Answers to some Quora questions
Page last updated 13 June 2021
I finally decided to ditch Quora after I received an email telling me that an answer I had made six months previously was being moderated because it infringed the rules. The question was:
My answer was: “Get a life”
Infringing the rules? And they bring this up six months after it was posted. Clearly, it was time to ditch Quora. By that time, I was already getting annoyed with Quora anyway, filling up my in-box with trivial questions. I only joined it originally to answer some questions regarding Gödel’s Incompleteness theorem. Quora has now turned into a mine of misinformation that is overloaded with people who think their opinions are more important than logic and evidence, and people who, when they disagree with a web-page, instead of posting their arguments on the comments section of that web-page, post it on Quora instead. So I deleted every Quora email, and deleted my Quora accounts. Note that if you have more than one email, you may have inadvertently joined Quora more than once, so it is best to check by putting in your different emails in the login page and click “Forgotten your password?”
Since Quora deletes your answers if you delete your account, I have kept a copy of my answers here.
There are many claimed proofs of incompleteness, besides Gödel’s original proof. But when most people assert that the proofs are correct, they do so not on the basis that they have checked them themselves, but they are relying on their belief that someone else must have checked them correctly and found no error.
So it depends on what sort of answer you want to the question of whether Gödel’s incompleteness theorem is 100% correct; you need to decide whether to:
- decide whether to base that answer on what most people believe, i.e., just follow the crowd, or
- examine the proofs yourself, and decide for yourself. To make this easier, you might want to first look at Gödel’s actual proof, you can see it online at On Formally Undecidable Propositions Of Principia Mathematica And Related Systems by Kurt Gödel.
For some guidance on following it, see online the Step by step guide to Gödel’s Proof.
You might also want to have a look at A Simplified Explanation of Gödel’s proof online at Gödel Simplified.
What should be changed in math to make Gödel’s incompleteness theorems false? Which axioms should we modify to render the Gödel theorems obsolete? How can we do it with the least impact on the rest of the math/logic?
Your question assumes that all proofs of incompleteness are correct. But any that I have seen have errors that render them invalid. So you don’t need to ‘fix’ anything in the basis of math at all.
For the flaw in Gödel’s original proof, see The Flaw in Gödel’s proof.
Totally weird. He was a creationist and believed in intelligent design. He believed that the brain is not sufficient to explain human thinking, and so he believed that there is a spirit that assists the brain. See Gödel quotations.
One very common misconception is that Gödel’s proof is completely rigorous. It isn’t. He simply assumed that a key proposition (Proposition V of his paper) could be rigorously proved, instead of actually proving it. The strange thing was that his paper was accepted even though his entire proof relies on that proposition. See Gödel’s Incompleteness Theorem .
When you say “fail to understand”, perhaps Wittgenstein did not understand how Gödel’s intuitive assertion that the Proposition V of his supposed “proof” could be correct. And since Gödel never actually provided a rigorous proof of that assertion (and either has anyone else), what does anyone else mean when they say they understand it?
How can it be lacking an understanding of mathematical logic to say that you don’t accept an intuitive leap in place of rigorous logic?
Kaiser Tarafdar deleted my comment:
The probability of an event A is the number of times that event A can happen divided by the number of times all events B can happen, where B includes events A. Presumably one isn’t talking about the likelihood of life as we know it emerging on a star or in space, which we know is zero. So we presume we are talking about the likelihood of life as we know it emerging on a planet that could actually support life as we know it - what we might call a habitable planet. So we would say that the probability of life emerging in our universe is the number of times that life can emerge on a habitable planet in the universe divided by the number of habitable planets that there have been in the universe.
The problem is that we have absolutely no data with which to obtain that probability.
Kaiser Tarafdar’s answer is simply an answer plucked out of the air with no evidence to support it. So it’s not really surprising that he deleted my comment.
What I find most counter-intuitive about much of mathematics is that when a paradoxical result occurs, mathematicians typically spend a little bit of time wondering about it, and then move on and simply accept it as the way things are (like John von Neumann’s approach: ‘In mathematics, you don’t understand things. You just get used to them’).
But mathematics should be based on a solid framework where paradoxes do not occur, and paradoxical results indicate that there is something not quite right about the foundations of the mathematics that is being used. And instead of ignoring the paradoxes and pretending that modern mathematics is just fine the way it is, the utmost effort should be made in discovering what are the root causes of those paradoxes, and building a foundation that does not contain those root causes.
