Axiom 2: The Axiom of Foundation
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- Опубликовано: 8 фев 2025
- In this video we talk about the second axiom of math.
Everything in math is based upon on 9 axioms.
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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.
Advice, I think you should focus on explaining only, random interview is disrupting.
Yes, thank you, I'm actually not doing that interviews anymore! Ciao!
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
I'm glad they did!!!
You are making simple things complicated.
Sorry, I'm allergic to thumbnails with open mouths.
Ok, fair
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.
thx for your comment ... yes, indeed, the reflexive property of equality is somewhat connected to axiom 2 ... at least intuitively, because if A belongs to B and B belongs to A ... we cannot help but think that they are closely interconnected ... but they may not be equal ... Maybe there's more to think about under the hood ...
@@guzmat Thanks for the reply. Because ellipsis means that something had been omitted, I've been trying to figure out what ideas you omitted from your reply that I was supposed to understand. I haven't come up with anything.
It seems to be that for now axioms in maths are made on the presumption that there is always a beginning and an end. But if that’s so then infinity does not exist? In the physical world this would mean for example that there is nothing tinier than the tiniest particule we know which means that at some point we end up with pure void. And some goes for what we call the universe, there is a thing that is bigger than everything. And at the same time those two concepts are intrinsically infinite since we could always go below each new particle we discover. I feel like there is some kind of paradox.
Not sure I make sense at all 😅
The illustration at @6:28 reminds me a lot of the activity of fractals. Can't what they do disprove this axiom?
Nice observation, yes fractals are very similar to this situation. The difference between these two situation is that:
- a set like the one shown would contain itself as an element of itself
- while a fractal does have a subset which is equal to the whole fractal...
so in the first case there is an element equal to the whole set,
in the second case there is a subset equal to the whole set ...
I hope this helps ... ciao!
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.
I really like your comment ... it is really interesting ...
I like that X is just X and nothing else ... but maybe you're mixing two concepts when you say that X has only X inside itself ... because X is just X and does not have X inside itself ... I mean: being something and having something are different concepts ...
@@guzmat But are being something and having only the elements of one self any different from one another?
No, ok, but X has not X amongst its elements ...
@@guzmat X has B and only B
B has 1,2,3,4
X has 1,2,3,4
Replace B with X
Looking at it further tho I’m unsure about how this works.
X has X and only X
X has 1,2,3,4
If X only has X then does that mean X can’t have 1,2,3,4?
@@guzmat basically what I was trying to get at was that if your going to say set B is inside set X and set X and set B are the same, then set X is indeed inside set X
I think it's kinda misleading to equate the special case with the axiom itself.
Yes, I totally get what you're saying ... but on the other hand explaining the axiom of foundation in its generality I think it was a bit too much for the general audience ... maybe I should have stated it near the end ...
@@guzmat I especially mean how you did it with the passersby
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.
Thank you very much for the comment and for the suggestions.
1 - Yes, you're right, this is not "THE" way to start math ... but it is the most common way nowadays ... so it makes some sense ...
2 - I've been thinking about that for a while. As you said, I simplified the presentation for the video but stating the actual axiom and explaining it, might make sense. I'm not looking for a very broad audience but rather for explaining some quality math ... so it might fit.
thx again ... cheers.
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.
I agree, perhaps your wording is better than what I was able to convey with my words ... thanks for the comment. Ciao!
The self-eating snake, the infamous Klein Snake.
yeah ... exactly
a = b, b = (c + a) then
a = (c + a)
a = (c + (c + a))
a = (c + (c + (c + a)))
Infinite logical paradox
C = 0 lol what paradox?
Greaaat
Is this series about these axioms? ( ZF (the Zermelo-Fraenkel axioms without the axiom of choice) ):
en.wikipedia.org/wiki/List_of_axioms
Yes, it's about ZF
@@guzmat
Thank you!
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.
I totally agree with what you say ... you need logic to start everything ... but logic somehow lives in a different level so that we can say that what we normally think as "math" starts after the logic level ... and rests upon it as you said ...
But yes, when explaining to general audiences you have to simplify a bit a take some decisions ...
Your videos are entertaining... except for all those interviews. I do not care what random people in the street think
Yes, thank you, I'm actually not doing that interviews anymore! Ciao!
Too many street interviews - please get to the point!
Ok
а мне понравились уличные интервью
Naw seeing peoples intuition is interesting.
I like to see people's reasoning. Keep with the good content.
the axioms of set theory are not the same as the axioms of math. derp.
Hi, math can be constructed in various ways but the most common way is to base everything on the axioms of sets.