The Real numbers are NOT larger than the Naturals: Information as a foundation of Maths.

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[@user-jn9hs5ry7h]

For some reason I can't reply in main comment thread. My answer is:
"For every undefinable real number x, x^2 > 0"

[@aleph1285]

I would say that since an undefinable number has an allegedly infinite number of decimal digits, x^2 would be a program that never halts and so does not have an output and so x^2>0 would be a meaningless statement

[@user-jn9hs5ry7h]

@@aleph1285 Ok. Let's approach it from axiomatic perspective. I have a bunch of axioms (ZFC and regular logic). This axioms form a formal system. Using this axioms I constructed an object. Let's call it "the set of real numbers". Now I say that there is no bijective function from this set to natural numbers. This statement also follows from axioms. Do you agree, that in this formal system the statement "the cardinality of reals is bigger than of naturals" is true?

[@user-jn9hs5ry7h]

@@aleph1285 Also, if you say an axiom of infinity doesn't make sense, you can't make proofs by induction. How would you work around that?

[@aleph1285]

@@user-jn9hs5ry7h I am not sure that is true, do you have a reference for that?

[@aleph1285]

@@user-jn9hs5ry7h I do not, I think the axiom of infinity in ZFC is a contradiction, because I think of all of these axioms and sets as informational objects, computer programs. the statement "the cardinality of reals is bigger than of naturals" is equivalent to saying "The area of my squared circle is 5"

[@matteovidali3829]

There may be a question of uniqueness when considering the constructor of an object as identical to the object - i.e f(x) = x^2 represents all squared numbers, and yet 25 is a unique object. Perhaps it can be said that f(x) = x^2 (and) x:=5 make it identical? I’m not particularly sold that a constructor and an object are identical…

[@aleph1285]

You do not need to be, the constructor is not the same as the output. All I am saying is that some constructor programs do not halt, and do not have a final output

[@matteovidali3829]

@@aleph1285 that makes sense, and follows. But if they aren’t the same, then the argument that pi has finite information is false.
And about the halting, it’s certainly a good point, but even for cantors argument, you can consider the diagonal as a similarly ever increasing number, and the simple fact that no matter how long you let it run, the diagonal will always differ from every other element by at least one numerical position holds even to infinity, thus leaving the halting somewhat unimportant?

[@aleph1285]

@@matteovidali3829 Oh no, the halting is crucial, the fact that the diagonal does not halt is the reason there is no Cantor's number

[@matteovidali3829]

@@aleph1285 I am not sure I follow there. Surely because every partial expansion of such a series holds that property, then the property is true of the entire infinite set, where does it go wrong?
If every subset of the diagonalization holds this property, then by induction the entire set does as well

[@aleph1285]

@@matteovidali3829 Oh no, that does not follow, see en.wikipedia.org/wiki/Fallacy_of_composition

[@user-jn9hs5ry7h]

The only thing this shows is that not every real number can be an output of a computer program.
Because the number of computer programs is countable.

[@aleph1285]

Mathematicians claim that undefinable numbers are real. Something that is undefinable seems as unreal as it gets

[@user-jn9hs5ry7h]

@@aleph1285 How would you define real numbers then?

[@aleph1285]

@@user-jn9hs5ry7h those that are definable. It seems crazy to me that mathematicians call things we cant even define “real”

[@user-jn9hs5ry7h]

@@aleph1285 I mean formal definition of the set of real numbers. For example, one way to do this is to consider all Cauchy sequences of rational numbers and define as equivalent those witch "converge to the same thing".
In your version how would you change the definition so that it only includes "definable" real numbers?

[@aleph1285]

@@user-jn9hs5ry7h watch this one for the formal definition of definable
czcams.com/video/RTcXiutDxYY/video.htmlsi=Gx_e_Wo8WRdzQMs2

[@bobbobbybobson2282]

In terms of a mathematical object that doesn't have finite kolmogorov complexity consider a random binary sequence. Also you may want to refine your argument about halting programs, in it current form it forbids basically the whole of calculus. I think it fails when you consider that the non-halting property doesn't applying to subsets of the programs output. Any finitely indexed digit of the program output can be found with a finite amount of computation. If cantor's diagonal argument is applied to numbers between 0, 1 then one can argue that the program output converges. If you don't like convergence then I refer you to my comment about calculus.

[@bobbobbybobson2282]

Also you didn't offer any proof that the cardinality of the naturals and real are equal. If they are then you should be able to construct a bijective map between them.

[@aleph1285]

clearly a random binary sequence has a finite Kolmogorov complexity because you just conveyed the concept to me. You could have a computer program that computed the sha-256 of all the natural numbers one buy one and added it to a sequence and that would give you a random sequence of 1s and 0s. I do not see why it would forbid calculus, could you explain further. I agree all the finite states of the computer program exist, what I am saying is that some programs, like Cantor's diagonal, do not have a final output.

[@aleph1285]

@@bobbobbybobson2282 Oh, no I did not, I thought I was clear at the end, I think speaking about the cardinality of infinite sets is like speaking about the cooking skills of a married bachelor.

[@bobbobbybobson2282]

@@aleph1285 I was only able define what random sequence is - not any specific example of one. The sha-256 ideas isn't random, it's completely predictable, just annoying to work out. If I give you a finite subsequence you cannot tell if it's random or not with any certainty weather its is random or just a complicated pattern, you need an infinite subsequence if you want to determine with certainty if a sequence is random. Calculus relies on of limits which would be classed as "non-halting programs", since they never achieve their final value only get arbitrarily close to it. The distinction between countable and uncountable infinities becomes import all over the place, not limited to the discrete vs. continuous spectra in quantum mechanics - whose existence can be verified empirically.

[@aleph1285]

@@bobbobbybobson2282 Sure, Sha-256 is not random is pseudorandom, if you want something really random you can capture background noise or something. Regarding Calculus we can have those programs stop at arbitrary degrees of precision, and we may even prove they converge to some value I am not against that. I dont know about these discrete vs. continuous spectra in quantum mechanics, could you explain how they rely on transfinite arithmetic?