Axiom 2: The Axiom of Foundation

In this video we talk about the second axiom of math.
Everything in math is based upon on 9 axioms.

New Puzzles every Sunday and Thursday.
I post math puzzles and their solutions with animations.

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[llf]

You make axioms like so easy, when I first entered the axioms of ZFCs, I had a lot of trouble memorizing them and understanding them. Please continue make videos on this. Especially on the axiom of choice, Zorn's lemma, that I still don't quite get it.

[vonjuez]

You are making simple things complicated.

[luojihencha]

Advice, I think you should focus on explaining only, random interview is disrupting.

[HunterHogan]

6:14 X cannot be an element of X made me think of the reflexive and symmetric properties of equality, which is mildly important. It seems that this axiom gives rise to those properties.

[emmanuelnwafor]

The illustration at @6:28 reminds me a lot of the activity of fractals. Can't what they do disprove this axiom?

[dsy-wklj]

I dint know what to do after my teacher said we will have a test and there will be aksioms on the test these videos helped me alot to understand them thankyou

[nicknolder]

Are there different systems of math with different axioms? I only ask because I have this intuition that every set is an element of itself with nothing extra. When I say this I don’t mean X and something extra is inside of X. I mean that whatever is inside of X is X itself. So X is the only element of X which is the only element of X and so on. I find this to just be equivalent to saying X=X=X=X=X= …

Now u might be asking “but how can u have the only element that is inside of X be only X? Can’t u have other elements?”

Well when I say X I’m talking about all the elements inside of X. So like if all the elements in set X are 1, 2, 3 and 4 then when I say only X is inside of X and we defined X as 1, 2, 3 and 4 then X and 1, 2, 3 and 4 are interchangeable. So if 1, 2, 3 and 4 are interchangeable with X since that’s how we defined set X, then we can say X is indeed inside set X, but that nothing extra with X is inside X because that would be like saying X = X + 1.

Idk I might be way off base here but that’s how I currently see it.

[Tommy_]

Sorry, I'm allergic to thumbnails with open mouths.

[alittax]

Is this series about these axioms? ( ZF (the Zermelo–Fraenkel axioms without the axiom of choice) ):

[worldnotworld]

These are very amusing videos, and I think you might be on to something! Two suggestions: 1. I'm not sure you should call set-theoretic axioms "_the_ axioms of math." You can "start" math in any number of ways, and even if you do start with set theory, there are different axiomatizations that will do the trick. 2. The way the Axiom of Foundation is often formulated (in contrast to your depiction and the basic intuition of banning infinite regress) is very hard to understand ("there is an element of the set that shares no member with the set "); maybe you could do a video on how it works? It might be too advanced for your intended audience; I don't know.

[whocareswho]

The self-eating snake, the infamous Klein Snake.

[lyingcat]

a = b, b = (c + a) then
a = (c + a)
a = (c + (c + a))
a = (c + (c + (c + a)))
Infinite logical paradox

[antoniusnies-komponistpian]

I think it's kinda misleading to equate the special case with the axiom itself.

[Lex-jjpw]

I understand the point of your videos.
Still, I don't agree with them in terms of foundations upon which we can start building up maths.
Indeed, to formulate those axioms we need something even more fundamental that we need to agree on : first order logic.
You couldn't even formulate your first axioms without the "if and only if".
It is so simple that it is taken for granted but it shouldn't because there I so much logic that intuition alone can provide. Hence the difference between logic and common sense. Once implication is involved, things start to become messy and without that knowledge, building up maths will quickly become a dead end.
But yeah, I understand the simplification here since it's hard to start with logic without knowing a thing about sets and axioms and it's hard to introduce sets and axioms without knowing logic.
I firmly believe those 3 should be taught in parallel but to grab people's interest, it's a cool place to start.

[mrslave]

6:50 “An axiom is something that does not need a proof“ 😂😂😂😂😂 no. And axiom is a ground truth by definition. Meaning we say that it’s true. Or we assume that it’s true. And then we build the rest of our system on this and other assumptions. Its truth is assumed by definition. We might be able to prove it we might not. We just assume for now that it is true.

[osvaldo]

Your videos are entertaining... except for all those interviews. I do not care what random people in the street think

[armantookmanian]

Too many street interviews - please get to the point!

[oversquare]

the axioms of set theory are not the same as the axioms of math. derp.