggings]

Is the minimum element of the natural numbers not 1 instead of 0?

[qianadam]

why considering the P(0) instead of P(1) {first number of N}

[jamusphillips]

Hey. Thanks for the video. Is the minimum element of a set the element that is lowest in value or the first element? For example what is the minimum element of the sequence 1/n?

[dineshdeshmukh]

Hi, nice video-i have a doubt-why p(k) is true and why that implies p(k+1) to be true? If i.e is assumption-why dont we assume that p(n) is true..

[soz]

Excellent explanation, thank you so much!

[miguelangelmerma]

Thank you so much. I understand that if we accept WOP, then we have to accept Induction. However, since WOP is logically equivalent to Induction, it seems that this proof is an instance of petitio principii. I will appreciate your response. Greetings from Lima, peru.

[ericjohn]

Induction works for all base cases not just for base case 0.
Proof by Contradiction:
Take the set of all such n greater than base case in the conclusion that does not hold.The min here is at least base case + 1(note min exist by well ordering). Let min be called as x. As x-1 is true but x is false we are contradicting the p(k)=>p(k+1) part.So the assumption is false .

[jackbinding]

Powers of 3. Does 3^0= 1 not count? I'm more curious than just trying to nit pick at irrelevant things haha. Thanks for the video though.

[berkayan]

Can't we conclude the proof at P(0) is false?

[hpp]

Assume S = {0} which clearly is a subset of non-negative integers. So, the cardinality of S is one, i.e., |S| = 1. Does it make sense to speak of a least element in this case since it wouldn't compare to anything? By saying 0 is the least element of S one could also argue it is the greatest element of S! Thus, 0 would be the least and greatest element at the same time which sounds quite nonsensical!

[tennisslayer]

Confusing, but makes sense! Thanks :P :)

[iosonoi.]

you jump over many important parts that makes it hard for someone who doesn't understand.