The Natural Numbers
There are two major constructions of the set of natural numbers in ZF set theory, namely the Zermelo construction and the Von Neumann construction. In this video we use the Von Neumann construction to construct the set and we then equip it with the classical ordering, which comes very naturally using the Von Neumann construction. After this we look at how to define the familiar addition and multiplication laws for the natural numbers and finally we end with a discussion of the axiom of induction, which is the logical axiom that allows proof by induction as a valid mathematical argument.
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I love your videos, thank you for being so helpful for me and all the people seeking to learn more
Would it be possible to increase the volume, on your end, for future uploads? Please, my volume on the KZread player and on my Laptop are maxed out, and it's still difficult for me to hear the playlist.
@ptmad7910
3 жыл бұрын
Mine with youtube player full and laptop 30% volume is enough for me in a quiet room. If you try a headphone, it could be better? Just personal suggestion :0
Thank you Elliot your video is amazing.
Love this stuff bud! Best way to keep sharp and maintain a strong fundamental basis for all maths and reasoning in general! Keep it up!🎓💪🏼😎
just me or the volume of this playlist is very low?
I read about a proof in Oxford maths note, it is saying: 1. By definition N is the smallest set such that (i) 0 ∈ N (ii) if n ∈ N then n + 1 ∈ N. and Let P(n) be a family of statements indexed by the natural numbers n = 0, 1, 2, . . .. Let S be the set of natural numbers such that P(n) is true. Then 0 ∈ S as we know P(0) is true. Further if n ∈ S then P(n) is true, so that P(n + 1) is true or equivalently n + 1 ∈ S. Thus S has the properties that (i) 0 ∈ S and (ii) if n ∈ S then n + 1 ∈ S. As N is the smallest such subset with these properties then N is contained in S. Hence S = N.
@ptmad7910
3 жыл бұрын
It also seems to be a nice and neat proof, I just don't know where it got the definition from :)
By the statement k belongs to N implies that s(k) belongs to N, are we essentially getting the s(k) of every element in the set? If that was what's happening, then all the elements being repeated get their second copy cancelled out leaving the s(k) element which previously wasn't in the set. Is that what's happening? I know this question is stupid but I wanted to be sure.
Von Neumann is similar to how consciousness builds itself. Have a look at my paper "The Self-Referential Aspect of Consciousness", author Cosmin Visan, and tell me what you think. I wrote it without having any knowledge about set theory.