As I witness my mother's body deteriorate in painful slow motion, feeling helpless to do anything about it - as she is in an Assisted Living facility (and made a ward of the State due to her relatives doing all they could to remove me from the picture), I also witness a complicated and contradictory bundle of emotions in my "mood and temerament" since part of my nervous system is stimulated by this subjective deepening of mathematical insight into the parallels between Euclid's "division algorithm" for finding the "Greatest Common Divisor" and the representation of ratios in "continued fraction" form.
That example in the Wildberger video above is
BEAUTIFUL:
NJ Wildberger shows ( 1001 / 203 ) :
Using a program I built while in "codemode" cohabitating with my mother (when we were getting along very well, in harmony even):
cf_rat 1001 203gcd_count returns 4
array created with 4 elements?
1001/203 =
1
4 + -----------------
1
1 + ------------
1
13 + -------
2
1001/203 = [4; 1, 13, 2]
Then we see the connection as we apply Euclid's division algorithm to find GCD:
1001 =
4*203 + 189
203 =
1*189 + 14
189 =
13*14 + 7
14 =
2*7 + 0
You see the 4, 1, 13, and 2?
___________________________________________________
This morning, after witnessing my body shed some quiet tears over my mother's existential crisis, I applied these techniques to our estimation of pi.
cf_rat 31415 10000gcd_count returns 6
array created with 6 elements?
31415/10000 =
1
3 + -------------------------
1
7 + --------------------
1
14 + ---------------
1
1 + ----------
1
8 + -----
2
31415/10000 = [3; 7, 14, 1, 8, 2]
Applying Euclid's division algorithm to find GCD:
31415 =
3*10000 + 1415
10000 =
7*1415 + 95
1415 =
14*95 + 85
95 =
1*85 + 10
85 =
8*10 + 5
10 =
2*5 + 0
Do you see the 3, 7, 14, 1, 8, and 2 ?
There is an undeniable subjective beauty in this parallel, I think; and it remains beautiful even with painful (and personal/private/subjective) emotions all mixed in with it.

No axioms or set theory necessary for this ... It is from the ancient branch of mathematics loosely called Number Theory. This may or may not help one endure a life not worth living.