Is the following text about the Incompleteness Theorem corre

Is the following text about the Incompleteness Theorem corre

Postby Guest » Sun May 17, 2020 9:23 pm

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!

* * *

[text]

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.
Guest
 

Re: Is the following text about the Incompleteness Theorem c

Postby HallsofIvy » Sun Jan 31, 2021 1:19 pm

I think your wording can be improved. For one thing, I don't believe that Godel's theorems means that math is "flawed". That's a value judgement. More importantly, you say that Godel proved that "no logical system has statements that are both provable and true". That is not at all what he proved and, I am sure, not what you intended to say! In the logical system of integers, the statement "1+ 1= 2" is both true and provable.

What Godel proved was that, in ANY logical system large enough to include the natural numbers there EXIST statements such that neither the statement nor its negation can be proved.

The condition "large enough to include the natural numbers" is important. It is easy to construct "toy" systems with just a few objects and postulates that can be proven to be "complete" (for every statement either the statement or its negation can be proved).

Further every logical system has statements that can be proved true. There wouldn't be much point to "logical sytems" if there weren't! Godel's theorem is about statements a logical system contains, not what it doesn't contain. There exist, first, statements that can be proven true, of course, statements that can be proven false (their negations can be proven true) AND statement that cannot be proven either way.

(I prefer not to use the words "true" and "false" which are also "value judgements". I Prefer to say that there are statements that are "provable", statements for which their negations are "provable", and, per Godel, statements for which neither are "provable". For completeness we might include statements such that both the statement and its negation are provable but then every statement is provable, so that "provable" has no useful meaning, so we call those systems "inconsistent" and don't use them.)

HallsofIvy
 
Posts: 340
Joined: Sat Mar 02, 2019 9:45 am
Reputation: 127

Re: Is the following text about the Incompleteness Theorem c

Postby Raiden Mitchell » Tue Apr 27, 2021 2:49 am

HallsofIvy wrote:I think your wording can be improved. For one thing, I don't believe that Godel's theorems means that math is "flawed". That's a value judgement. More importantly, you say that Godel proved that "no logical system has statements that are both provable and true". That is not at all what he proved and, I am sure, not what you intended to say! In the logical system of integers, the statement "1+ 1= 2" is both true and provable.

What Godel proved was that, in ANY logical system large enough to include the natural numbers there EXIST statements such that neither the statement nor its negation can be proved.

The condition "large enough to include the natural numbers" is important. It is easy to construct "toy" systems with just a few objects and postulates that can be proven to be "complete" (for every statement either the statement or its negation can be proved).

Further every logical system has statements that can be proved true. There wouldn't be much point to "logical sytems" if there weren't! Godel's theorem is about statements a logical system contains, not what it doesn't contain. There exist, first, statements that can be proven true, of course, statements that can be proven false (their negations can be proven true) AND statement that cannot be proven either way.

(I prefer not to use the words "true" and "false" which are also "value judgements". I Prefer to say that there are statements that are "provable", statements for which their negations are "provable", and, per Godel, statements for which neither are "provable". For completeness we might include statements such that both the statement and its negation are provable but then every statement is provable, so that "provable" has no useful meaning, so we call those systems "inconsistent" and don't use them.)

Thank you for such a valuable information. I was looking for a long time detailed explanation. Really appreciate your reply! I once wrote an explanation of this concept together with Edubirdie and also came to the conclusion that in fact Gödel proved that in absolutely any logical system large enough to include natural numbers, there are statements that neither a statement nor its negation can be proven. It was quite interesting and strange to understand.

Raiden Mitchell
 

Re: Is the following text about the Incompleteness Theorem c

Postby ChanelLeuschke » Thu Jul 27, 2023 4:36 am

Godel showed that if a logical system is large enough to contain the natural numbers, then it MUST CONTAIN assertions for which neither the statement nor its negation can be proved.

For this to work, the stipulation that the number be "large enough to include the natural numbers" must be met. Constructing "toy" systems with a small number of objects and postulates that can be shown to be "complete" (every statement can be proved, or its negation) is trivial.

What's more, assertions can be proven true in every logical system. If there weren't, the term "logical sytems" would be meaningless. Not the negation of claims, but the statements themselves are at the heart of Godel's theorem. First, there are propositions that can be proven true, incorrect (their negations also), and unprovable. wordle today

ChanelLeuschke
 
Posts: 10
Joined: Wed Dec 21, 2022 7:38 am
Reputation: 0


Return to Math History



Who is online

Users browsing this forum: No registered users and 1 guest