How can a Gödel number have any meaning in the incompleteness theorem proof other than that propositional signs are expressed in numbers? How can Gödel numbers be meaningfully manipulated along with natural numbers in general to produce a theorem?
Gödel numbers are simply numbers. They do not "express" anything other than their own numerical value. But one can use a system of encoding that associates a number with some sequence of symbols such as a proposition. That does not mean that the number expresses the proposition. It doesn’t.
I think that by the second part of your question you are questioning how Gödel numbers, if they are only numbers, and not expressions of propositions, can be used to produce the self-referencing sequence that is supposed to represent a statement that is true but not provable. Well, that result depends on assuming that the purely numerical system can in fact extract the information given by the Gödel encoding system, which itself is not purely numerical. The fact is that Gödel never published a rigorous proof of this part of his paper, substituting proof by intuitive assumptions, and everyone else since has followed this path. You can see more on this at my website.
Why attach philosophical labels? In the field of mathematical logic, it doesn’t matter what my philosophical outlook is, since a philosophical outlook cannot ever override strict logic.
Yes. Assume that the formal system for which the proof is to apply can self-reference. Once that is assumed the proof is trivial.
Note that this is the method used by nearly all proofs of incompleteness. It neatly avoids the problem of how a logically valid formal system can logically self-reference itself.
That you can pretend that you have a real proof that proves something by taking a step of your proof and saying, well this step is perfectly obvious, so I’ll not actually prove it, but let’s all assume it’s correct, and lo and behold, this is my proof.
This is what he did with his incompleteness proof, assuming his proposition V is correct without actually proving it. And amazingly nearly everyone accepts this as a real proof.
Many things. One of them is promoting the idea of “intelligent design” by a probability argument:
“I believe that mechanism in biology is a prejudice of our time which will be disproved. In this case, one disproof, in my opinion, will consist in a mathematical theorem to the effect that the formation within geological times of a human body by the laws of physics (or any other laws of a similar nature), starting from a random distribution of the elementary particles and the field, is as unlikely as the separation by chance of the atmosphere into its components.”
- he doesn’t fully understand Gödel’s paper and
- he doesn’t clearly define what consciousness might be - trying to talk about something you can’t define is always going to be problematical.
Of itself, the term “infinity” is not well defined, being a noun which implies that it refers to a specific thing or one of a set of things.
On the other hand the adjective “infinite” is well defined, indicating the absence of a limit of a quantity. No-one has proven the existence of any physical thing that has such a property.
The Wikipedia definition of transfinite numbers are “numbers that are ‘infinite’ in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.” I can’t reconcile this. How can there be a largest finite number?
There isn’t a largest finite number. In relation to numbers the term infinite is commonly used to refer to the concept of being limitlessly large. So if a quantity is described as limitlessly large, it is larger than any finite number.
I think your question is really as to how a "number" can be limitlessly large and yet at the same time not "absolutely infinite", which Wikipedia defines as "can be thought as a number which is bigger than any conceivable or inconceivable quantity, either finite or transfinite", which is saying that "absolutely infinite" is limitlessly large, which shows the Wikipedia definition of transfinite number to be contradictory.
Although there is widespread acceptance of the notion of transfinite numbers, the simple fact is that no-one has ever provided a mathematical proof that there can be transfinite numbers, see Proving that there are more Real numbers than Natural numbers?
Well, if you think that an alleged proof which simply assumes the crucial part of it to be correct (as is the case in Gödel’s proof) is an acceptable way to prove mathematical truths, then it isn’t a cheap trick. On the other hand, if you think that isn’t an acceptable way to prove mathematical truths, then maybe it is a cheap trick. See Gödel’s Incompleteness Theorem.
(See below for a response to Edward Cherlin’s answer, and for a response to response to Bernard Leak’s answer)
Because Gödel never actually proved it. In his paper, he simply assumes that the crucial step in his proof is correct, evading having to prove it by saying that he only gives an outline of a proof “since it offers no difficulties of principle and is somewhat involved.”
