English: Foto of blackboard picture of a const...

logic is a funny thing. there’s syntax and there’s its meaning. totally independent.
it is quite frequent in maths that while syntax stays the same, meaning changes. the reason why syntax stays the same is: so you can make the same claims. write the same thing over and over again, only the meaning changes. this way the deep insight is shared that abstractly it’s the same thing.
for example numbers. we’re used to a decimal system, 10 digits. abstractly it’s the same as a binary system though. instead of denoting which number-system is meant, be sloppy. adding the b to every number over and over again is redundant. write once you’re talking of binary numbers, and everybody will understand.
only problem is when quoting someone. you’d need to repeat the definitions. so a non-mathematician might write 1+1=10 because someone wrote that. more correct would be to first define what the numbers mean. i.e. say: a number a_na_{n-1}\ldots a_1a_0 means \sum_{i=0}^n 2^i\cdot a_i

 this will be the 1+1=10 kind of proof. i.e. change the definition to get something unexpected.
the definition of real numbers based on rational ones is to look at limits of open rational intervals. every real number is a (non-unique) sequence of open intervals, each containing the next. and every such sequence which converges to an empty set represents some real number.
this real number then will be contained in every single interval of that sequence.

more exactly, every real number is a whole lot of such sequences, a whole equivalence class. two sequences are in the same equivalence class if eventually the intervals mutually contain eachother.
i.e. there exists a sub-sequence of one such that they’re inside the intervals of the other, at same index. and also the other way around, as both are open intervals, by definition containing points other than their border.

in a way real numbers are infinitesimal intervals, while rational numbers are actual points you can draw.
but then, \sqrt 2 is also a point you can draw! just take a  1×1 square, it will have diameter \sqrt{1^2+1^2}.
well, ability to draw means it’s constructible on paper with a ruler and compass. so, no, this can’t be the proof.
we need to show, \left(\frac pq\right)^2==2 or p^2==2\cdot q^2 for whole numbers p,q:
obviously p must be an even number, but then squaring it makes it divisible by 4. hence q must be an even number, and p^2 must be divisible by 8. and so on.
in the end both, p and q must contain all powers of 2. i.e. they must be infinite in size.

is that allowed? in the axioms of ZFC it isn’t. actually it isn’t the axioms on their own. what isn’t allowed is to create an expression that is infinite in length. in the language of logic this is clearly forbidden, you can’t have logic-expressions “converge” to something.
but why not? in fact, there are several approaches to create a set \sqrt 2 is contained in, as a point!

i.e. you can actually turn some real numbers into actual constructible points. apart from circle also other shapes could be used for construction. naturally construction of those other tools require some kind of approximation.
but then, if you restrict yourself to the 1-dimensional line, rational numbers are an approximation of their own already!
it’s possible to check if a rational number is what it claims to be. just use the compass to copy the rational number q times along the straight line. in the same way also other approximations can be verified for accuracy.

math isn’t just geometry. it’s also about formulas and functions. most prominent example is derivatives and their inverse, integration.
every formula has a formally definitive derivative. but formal integration is not always possible. so, while derivatives are constructible by formulas, integration is an approximation.
hence, real numbers created by integrals of formulas can be verified by derivative. but as language allows only finite expressions, these real numbers are countable.

English: Illustration of how the rational numb...

English: Illustration of how the rational numbers may be ordered and counted. Illustration contains a grid and an arrow, and two Swedish words (the words for “nominator” and “denominator”). (Photo credit: Wikipedia)

the notion of countable means: there exists a mapping onto the natural numbers.
for example whole numbers are countable: map every positive number to the odd numbers. negative numbers and zero are mapped to even numbers including zero.
a similar method works also for rational numbers. just interpret them as integer-coordinates on a half-plane.
then use some distance-function mapping them on natural numbers. and then add some ordering to points with the same distance.
(well, maybe this will skip some natural numbers. but it shouldn’t be a problem to re-index.)

no such strategy will work for infinite-length expressions. polynomials are countable, but power-series \sum_{i=0}^{\infty} a_i\cdot x^i aren’t. for every method to count them, one can create an element that isn’t counted.
i.e. a map from natural numbers to real numbers has no inverse! isn’t surjective!

thanks to this limitation of logic, only countably many reals are constructible. an uncountable amount remains as intervals.
in logic the constuction of a point cannot take up infinitely many steps. if it would, maths would be an intangible mess,
you cannot prove consistency by finite-size proofs then.

Gödel’s incompleteness theorem says in our mathematical logic you cannot prove consistency.
the proof starts out with the observation that expressions can be counted. same with proofs. then it goes on showing how this counting can be implemented in the language of logic.
the problem arises when you start proving stuff about these numbers. then you get into the same mess as with my proof above. some things require expressions of infinite length. but as history has shown, then inconsistency easily would creep in.
when dealing with infinity, a lot of additional information is required. literally, an infinite amount of information, infinite sets of axioms and such. talking of infinity is always alike to leaning out of a window: easy to lose your grip on the ground.

 

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