Why Does Mathematical Induction Work?

Why Does Mathematical Induction Work?

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[vedkorla]

Here's another beautiful yet elegant analogy in Rosen.

Imagine you're trying to climb a ladder of n steps. You know that you can reach step number 1. You also assume that you can reach any step 'k', k <= n. If you can deduce you can reach step 'k+1' you can reach any step of the ladder. How? Since you're on step 1, you can reach step two. If you can reach step two then you can reach step three. So on and so forth till n-1. Since you have reached n - 1, you can also reach n because it's a direct application of your result for S(k+1).

[aitanapalomanespardos]

Thank you for sharing your intuition with the community. I love when people explain the real mechanics of Mathematics, instead of just a formula you have to learn off.

[lots]

This makes so much more sense now. Thank you

[SimchaWaldman]

It took me MANY years for the mathematical induction to click into place.
02:30 This is Strict Induction.

[gustavosolano]

you sir as a gem thank you for all these videos

[Risenoph]

I don’t know if you have already, but a video going over weak induction vs strong induction would be interesting.

[Amantheparadise]

I also did same explanation to satisfied me, after learning pmi

[FROSKYX]

4:40 at this exact second my mind blow!, it was easy yet i didn't see it before, the moment you said by 2 I thought k = 1 then i knew s2 is true and s3 and finally understood the Damion reference, thank you a lot it's such nice thing to understand this ❤️ so don't worry your explanation was so good i saw everything you gonna say before you even said it

[viktorpiktor]

I apologize for my horrible english. First, thanks for this video. Well done. Im confused that you need two variables n and k. Its correct, but its confusing. All you need is a first proof with a special integer, then you have to assume that this statement is also correct for any n and then you have to show that you can induce n+1 by n. This is how i see it in german literature everywhere. In english books i saw your style often, but i dont see why its usefull.

[shreyankranganath]

What about the proof for condition 2? Why is Sk+1 true when Sk is true (for k<=n)?

[karamkassem]

I am sorry but didn't you just used the induction to explain induction itself??
So the "why" question is saying that if S1 is true why S2 will be true ?
( but instead you used that normaly without asking why and generalise it on Sn and Sn+1)

The real question is why if something is true the next will be ?

[cgfam]

prof! Do you have video(s) for strong induction?

[humanvv]

I think its nothing more than "modus ponens" . if one has studied propositional calculus prior to proofs, than it becomes way easier to understand.

[eeee]

While I was watching this video, I was enlightened by the fact that by parity property, if k is even, k+1 is odd and if k is odd, k+1 is even and an integer is either even or odd. Therefore, because if the basic hypothesis is true, then it is also true for all the input integer n that have the same parity property with that lowest value domain of that statement which is the input integer of basic hypothesis. That's why if we can also prove conclusion which is P(k+1), it is true for all the range.(MY QUESTION IS HOW DO I USE MATHEMATICAL INDUCTION FOR NUMBERS THAT ARE NOT INTEGERS)😢😢😢😢😢😢😢😢😢😢😢😢😢😢😢😢🎉

[johnperez]

So the reason this works is because k can be any positive integer then the integer right after that will work?

[azizghalib]

I don't get how an assumption becomes part of the proof!
If there is a axiom in mathematics that says that an operation that works for the value of "1" works also for any value that is made of any multiple of "1", then this is true by definition; in other words "because we say so!" not because it is evident in the premise which in this case uses an assumption which is by definition a belief without proof.

[arnavpandey]

I was hoping for something better and a lot more rigorous