Thinking that someone would surely have tried to actually complete the full proof, I searched for such a proof, but could not find any. So I did the proof myself following Gödel’s brief outline of how that should proceed, and when I did so, I found that the outline concealed a hidden assumption that was a logical error and which renders the proof invalid, see https://www.jamesrmeyer.com/pdfs/FFGIT_Meyer.pdf
Since that time, I have subjected several other claimed incompleteness proofs to rigorous logical analysis and have found errors in all of them that render them invalid, see Errors in Incompleteness Proofs.
I have included below the response I made in a comment to Edward Cherlin’s answer, but since the comment has been collapsed, I am including it here. First of all, Edward Cherlin refers to my page Gödel’s Incompleteness Theorem which is only an overview, and which is not, and was never intended to be a complete logical analysis of Gödel’s proof of incompleteness. I understand very well what the statement of the theorem states, and Cherlin’s claim that I fall "into the common philosophical trap of supposing that the Theorem proves an undecidable Gödel sentence" is completely false, and I will explain why.
First of all, one needs to realize that there are several versions of what people consider to be a statement of an incompleteness theorem. A common version is: "In all formal systems that include sufficient arithmetic, there exist propositions that exist in that system that that system cannot prove either the proposition or its negation."
Gödel’s own theorem for a particular formal system is: "To every ω-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r)."
Gödel’s proof of this refers to a formula (which we may refer to as the Gödel sentence) of a formal system P and his proof asserts that this Gödel sentence is provable within the formal system P, nor is its negation.
Gödel himself says regarding this Gödel sentence that he says is unprovable in its own formal system "turns out to be decided by metamathematical considerations", i.e., is provable outside of the formal system.
If Cherlin wants to claim that I do not understand Gödel’s theorem and/or its proof, rather than make vague assertions about my overview pages, he is welcome to point out errors in my paper on it at: https://www.jamesrmeyer.com/pdfs... or in my Step-by-Step Guide to Gödel’s proof
Bernard Leak: My response to Bernard Leak’s lengthy answer follows:
Leak claims that I am "complaining that one can’t actually construct the function which maps a formula to its status as a theorem" Incorrect. In my paper I go through Gödel’s proof and I construct the formula of Gödel’s paper. I then demonstrate that there is an illogical flaw in that construction.
Leak complains that I state that "mathematics ought to be done with proper rigour". Would he not complain if someone stated that mathematics ought not to be done with proper rigour? Leak’s complaint is hyperbole devoid of content.
Leak says of my point that Gödel did not actual provide a detailed proof of the construction of the self-referencing formula that: "Expecting a detailed specification of how to do it is fairly silly"
Isn’t that very convenient? Anytime you want to have a proof that includes a self-referencing formula, you tell everyone it would be silly to provide a rigorous proof of how the formula is constructed. And yet Leak has the audacity to complain that I am "self-indulgent and self-congratulatory" to ask for a rigorous proof.
Leak on "proof-schema".
The word proof-schema can have at least two meanings:
(a) as a well-defined sequence of all formal steps in a fully formal proof
(b) as a brief representation of a proof in the form of an outline.
Gödel’s use of the term "proof-schema" in his paper as meaning (a) above, and I refer to Gödel’s use of it as (a). I also use the term "proof-schema" in one place as the meaning (b). But the meaning is readily apparent by context. Leak appears to claim that by "ignoring the difference" between a proof-schema and a proof, I am not being rigorous, but he has provided no logical reason whatsoever for that assertion.
Leak’s assertions re constructing a self-referential sentence in a formal system amount to: “I can’t see why it can’t be done, therefore I don’t need to prove it can be done”
Contrary to Leak’s assertions, I do not try to hide anything on my website. It’s all there with a sitemap to every page, and many links between pages, and I include a search box to facilitate searches for words or phrases. I actively ask for people to attend to facts and logical argument rather than lengthy pejorative assertions.
Michael Harrison’s answer says "if someone in the STEM fields promotes nonsense, then ...". But first you have to prove that the person promotes nonsense, so Harrison’s comment is vacuous.
Because mathematics arose from activities such as measurement of physical things, so it didn’t arise independently of experience. Contrast that with certain aspects of modern math such as set theory and which have never found any real world application. Einstein grew up on a world where modern set theory had not developed.
It’s interesting that the featured answer doesn’t even start to address the question how was it proved that most numbers are incomputable. The question asked "How was it proved..."
The simple fact is that there has never been a rigorous logical proof that there can be numbers that are not definable, see for example: