It is terrible. I mentioned why earlier in this thread. — TonesInDeepFreeze
I'm interested in why Godel's Theorems 'are unquestionably among the most philosophically important logico-mathematical discoveries ever made' (says this article.) — Wayfarer
The philosophical implications of Godel's theorems are usually very overblown. — Pfhorrest
What is mathematical truth is an open question in the philosophy of mathematics that has been much debated over the last 100 years, since Tarsky resurfaced it. This exchange illustrates the traditional axis of controversy between Platonists and formalists, realists and anti-realists. — SophistiCat
I seem to recall reading somewhere that Gödel was a mathematical Platonist — Janus
Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason. — Godel and the nature of mathematical truth, Rebecca Goldstein
I'm interested in why Godel's Theorems 'are unquestionably among the most philosophically important logico-mathematical discoveries ever made' (says this article.)
— Wayfarer
That article gives an answer. It's a great article. (By the way, Panu Raattkainen is a top notch source on the subject.) — TonesInDeepFreeze
All of the various self-reference paradoxes have always seemed trivial to me, e.g. "This sentence is false." Who cares? Russell's paradox seems just the same, just dolled up in mathematical/logical language. — T Clark
I don't mean to be too intrusive but I do want to pick your brain regarding some interesting aspects of Godel's theorems but in a much broader context. — TheMadFool
When Bertrand Russell told Gottlieb Frege about the 'barber paradox' it had a momentous impact on Frege's whole life work. — Wayfarer
You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning. — Russell
All of the various self-reference paradoxes — T Clark
Do these "paradoxes" really have a significant, real-time, practical impact on the effective use of mathematics and computer science in the real world? — T Clark
Are you [SophistiCat] suggesting that Gödel's incompleteness theorem would be trivially true on a formalist understanding of mathematics because to be true in a language just is to be proven in that language? — Janus
Would a formalist allow that there could be mathematical truths that cannot be proven? — Janus
I thought that was the point I was trying to make in quoting from his book, but thanks for spelling it out to clear up any ambiguity. — Wayfarer
Would you say the below is a fair description of what Gödel is saying? Whilst the equation is true according to the rules of the math. The rules of the math cannot prove the equation true. To prove the equation true we need to look outside the rules of the math. — Pop
From the intro to Gödel's Theorem: An Incomplete Guide to Its Use and Abuse.
You might find the answers in there. — Wayfarer
Relationship with the liar paradox [of Godel's Incompleteness Theorems]
Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a system F makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the system F." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence. — Wikipedia
Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result — Wikipedia
I suspect there's something you're not seeing here. — Wayfarer
When Bertrand Russell told Gottlieb Frege about the 'barber paradox' it had a momentous impact on Frege's whole life work. I think, in layman's terms, what is at stake is elucidating a set of mathematical and logical principles which are both consistent and complete 'all the way down', so to speak. — Wayfarer
Basic mathematical curiosity alone leads to the question whether there is a mechanical procedure to determine whether any given Diophantine equation has a solution. And there are other answers in mathematics that incompleteness elucidates. — TonesInDeepFreeze
Where do they meet the world? — T Clark
See sections 4.4 and 4.5 here: — TonesInDeepFreeze
But why, in any practical sense, does that matter? Does it make mathematics less useful or effective in any significant way? It seems Platonic. Forms existing without relation to physical reality. — T Clark
Then I'll come back later and demonstrate more of my mathematical ignorance. — T Clark
There's an SEP article on Russell's paradox here. So it may not matter in the practical sense of you and I carrying on with our lives, but it is a philosophical issue of great significance. — Wayfarer
Hey we're all in the same boat! In fact I bet my ignorance is bigger than yours! — Wayfarer
Then I'll come back later and demonstrate more of my mathematical ignorance. — T Clark
Gödel's incompleteness theorems are two fundamental theorems of mathematical logic which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the first incompleteness theorem, Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable", called the "Gödel sentence G" — Wikipedia
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