There is a video (https://www.youtube.com/watch?v=S4zfmcTC5bM) from the PBS Infinite Series that covered this topic as well, for people that what a visual version of the explanation. I'm still sad that the series/channel was shutdown though, such a great series.
PS: this article/video essentially defines the naturals from the fundamental set theory axioms. This other video (https://www.youtube.com/watch?v=KTUVdXI2vng) from the same PBS series shows how you can then use the naturals to construct other types of common numbers, up to the reals.
PS: this article/video essentially defines the naturals from the fundamental set theory axioms. This other video (https://www.youtube.com/watch?v=KTUVdXI2vng) from the same PBS series shows how you can then use the naturals to construct other types of common numbers, up to the reals.