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I'm surprised this doesn't mention Cauchy sequences. It seems to get close, but never quite hit the mark. Anyway, a Cauchy sequence is one where the terms of t

by msla 7y ago

I'm surprised this doesn't mention Cauchy sequences. It seems to get close, but never quite hit the mark. Anyway, a Cauchy sequence is one where the terms of the sequence eventually become arbitrarily close to one another; that is, you can pick an epsilon, an arbitrary small number, and after a certain point, the difference between elements of the sequence is less than epsilon. The intuition is an infinite list of rational numbers which, when plotted, look like a ping-pong ball endlessly bouncing around a given point, getting closer and closer... to what? Well, it depends on the precise sequence, but that end-point might not exist in the rationals. If it's a_n/2 + 1/a_n, it converges to the square root of 2, but we've known for a long time that that's irrational, so the ping-pong ball bounces forever, from one rational to another, getting closer and closer to a spot that doesn't exist... What if it did exist? What if all those spots existed? Well, if you have that, the set is now Cauchy-complete, also known as a complete metric space; if you have that, plus the field axioms (so addition, subtraction, multiplication, and division all work as you expect), plus ordering (so greater-than and less-than always work), plus the Archimedean axiom (no infinitesimal or infinite numbers), you have the real numbers. The reals are the unique Cauchy-complete Archimedian ordered field; every other set with those properties is isomorphic to the reals.

heyitsguay 7y ago
Great layman explanation! I'd suggest one extremely minor modification to your ping-pong metaphor: Cauchy sequences don't necessarily have to bounce back and forth between being smaller than your target point and being larger than your target point. They can always increase, or always decrease, too. To give a concrete example, one possible Cauchy sequence for the square root of 2 would just be {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...}
addcninblue 7y ago
I'd also like to add one more clarification: Cauchy sequences require _any two_ elements after N (the "certain point" mentioned above) to be within epsilon of each other. Why? I won't explain it fully, but hopefully a counterexample can convince you so: Take the common harmonic series (a_n = 1+1/2+1/3+...+1/n) Each successive element differs from the previous by 1/n. Therefore, informally we can see that their pairwise differences gets arbitrarily small. However, this sequence is _unbounded_: The infinite sum of the harmonic series tends to infinity. It therefore cannot be Cauchy. (Cauchy "bounces around a fixed point")
- msla 7y ago
  That is a very good point.
steerablesafe 7y ago
> The reals are the unique Cauchy-complete Archimedian ordered field; every other set with those properties is isomorphic to the reals. One can also arrive at the reals by constructing from the rationals through completion. https://en.wikipedia.org/wiki/Complete_metric_space#Completion https://en.wikipedia.org/wiki/Complete_metric_space#Completi... Edit: There is a nice wiki page about the different approaches https://en.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...

by msla 7y ago

heyitsguay 7y ago

Great layman explanation! I'd suggest one extremely minor modification to your ping-pong metaphor: Cauchy sequences don't necessarily have to bounce back and forth between being smaller than your target point and being larger than your target point. They can always increase, or always decrease, too. To give a concrete example, one possible Cauchy sequence for the square root of 2 would just be {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...}

addcninblue 7y ago

I'd also like to add one more clarification: Cauchy sequences require _any two_ elements after N (the "certain point" mentioned above) to be within epsilon of each other. Why? I won't explain it fully, but hopefully a counterexample can convince you so: Take the common harmonic series (a_n = 1+1/2+1/3+...+1/n) Each successive element differs from the previous by 1/n. Therefore, informally we can see that their pairwise differences gets arbitrarily small. However, this sequence is _unbounded_: The infinite sum of the harmonic series tends to infinity. It therefore cannot be Cauchy. (Cauchy "bounces around a fixed point")

msla 7y ago
That is a very good point.

steerablesafe 7y ago

> The reals are the unique Cauchy-complete Archimedian ordered field; every other set with those properties is isomorphic to the reals. One can also arrive at the reals by constructing from the rationals through completion. https://en.wikipedia.org/wiki/Complete_metric_space#Completion https://en.wikipedia.org/wiki/Complete_metric_space#Completi... Edit: There is a nice wiki page about the different approaches https://en.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...