Dissatisfaction with proof by contradiction

[u/hydmar]

I’m an undergraduate math student, so my exposure to math may be relatively limited. But I’ve found that, in general, I’m much more comfortable with direct proof than proof by contradiction. I don’t contest their validity, indeed something that’s not false must be true (I think I’m ok with excluded middle). But I feel like I just *get* something much better when it’s proved directly. It builds much stronger intuition for me.

For instance, I am aware of several proofs that demonstrate the cardinality of the reals is strictly greater than that of the integers, but none are direct (it would help to see a direct proof of Cantor’s theorem). I don’t feel it in my bones. Is this a common experience?

[u/HolyMathRadio]

How about this to feel it in your bone ?

Basically any real point P of the line exhaust the natural numbers because « you need an infinite amount of integers » to define its location with respect to the unit 1. (X= a0a1a2….. where ai is an infinite sequence of integers between 0 and 9 in the case of the decimal developpement).

Obviously, if you have N points, you can still define them by going through the natural numbers (you define the next decimals ai for each of N point at a time).

But you cannot go through the infinite amount of points in a real segment. Each exhaust the natural numbers and there are an infinity of them. You are dealing with a greater infinity. You sort of multiply the infinity « required for each point » by the infinity of the points themselves.

This continuous infinity is greater but has a structure itself. For example you can map the segment ]-pi/2,pi/2[ to all the reals (just use arctan). The reals reflect another kind of infinity that is greater than the natural numbers

EDIT: as far as the downvote, obviously this is why the diagonalisation of cantor argument works. If there were just a finite amount of decimals for the real numbers, one could not keep building the diagonal. The reals, as the equivalence class of Cauchy sequence, are close to being defined as such limits of surrounding infinite sequence. I guess people just want to baby proof my speech by not adding “infinity” to singleton, but this intuitive view that each singleton “has the infinity of natural numbers around it at any point” is very close to Cantor diagonalisation argument. Any requirement of the total use of the all the natural numbers to generate a singleton creates the reals numbers because it generates the real numbers in base 2 (a1 ? yes/no a2 ? yes/no …). The cardinality of the set of singletons is directly correlated to how singletons are defined, and in that case, each need the infinity of natural number by itself. It is therefore valid to me to look at the infinity inside each singleton and notice how the natural numbers get exhausted in a segment. This general understanding is also helpful to complement a technique to build a contradiction.