KJW

I assume that. However, it is reasonable to assume that a downvote came from the one antagonistic person who did post in the thread.

some bigotries which very common in this forum. step 1: one demonstrates or proposes an opinion. step 2: a well known member attempts to disagree to that opinion. step3 : there is occuring of existence of many members downvoting that opinion (regardless the reality in that opinion, in fact this is a weakness of opinionating). And this is bigotry, isn't it? I don't understand who is who in the above. In the "Today I Learned in Mathematics" thread, only Genady and studiot were downvoted, and they were presumably by you (as the only person with the motive to downvote these two posters).

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That's it!! Just to clarify, the definition says, "for each y in S", which includes x0, whereas y must not be equal to x0.

I think I see it: y has to be not equal to x0.

@Genady, is the mistake you see that subset S requires at least two elements and not merely be non-empty?

[math]\dfrac{\partial \star}{\partial \overline{x}^{\mu}}[/math]

Q: What's the difference between Iran and Vietnam? A: Trump had a plan to get out of Vietnam!

Have you considered ammonia solution? You could also try (if you can obtain it) ethylenediaminetetraacetic acid (EDTA).

I think one thing that says they lean to the right is if they have a national flag in their front lawn. (This probably isn't limited to the US.)

I have to admit that this actually took me by surprise. After taking some time to think about this, I realise that it is ironic that what I said about two different types of axioms, the "deeper stuff" that you said was "not even wrong", appears to be key to my misunderstanding of the notion of completeness. Yes, there are two different types of axioms: one that defines a mathematical universe, and another that constrains that universe. It would seem that I neglected the mathematical universe. That would be because of the way I view mathematics, which is that everything exists unless proven otherwise. That means, for example, I assume the existence of multiplication even if it has not been explicitly defined. My mathematical universe contains multiplication, contains infinity, contains transfinite numbers, contains the axiom of choice, the continuum hypothesis has a definite answer, etc. But of course, that's not how this subject in mathematics is done. The mathematical universe is defined explicitly by the axioms, and notions such as completeness are based on it. So, I actually can see how a system with few axioms can be complete. And I can see that Gödel's incompleteness theorem is not trivial.

Surely this statement cannot be right. Say there are G axioms and axiom G is found to be provable from the other axioms A to F. Unless the proof of G is independent of one or more axioms, say B, how does this lead to an inconsistent set ? The point is that if an axiom is proven or disproven, then it is no longer an axiom. If the "axiom" is proven, it is a theorem. If the "axiom" is disproven, the set of axioms is inconsistent. Sometimes this is difficult to avoid. For example, in group theory, the existence of inverse elements can be expressed as: For each element [math]g[/math], there exists an inverse element [math]g^{-1}[/math] such that [math]g^{-1} g = g g^{-1} = e[/math] (the identity element). However, only one of the statements [math]g^{-1} g = e[/math] or [math]g g^{-1} = e[/math] is actually an axiom as the other statement can be proven. Which one is the axiom, and which one is the theorem is an arbitrary choice, so it seems natural to include both in a single statement so as to not arbitrarily break the symmetry.

This understanding is wrong. Well, it certainly is true that "any finite set of axioms is limited in what can be proven (or disproven) from them", so why is this not what (in)completeness is about? However, this is what axioms are. The problem is that it doesn't actually differentiate between statements that are axioms and statements that are theorems.

I believe what I said went into greater depth than that covered by this. To me, axioms are definitions, and as such are not subject to proof. I do reject the commonly held notion of axioms being "self-evident truths". What is completeness or incompleteness of a theory in your understanding? My understanding of (in)completeness is that any finite set of axioms is limited in what can be proven (or disproven) from them, and that additional axioms must be included to extend what can be proven (or disproven), ultimately requiring an infinite number of axioms to cover everything. But if any axiom can actually be proven (or disproven) from the other axioms, then there are too many axioms, and the set of axioms is potentially inconsistent. I see Gödel's Incompleteness Theorem as a statement that the set of all axioms is an open set in the analogous sense that the set [math]\{x \in \mathbb{R}: 0 < x < 1\}[/math] is an open set for which there is no largest or smallest number, and that there is no final axiom that completes the set of axioms, with the closure of the set [math]\{0,1\}[/math] rendering the axioms inconsistent. I say Gödel's Incompleteness Theorem is trivial because it is obvious that if one has a set of axioms that define a class of objects, then any particular object from that class will have properties that are unique to that particular object and not derivable from the axioms that define the class as a whole. To derive those unique properties require axioms that define that particular object within the class. And because the number of objects in the class are infinite, the total number of axioms that define all the individual objects in the class are also infinite. Thus, completeness becomes impossible.

For me, this seems to raise the question of what precisely is an "axiom"? Consider, for example, the axioms of a group. Then the commutative axiom of an Abelian group acts as a constraint on the structure of a general group. More can be said about Abelian groups than about general groups because everything that can be said about general groups can also be said about Abelian groups, whereas there are things that can be said about Abelian groups that cannot be said about general groups. But the axioms of multiplication do not constrain the structure of arithmetic. Instead, they make explicit an operation that already implicitly exists. The mere fact that addition exists implies that multiplication exists. The notation associated with the operation still needs to be defined, but this doesn't change the intrinsic properties of numbers (for example, whether they are prime) in the Platonic sense. Admittedly, I subscribe to the philosophical view that everything in mathematics exists unless proven otherwise. In the case of group theory, the notion of automorphisms add to what can be said about groups, though they do not act as constraints on groups. The automorphisms of a given group existed even before the notion of automorphisms were discovered. And yes, they were discovered, not merely invented. Although I'm not sure I properly understand this theorem, I am inclined to think it is rather trivial.