A set of mathematical axioms (of real numbers?) is not complete or consistent.
For instance, take all axioms such as a=b/b=c/a=c or 1 =/= 0 and assign them a number ie 1 (a=b/b=c/a=c) and 2 (1 =/= 0) respectively. Then also include the statement "Not all axioms can be proved by this set." If you include this statement and assign it a number say 237, then the set will not be complete (either with validating the statement or not validating) and also cannot be consistent.
You can add another axiom to take care of the problem, but then the old set + new axiom, falls into the same trap.
Implications: 1) Not all mathematical truths can be proved. 2) One cannot prove math has no contradictions.
Turing Machine
How do algorithms or programs work? It is not the mechanism of the computer that is ultimately important it is the algorithm that it is following. All problems can then be solved with 0s and 1s.
Turing Halting Problem
Turing used proof by contradictions
A computer cannot know if a program will halt.
"Taking a solemn oath to promise never to write a program that analyses other programs? - Professor Brailsford"https://www.youtube.com/watch?v=lLWnd6-vSGo
This is a self-referential problem. A program cannot write a program without knowing if it will loop infinitely.
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Gerog Cantor Transfinite Numbers
"Diagonlization"
Zeno's paradox going halfway between 2 points can go on for infinity
Galileo's paradox (Bijection)
"Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the "reals." (This set comprises all numbers that can be represented as a decimal, even if that decimal representation is infinite in length. Hence, 27 is a real number, as is π, or 3.14159….). In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals... Continuing down the list, this mathematical method (called "diagonalization") generates a real number p between zero and one that, by its construction, differs from every real number on the list in at least one decimal place. Ergo, it cannot be on the list.
In other words, p is a real number without a natural number partner—an apple without an orange. Thus, the one-to-one correspondence between the reals and the naturals fails, as there are simply too many reals—they are "uncountably" numerous—making real infinity somehow larger than natural infinity." From https://www.scientificamerican.com/article/strange-but-true-infinity-comes-in-different-sizes/
Real Numbers < N(0,1)
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For instance, let's say I have this Turing machine, H, which tells us whether or not a program and input will halt. Let's say we call H on itself. It has to give an answer, so if it prints out "does not halt" then didn't it technically halt to print that statement? Or would it just always in theory print out "does halt"? I'm having trouble wrapping my head around calling H purely upon itself, without negation, and what it would do. I get why the negation leads to a contradiction, but I am just wondering if the following scenario also leads to a contradiction."
You need to prove that H does not exist. You have shown that H applied to itself cannot print "does not halt". But, as you rightfully point out, the possibility that it prints "does halt" is not excluded. There's no apparent contradiction in this. So this application of H to itself is not sufficient to prove that H does not exist, we need to use other techniques. It is incorrect to say that this scenario does not lead to contradiction. It probably will if you explore it further. It just doesn't do so immediately.
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"Diagonlization"
Zeno's paradox going halfway between 2 points can go on for infinity
Galileo's paradox (Bijection)
"Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the "reals." (This set comprises all numbers that can be represented as a decimal, even if that decimal representation is infinite in length. Hence, 27 is a real number, as is π, or 3.14159….). In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals... Continuing down the list, this mathematical method (called "diagonalization") generates a real number p between zero and one that, by its construction, differs from every real number on the list in at least one decimal place. Ergo, it cannot be on the list.
In other words, p is a real number without a natural number partner—an apple without an orange. Thus, the one-to-one correspondence between the reals and the naturals fails, as there are simply too many reals—they are "uncountably" numerous—making real infinity somehow larger than natural infinity." From https://www.scientificamerican.com/article/strange-but-true-infinity-comes-in-different-sizes/
Real Numbers < N(0,1)
