I am new to this forum (I just signed up) but would like to pose a question regarding Gödel's Incompleteness Theorem. I have written a couple of paragraphs that I intend to use and would like someone who knows the two theories to comment on whether there is anything that is incorrect in what I have written. It is a summary of the more famous of the two theories and is mostly gathered from professors' writings about the Incompleteness Theorem as well as some books that I have. It is a two-paragraph, non-mathematical summary of his ideas. Please feel free to comment if you are knowledgeable in Gödel's theories.

I am most obliged for any help!

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The final example used here, but certainly not the last that could be given, deals with one of the greatest mathematical discoveries of the 20th century. In fact, its discovery altered the way that scientists look at their discipline, how philosophers see their own subject, and in fact, every subject area that we know. The discovery is referred to today as Gödel's Incompleteness Theorem. In the late 1800s, it was assumed that mathematics would ultimately discover every truth that could be proved using math. As the nineteenth century came to a close, the belief that mathematics could solve any problem that it was capable of solving began to crumble. In the early twentieth century, even the most self-assured mathematicians realized that math itself had fundamental flaws. Then, in 1931 a Czech-born mathematician named Kurt Gödel submitted a paper that showed no logical system has statements that are both provable and true; there are only statements that are unprovable and true or provable but not true. His discovery means that all logical systems of any complexity are, by definition, incomplete; no logical system can ultimately be proved to be true.

To understand this concept, imagine for a moment you have a blank sheet of paper in front of you divided into four quadrants by a vertical and a horizontal line: upper left, upper right, lower left and lower right. In the upper left quadrant is written “Proven and True,” in the upper right quadrant is written “Proven and Untrue,” in the lower left quadrant is written “Unproven and True,” and in the lower right quadrant is written “Unproven and Untrue.” Kurt Gödel showed through a series of mathematical proofs that propositions and ideas can exist in any of the three quadrants except the upper left. Ideas can be shown to be proven but untrue for example, or unproven but true. However, in the end Gödel showed that there is no idea, mathematical system or logical system that can inhabit the upper left quadrant where it is both proven and true.