Have any of you heard of kurt Godel's proof, that arithmetic is "Incomplete"?

Question

i believe that this should be answered in "philosophy",as
distinct from math.
Now let me word this carefully: If the above is true-that higher
arithmetic can never be rational like 1+1=2, -there must be
"a Change-over",where the lower-becomes-the higher(!)
An example in math would be the change from mathematical
terms to metamathematical terms.
Im interested in what,if any, has Godel's proof in philosophy
and thus science,et al.

Answer 1

Godel's incompleteness theorm not only speaks to mathematics but to all formal systems. Godel's theorm is not directed at arithmatic per se, but rather at Russell and Whitehead's formalizaion of arithmatic through logic expounded in their magum opus, Principia Mathematica. Godel showed that any system which is sufficiently powerful to handle arithmatic is also necessarily incomplete.

The holy grail of logicians (and presumably mathematicians?) is a formal system that is both consistant and complete, but Godel's theorm is a powerful challenge to that even being possible, that to my knowledge has not been fully answered even today.

This finding has profound ramifications not only for math but for information processing in general, which drives deep into the heart of philosophy and science as well. Perhaps a new radically skeptical argument could be advanced that uses the incompleteness theorm to state that certain knowledge is impossible...

For a thourough, interesing, incicive, and beautiful discussion of Godel's Incompleteness Theorm and its ramifications for both Philosophy and Mathematics I reccomend you read:

Godel, Escher, Bach & I Am a Strange Loop
both by Douglas Hofstadter.

Answer 2

Yes. I have heard of Godel's "incompleteness" theorem.

The incompleteness theorem is a "higher mathematics" parlance. It says that there are (many) statements in arithmetic that could not be proven or constructed using mere "axioms".

You may have probably heard of "mathematics is axiomatic" from you professors. Well, that statement is a half-lie.

That is, Godel says that by using simple axioms to prove very deep conjectures in arithmetic would likely fail. In other words, axioms aren't eveything in arithmetic.